methods and applications to dsge models

Methods and Applications to

DSGE Models

2016-5

Anders Kronborg

PhD Thesis

DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS

AARHUS UNIVERSITY DENMARK

METHODS AND APPLICATIONS TO DSGEMODELS

By Anders Kronborg

A dissertation submitted to

Business and Social Sciences, Aarhus University,

in partial fulfilment of the requirements of

the PhD degree in

Economics and Business Economics

JANUARY 2016

CREATESCenter for Research in Econometric Analysis of Time Series

PREFACE

This dissertation is the tangible result of my PhD studies at the Department of Eco-

nomics and Business Economics at Aarhus University from February 2013 to January

2016. I am grateful to the department and to Center for Research in Econometric

Analysis of Time Series (CREATES) for providing facilities and an excellent research

environment during my studies. The generous financial support has allowed me

to participate in numerous courses and conferences both in Denmark and abroad.

Further, financial support received from Augustinus Fonden, Knud Højgaards Fond,

Konsul Axel Nielsens Mindelegat, Norges Bank, and Oticon Fonden is gratefully ac-

knowledged.

I would like to take the opportunity to thank a number of people for contributing

to this dissertation. Special thanks go to my two supervisors Torben M. Andersen and

Martin M. Andreasen for encouragement, helpful comments and valuable insights. It

has been a privilege to have such inspiring advisors. I am particularly pleased to have

worked together with Martin on one of the chapters in this dissertation.

I am thankful to Solveig N. Sørensen for help of all sorts and for proof-reading the

dissertation chapters.

From February to June 2015 I had the privilege of visiting Professor Frank Schorf-

heide at the Department of Economics at University of Pennsylvania. I would like to

thank both him and the department for the hospitality during my stay. The discus-

sions I have had with Frank have been enormously helpful and have contributed to

the third chapter of this thesis. Further, getting the opportunity to present my research

to several leading researchers in my field was inspiring and greatly appreciated.

I have been blessed with many great colleagues and a friendly work environment

during my time as a PhD student at Aarhus University. In particular, all my fellow PhD

students deserve a big thanks. First and foremost, I would like to thank Simon for the

years of laughs and interesting conversations, both related to economics and not. It

is my hope and expectation that we will continue to do so. Further, special thanks

go to Anne, Carsten, Jakob, Jonas, Mikkel, Morten, Niels, Palle, Sanni, and Silvana for

making it enjoyable to come to work every day.

A deep thank you to my family for supporting me and for taking an interest in

i

ii

my work even if it was hard to comprehend or not particularly interesting. Finally,

and most importantly, I would like to express my heartfelt gratitude to Josefine. Your

unconditional love and support has truly been invaluable.

Anders Kronborg

Copenhagen, January 2016

UPDATED PREFACE

The predefense took place on March 7. The assessment committee consists of Pro-

fessor Wouter den Haan from London School of Economics, Jens Iversen Head of

Modelling Division at Sveriges Riksbank, and Associate Professor Allan Sørensen from

Aarhus University. I am grateful to the members of the assessment committee for

their careful reading of the thesis, and their constructive comments and insightful dis-

cussion of my work. Some of the suggestions have been incorporated in the current

version of the thesis, while others remain for future research.

Anders Kronborg

Copenhagen, March 2016

iii

CONTENTS

Summary vii

Danish summary ix

1 The Extended Perturbation Method 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The Extended Perturbation Method . . . . . . . . . . . . . . . . . . . 3

1.3 Stability Properties of the Extended Perturbation Method . . . . . . 8

1.4 The Neoclassical Growth Model . . . . . . . . . . . . . . . . . . . . . . 12

1.5 A New Keynesian Model . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.6 Efficient Implementation of Extended Perturbation . . . . . . . . . . 21

1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 New Evidence on Downward Nominal Wage Rigidity and the Implicationsfor Monetary Policy 55

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.2 The DSGE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.4 Econometric Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.5 Data and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.6 Model Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.7 Optimal Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

v

vi CONTENTS

3 Forecasting Using a DSGE Model with a Fixed Exchange Rate 853.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.2 The DSGE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.3 Econometric Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.5 Prior Distributions and Calibrated Parameters . . . . . . . . . . . . . 101

3.6 Posterior Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 102

3.7 DSGE Model-based Forecasting . . . . . . . . . . . . . . . . . . . . . . 106

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

SUMMARY

Dynamic stochastic general equilibrium (DSGE) models are macroeconomic models

which try to explain aggregate economic movements based on an explicit micro-

economic foundation such as utility maximization. As the name suggests, they are

dynamic meaning they are concerned with intertemporal decisions which are subject

to stochastic shocks, for example through productivity changes. This uncertainty

implies that the economic agents have to form expectations about the future when

making decisions today. General equilibrium means that they aim to describe quanti-

ties and prices for both the supply and demand side simultaneously.

This dissertation comprises three self-contained chapters that all relate to me-

thods and applications to DSGE models. Since each chapter can be read indepen-

dently this implies that some repetition of arguments is impossible to avoid. Further,

it might beneficial to read the first chapter if one wants to get a more thorough

understanding of the methodology later applied in the second chapter.

Generally, the solution to DSGE models is unknown and we need to use numerical

techniques to approximate the equilibrium dynamics resulting from optimal behavior.

Chapter 1 in this dissertation proposes a solution method that generally improves

the accuracy relative to some of the most widely applied methods while still being

feasible for estimation purposes.

DSGE models are true multi-tools with many possible applications. These include

the estimation of the structural parameters in a model in order to answer some em-

pirical questions or to analyze the effects of changes in macroeconomic policy. Both

are the subject of chapter 2. Chapter 3 considers using a DSGE model for predictions

of comovements in aggregate time series.

The first chapter "The Extended Perturbation Method" (joint with Martin M. An-

dreasen) proposes a new solution method to DSGE models. Even though the nonlin-

ear solution to DSGE models is frequently approximated by higher-order perturbation

it is well known that they often generate explosive sample paths and struggle to pre-

serve underlying characteristics of the true solution, such as convexity or monotonic-

ity. Our solution method combines perturbation with the Extended Path. Using the

vii

viii SUMMARY

neoclassical growth model and a New Keynesian model with Calvo pricing, we show

that this extended perturbation method improves the accuracy relative to standard

perturbation when using third order approximations. Further, we are able to gen-

erate stable approximations even when standard perturbation explodes. The main

drawback of this method is that it may be computationally demanding, as it requires

solving a rather large fixed-point problem. We analyze these costs and suggest several

ways to improve the algorithm without loosing substantial precision.

The second chapter "New Evidence on Downward Nominal Wage Rigidity and the

Implications for Monetary Policy" revisits the question of wage rigidity. Downward

nominal wage rigidity has received increased attention during the recent crisis. To

examine this a simple New Keynesian DSGE model with potential asymmetric wage

rigidity is estimated using U.S. data. The model solution is approximated by the

extended perturbation method which is particularly suitable for strong model non-

linearities as in this application. In line with previous studies I find that downward

nominal wage rigidity is present, meaning nominal wages are more downwardly that

upwardly rigid. However, the parameter governing this asymmetry is found to be

orders of magnitude smaller than in the literature, which can be attributed to the

change in solution methodology. The estimated model is subsequently used to find

the optimal inflation target when implemented in a Taylor rule. For the U.S., the

optimal inflation is approximately 0.25 percent per year which provides support for a

low but positive inflation target.

The third and final chapter is "Forecasting Using a DSGE Model with a Fixed Ex-

change Rate". While macroeconomic forecasters can choose between a wide range

of models to generate predictions, DSGE models are increasingly being applied by

economic institutions and policy makers. Using Danish data, this chapter examines

the forecasting ability of a DSGE model in which the exchange rate is fixed. While

previous studies document the performances of DSGE models in a closed-economy

setting and where the exchange rate is flexible, it is not clear a priori that the model

will produce similar results under a fixed exchange rate regime. The forecasting per-

formance of the DSGE model is examined by generating recursive out-of-sample

forecasts of several time series from one to eight quarters ahead. The accuracy of the

DSGE model is generally comparable to an AR(1) model and better than the random

walk. Consistent with previous research, the DSGE model largely underestimates the

severity of the recent crisis that hit Denmark in the third quarter of 2008. However,

the model correctly predicts a continued fall in GDP growth and a subsequent slow

recovery when forecasting from the beginning of 2009.

DANISH SUMMARY

Dynamisk stokastiske generelle ligevægtsmodeller (DSGE-modeller) er makro-

økonomiske modeller, der forsøger at forklare udviklingen i aggregerede økonomiske

variable baseret på et eksplicit mikroøkonomisk fundament som f.eks. nyttemak-

simering. Som navnet antyder, er modeltypen dynamisk, hvilket betyder, at de be-

tragter intertemporale beslutninger, der påvirkes af stokastiske stød for eksempel

via produktivitetsændringer. Denne usikkerhed medfører, at de økonomiske aktører

må danne forventninger om fremtiden, når de træffer beslutninger i dag. Generel

ligevægt betyder, at der søges at beskrive mængder og priser simultant for både

udbuds- og efterspørgselssiden.

Denne afhandling består af tre selvstændige kapitler, der alle relaterer sig til

metoder og anvendelser vedrørende DSGE-modeller. Idet hvert kapitel kan læses

uafhængigt, betyder det, at det er uungåeligt, at nogle argumenter går igen. Man kan

med fordel læse det første kapitel først, såfremt man ønsker en mere dybdegående

forståelse af den metodologi, der senere anvendes i andet kapitel.

Som oftest er løsningen til DSGE-modeller ikke kendt og vi er derfor nødt til

at anvende numeriske teknikker til at approksimere den ligevægtsdynamik, der er

resultatet af optimal adfærd. Kapitel 1 i denne afhandling foreslår en løsningsmetode,

der generelt set forbedrer nøjagtigheden i forhold til nogle af de mest anvendte

metoder imens den stadigt kan anvendes til estimationsformål.

En DSGE-model er et multiværktøj med mange anvendelsesmuligheder. Disse

inkluderer estimation af modellens strukturelle parametre for at besvare empiriske

spørgsmål eller at analysere effekten af ændringer i den makroøkonomiske politik.

Begge er temaet i kapitel 2. Kapitel 3 betragter, hvorvidt man kan bruge en DSGE-

model til at forudsige bevægelserne i aggregerede tidsserier.

Det første kapitel "The Extended Perturbation Method" (skrevet med Martin M. An-

dreasen) foreslår en ny løsningsmetode til DSGE-modeller. Selvom den ikke-lineære

løsning ofte approksimeres ved perturbation af højere orden, er det velkendt, at

de ofte genererer eksplosive stikprøver og har svært ved at bevare underliggende

karakteristika fra den sande løsning såsom konveksitet eller monotonicitet. Vores

ix

x DANISH SUMMARY

løsning kombinerer perturbation med Extended Path. Ved at anvende den på den

neoklassiske vækstmodel og en Ny-Keynesiansk model med Calvo priser viser vi, at

denne extended perturbation-metode forbedrer nøjagtigheden i forhold til standard

perturbation, når der anvendes tredjeordens-approksimationer. Derudover er vi i

stand til at generere stabile stikprøver, selv i de tilfælde hvor standard perturbation

eksploderer. Den primære ulempe ved metoden er de omkostninger, der er forbundet

med den krævede beregningskraft, idet den forudsætter, at der løses et temmeligt

stort fikspunktsproblem. Vi analyserer disse omkostninger og foreslår flere måder,

hvorpå de kan nedbringes betragteligt, uden at det får betydelige konsekvenser for

præcisionen.

Det andet kapitel "New Evidence on Downward Nominal Wage Rigidity and the Im-

plications for Monetary Policy" betragter spørgsmålet om lønrigiditet. Nedadgående

nominel lønrigiditet betyder, at lønninger tilpasser sig langsommere når de mindskes

end når de øges og har tiltrukket sig stigende opmæksomhed under den nylige krise.

For at betragte dette estimeres en simpel Ny-Keynesiansk DSGE-model ved brug af

amerikanske data. Løsningen approksimeres ved extended perturbation-metoden,

som er særlig brugbar ved stærke ikke-lineariteter som i denne anvendelse. I ov-

erensstemmelse med tidligere studier finder jeg, at lønningerne er mere rigide i

nedadgående retning end omvendt. Dog findes, at den parameter der styrer graden

af asymmetri er af en anden størrelsesorden end i literaturen, hvilket kan tilskrives

ændringen i løsningsmetodologien. Den estimerede model benyttes herefter til at

bestemme det optimale inflationsmål, når dette implementeres i en Taylor-regel.

For USA findes den optimale inflation at være omkring 0,25 procent årligt, hvilket

dermed støtter et lavt, men positivt inflationsmål.

Det tredje og sidste kapitel er "Forecasting Using a DSGE Model with a Fixed Ex-

change Rate". Mens makroøkonomer kan vælge mellem en lang række modeller til at

generere fremskrivninger, bliver DSGE-modeller stadigt mere populære. Ved brug af

dansk data undersøger dette kapitel præcisionen af fremskrivningerne for en DSGE-

model, hvor valutakursen er underlagt en fastkurspolitik. Mens tidligere studier

dokumenterer fremskrivningsevnen for DSGE-modeller, hvori økonomien antages at

være lukket eller hvor valutakursen er flydende, er det ikke klart a priori, at modellen

vil fremvise lignende resultater, når valutakursen er fast. Fremskrivningsevnen un-

dersøges ved rekursivt at generere out-of-sample fremskrivninger for flere tidsserier

fra et til otte kvartaler frem. DSGE-modellens præcision kan generelt sammenlignes

med en AR(1) model og klarer sig bedre end random walk. I overensstemmelse med

litteraturen på området findes, at DSGE-modellen kraftigt undervurderer omfanget

af krisen, der ramte Danmark i tredje kvartal i 2008. Dog forudsiger modellen korrekt

xi

BNP-vækstens fortsatte fald samt det langsomme opsving, når fremskrivningerne

startes fra primo 2009.

C H A P T E R 1THE EXTENDED PERTURBATION METHOD

Martin M. Andreasen

Aarhus University and CREATES

Anders Kronborg


Abstract

The exact solution to a broad class of DSGE models can be decomposed into a compo-

nent under perfect foresight and a component containing the effects of uncertainty.

We therefore propose to compute the perfect foresight component with arbitrary

precision by the Extended Path whereas the stochastic part of the solution is approxi-

mated by the perturbation method. Using the neoclassical growth model and a New

Keynesian model, we show that this alternative approximation is more accurate than

standard third-order perturbation and delivers stable dynamics even in cases where

standard perturbation explodes.

1

2 CHAPTER 1. THE EXTENDED PERTURBATION METHOD

1.1 Introduction

The solution to dynamic stochastic general equilibrium (DSGE) models is frequently

approximated by the perturbation method to obtain higher-order Taylor series expan-

sions of the policy function. The popularity of this approximation is mainly explained

by its ability to i) preserve non-linearities in the model such as asymmetries, ii) im-

prove parameter identification compared to a linearized solution, and iii) capture

effects of uncertainty to explore determinants of risk premia and implications of

uncertainty shocks (see An and Schorfheide (2007), Kim and Ruge-Murcia (2009),

Fernandez-Villaverde et al. (2011), Rudebusch and Swanson (2012), among others).

Despite the widespread use of second- and third-order perturbation approxima-

tions, it is well known that they often generate explosive sample paths even when

the corresponding linearized solution is stable. The perturbation approximation may

also struggle to preserve key properties of the true solution such as monotonicity and

convexity as emphasized by den Haan and de Wind (2012). These findings suggest

that the second- and third-order perturbation approximations currently applied in

the literature may not always be sufficiently accurate. Obtaining fourth- or even

fifth-order expansions is often computationally infeasible and may even in some

cases be insufficient to get an accurate approximation as shown by den Haan and de

Wind (2012) . A tractable alternative that preserves stability of the true solution, but

not necessarily monotonicity, is to apply a pruning scheme as proposed by Kim et al.

(2008) for models approximated to second order and extended to higher order by den

Haan and de Wind (2012), Andreasen et al. (2013), and Lombardo and Uhlig (2014).

The contribution of the present paper is to improve accuracy and stability of

the perturbation approximation by combining it with the Extended Path of Fair and

Taylor (1983). This is done based on a simple, yet powerful, decomposition of the

policy function into i) a deterministic component under perfect foresight and ii) a

component containing the effects of uncertainty, also referred to as the stochastic

component. The perturbation method is currently applied to approximate both parts

of the policy function, although the perfect foresight component may be approxi-

mated with arbitrary precision by the Extended Path. Based on this observation, we

propose to compute the perfect foresight component by the Extended Path, whereas

the stochastic part of the policy function is approximated by the standard perturba-

tion method. We name this combined solution procedure the extended perturbation

method and it improves accuracy and stability of standard perturbation by remov-

ing approximation errors under perfect foresight. The approximation order in the

extended perturbation method is thus determined by the order of the polynomial

used to approximate the stochastic part of the policy function.

For a second-order approximation, we show that extended perturbation always

gives a stable solution without explosive sample paths, provided the model is stable

1.2. THE EXTENDED PERTURBATION METHOD 3

under perfect foresight. This result does not generalize beyond second order and we

therefore present a stability test to numerically evaluate if a given approximation is

stable.

Using the neoclassical growth model and a New Keynesian model with Calvo

pricing, we show that extended perturbation achieves higher accuracy than standard

perturbation when using third-order approximations. We also show that extended

perturbation generates stable approximations even when standard perturbation

explodes. For the neoclassical model, the explosive behavior is caused by a lack of

monotonicity and convexity of the approximated consumption function, whereas

standard perturbation in the New Keynesian model generates an explosive price-

inflation spiral because the approximation does not account for the upper bound on

inflation induced by Calvo pricing.

A potential drawback of extended perturbation relates to the repeated use of the

Extended Path which may be computationally demanding. We address this poten-

tial concern by presenting three modifications of the Extended Path to improve its

efficiency: i) deriving good starting values using a third-order perturbation approx-

imation under perfect foresight, ii) appropriately setting the terminal condition in

the Extended Path, and iii) occasionally using the perturbation approximation of the

perfect foresight component in the policy function if it is sufficiently accurate. When

adopting the third improvement, extended perturbation reduces to the standard per-

turbation approximation close to the steady state, but otherwise applies the Extended

Path to improve accuracy and obtain a stable approximation. We show that each of

the three improvements may be combined to substantially reduce the computational

cost of extended perturbation. Depending on the required degree of precision, we

are able to simulate 1,000 draws from a medium-sized New Keynesian model with

nine state variables in 10 to 20 seconds using MATLAB on a standard desktop.

The remaining part of the paper is structured as follows. Section 1.2 derives

the extended perturbation method and compares it to existing solution methods.

The stability properties of extended perturbation is analyzed in Section 1.3 which

also presents a stability test. Numerical evidence on the performance of extended

perturbation is provided for the neoclassical growth model in Section 1.4 and a New

Keynesian model in Section 1.5. We discuss how to efficiently implement extended

perturbation in Section 1.6, while Section 1.7 concludes.

1.2 The Extended Perturbation Method

We start by presenting the considered class of DSGE models in Section 1.2.1. The

extended perturbation method is derived in Section 1.2.2 and compared to standard

perturbation in Section 1.2.3 and a stochastic version of the Extended Path in Section

1.2.4.


1.2.1 DSGE Models

We consider the broad class of DSGE models which can be expressed as

Et

[f(xt ,xt+1,yt ,yt+1

)]= 0, (1.1)

where Et denotes the conditional expectation given information available in time

period t . The state vector xt with dimension nx ×1 belongs to the set Xx , denoting

Borel subsets of Rnx . The control variables are stored in yt with dimension ny ×1

and yt ∈Xy , where Xy refers to Borel subsets of Rny . We further let nx +ny = n. The

function f maps elements from Xx ×Xx ×Xy ×Xy into Rn , and we assume that this

mapping is at least m times differentiable, where m will be used below to indicate

the approximation order of the DSGE model.

It is useful to consider the partitioning xt ≡[

x′1,t x′2,t

]′, where x1,t contains the

endogenous state variables and x2,t denotes the exogenous states. The dimensions

of these vectors are nx1 ×1 and nx2 ×1, respectively, with nx1 +nx2 = nx . We further

assume that the dynamics of the exogenous state variables belong to the general class

x2,t+1 = Γ(x2,t

)+σηεt+1, (1.2)

where εt+1 ∈Xε and has dimension nε×1. We also assume εt+1 to be independent

and identically distributed with zero mean and a unit covariance matrix, i.e. εt+1 ∼IID

(0,I

). The function Γmaps elements from Xx2 into Xx2 and is required to be at

least m times differentiable. We further assume that Γ generates a stable process

for x2,t .1 In linear systems, this corresponds to requiring that all eigenvalues of the

Jacobian ∂Γ/∂x′2,t lie inside the unit circle. For non-linear systems, Γmust satisfy the

general stability condition for nonlinear first-order Markov processes provided in

Section 1.3.1.

As in much of the perturbation literature, we focus on models with a unique

solution. The exact solution may then be expressed as (see Schmitt-Grohe and Uribe

(2004))

yt = g(xt ,σ

)(1.3)

xt+1 = h(xt ,σ

)+σηεt+1 (1.4)

η≡[

0nx1×nε

η

].

The assumption that innovations enter linearly in (1.2) and (1.4) is without loss of

generality, because the state vector may be extended to account for non-linearities

1This implies that trends may only be included in the class of DSGE models considered if a givenmodel after re-scaling has an equivalent representation without trending variables. A similar requirementis needed to apply the standard perturbation method. The procedure of re-scaling a DSGE model withtrends is carefully described in King and Rebelo (1999).


between xt and εt+1, as shown by Andreasen (2012b). The perturbation parameter

σ ≥ 0 scales the square root of the covariance matrix for the innovations η with

dimension nx ×nε and enables us to capture effects of uncertainty in the policy

functions. In particular, when σ= 0 we get a model under perfect foresight, i.e.

gPF (xt

) ≡ g(xt ,σ= 0

)(1.5)

hPF (xt

) ≡ h(xt ,σ= 0

),

whereas the model with uncertainty is obtained by letting σ = 1. Unfortunately,

the policy functions g and h in (1.3) and (1.4) are generally unknown and must be

approximated.

1.2.2 The Extended Perturbation Method

Our paper builds on the key observation that the policy functions can be decomposed

into

g(xt ,σ

) ≡ gPF (xt

)+gstoch (xt ,σ

)(1.6)

h(xt ,σ

) ≡ hPF (xt

)+hstoch (xt ,σ

),

where gstoch and hstoch capture effects of uncertainty when the perfect foresight

component is removed from the policy function. We also refer to gstoch and hstoch

as the stochastic part of the policy function, as indicated by the superscript. The

decomposition in (1.6) shows that the stochastic part of the policy function is zero

under perfect foresight, i.e. gstoch(xt ,σ= 0

)= 0 and hstoch(xt ,σ= 0

)= 0 for all values

of xt . This in turn implies that all derivatives of g and gPF solely with respect to the

states are identical at σ= 0, and similarly for h and hPF . That is,

g(xt ,σ= 0

)xm = gPF (

xt)

xm for all xt ∈Xx (1.7)

h(xt ,σ= 0

)xm = hPF (

xt)

xm for all xt ∈Xx

for m = 0,1,2, ...

where subscripts refer to partial derivatives taken m times with

respect to xt . We also note that all derivatives involving the perturbation parameter σ

are identical for g and gstoch because σ does not appear in gPF , and similarly for hand hstoch . That is,

g(xt ,σ

)xmσ j = gstoch (

xt ,σ)

xmσ j for all xt ∈Xx ,σ ∈R+ (1.8)

h(xt ,σ

)xmσ j = hstoch (

xt ,σ)

xmσ j for all xt ∈Xx ,σ ∈R+

for m = 0,1,2, ...

and j =

1,2, ...

, where subscripts refer to partial derivatives taken

m times with respect to xt and j times with respect to σ. Thus, our observations in

(1.7) and (1.8) imply that the standard perturbation method can be used to compute


gstoch(xt ,σ

)xmσ j and hstoch

(xt ,σ

)xmσ j at the deterministic steady state, i.e. at xss =

xt+1 = xt and σ= 0.

Inserting the decomposition in (1.6) into (1.3) and (1.4), the exact solution may

be expressed as

yt = gPF (xt

)+gstoch (xt ,σ

)(1.9)

xt+1 = hPF (xt

)+hstoch (xt ,σ

)+σηεt+1.

Following the work of Guu and Judd (1997), the perturbation method is usually ap-

plied to approximate both(gstoch ,hstoch

)and

(gPF ,hPF

)at the deterministic steady

state. However, a finite Taylor series expansion of gPF and hPF may generate unneces-

sary approximation errors, given that gPF and hPF can be approximated to arbitrary

precision by the Extended Path. We therefore suggest to compute the perfect foresight

components gPF and hPF by the Extended Path, whereas the stochastic part of the

policy function, i.e. gstoch and hstoch , remains approximated by the standard pertur-

bation method at the steady state. We name this combined solution procedure the

extended perturbation method, and it improves accuracy and stability of standard

perturbation by removing approximation errors in the perfect foresight component

of the policy function.

The approximation order in the extended perturbation method is determined

by the order of the Taylor series expansion used to approximate gstoch and hstoch .

Hence, a first-order approximation simply reproduces the perfect foresight solution,

whereas the second-order approximation is

yt = gPF (xt

)+ 1

2gσσ (1.10)

xt+1 = hPF (xt

)+ 1

2hσσ+σηεt+1.

The third order approximation reads

yt = gPF (xt

)+ 1

2gσσ+ 3

6gσσx

(xt −xss

)+ 1

6gσσσ (1.11)

xt+1 = hPF (xt

)+ 1

2hσσ+ 3

6hσσx

(xt −xss

)+ 1

6hσσσ+σηεt+1.

In (1.10) and (1.11) derivatives of gstoch and hstoch known to be zero are omitted for

simplicity (see Schmitt-Grohe and Uribe (2004) and Ruge-Murcia (2012)). Thus, it is

straightforward to form the m’th-order approximation by the extended perturbation

method. The required steps are:

Step 1: Run the standard perturbation method to obtain all required derivatives

of gstoch(xt ,σ

)and hstoch

(xt ,σ

)at the steady state to order m. Use these

derivatives to construct the perturbation approximations of gstoch(xt ,σ

)and

hstoch(xt ,σ

), denoted gstoch

(xt ,σ

)and hstoch

(xt ,σ

).


Step 2: In every time period, use the Extended Path to compute gPF(xt

)and hPF

(xt

)and approximate gstoch

(xt ,σ

)and hstoch

(xt ,σ

)by gstoch

(xt ,σ

)and hstoch

(xt ,σ

),

respectively.

Appendix A summarizes the Extended Path and explains how we numerically

obtain the perfect foresight solution. It should be emphasized that all derivatives of gand h solely with respect to the state variables obtained in Step 1 are not redundant,

as we use these derivatives to efficiently implement the Extended Path as described

in Section 1.6.

1.2.3 Comparing Extended and Standard Perturbation Approximations

To see how extended perturbation differs from standard perturbation, consider an

infinite Taylor series expansion of the perfect foresight solution at the steady state,

i.e.

gPF (xt

) = g(xt ,0

)= ∞∑k=0

g(xss ,0

)xk

k !

(xt −xss

)⊗k

hPF (xt

) = h(xt ,0

)= ∞∑k=0

h(xss ,0

)xk

k !

(xt −xss

)⊗k

where (xt −xss

)⊗k ≡ (xt −xss

)⊗ ...⊗ (xt −xss

)︸︷︷︸k times

.

Here, g(xss ,0

)xk is expressed as an ny ×

(nx

)k matrix and h(xss ,0

)xk as an nx ×

(nx

)k

matrix. We next analyze a third-order approximation by the extended perturbation

method, which may be written as (for σ= 1)

yt =∞∑

m=0

g(xss ,0

)xm

m!

(xt −xss

)⊗m + 1

2gσσ+ 3

6gσσx

(xt −xss

)+ 1

6gσσσ (1.12)

xt+1 =∞∑

m=0

h(xss ,0

)xm

m!

(xt −xss

)⊗m + 1

2hσσ+ 3

6hσσx

(xt −xss

)+ 1

6hσσσ+σηεt+1

Equation (1.12) shows that our new approximation may be interpreted as adding the

higher-order terms∑∞

m=4g(xss ,0)xm

m!

(xt −xss

)⊗m and∑∞

m=4h(xss ,0)xm

m!

(xt −xss

)⊗m to a

standard third-order perturbation solution. It is also evident that we only include

some of the additional terms in a higher-order Taylor series expansion. Comparing

to a fourth-order approximation for the control variables, we include g(xss ,0

)x4 but

not the additional terms gσ4 , gx2σ2 , gx,σ3 correcting for uncertainty. These terms and

further risk corrections at fifth and sixth order are typically small in DSGE models,


and we therefore conjecture that a relative low approximation order of gstoch and

hstoch will be sufficient for most models.

A further inspection of (1.12) reveals that extended perturbation at third order

has at least two attractive properties. Firstly, the policy functions include gσσx and

hσσx from standard perturbation that capture time-varying precautionary saving

and state-dependent risk premia in equity and bond prices. Secondly, the linear

approximation to the stochastic part of the policy function implies that extended

perturbation at third order is likely to preserve monotonicity and convexity of the true

policy function. This is because the approximation always captures these properties

for the perfect foresight component, and the linear approximation to the stochastic

component (obtained at the steady state) has the same monotonicity and convexity

properties for all values of xt . Thus, it is only if changes in monotonicity and/or

convexity in the exact solution arise from the stochastic part of the policy function

that extended perturbation at third order is unable to reproduce these properties of

the policy function.

1.2.4 Comparing Extended Perturbation to the Stochastic Extended Path

The extended perturbation method is also related to the stochastic version of the

Extended Path in Fair and Taylor (1983), where the Extended Path is computed S

times in every time period using sample paths for the structural innovations. That

is, the innovationsεt+i

∞i=1 are not restricted to zero as under the perfect foresight

solution, and this generates a distribution for the control and state variables, de-

noted

y(s)t ,x(s)

t+1

Ss=1

. An approximation that accounts for uncertainty is then given by

yt = 1S

∑Ss=1 y(s)

t and xt+1 = 1S

∑Ss=1 x(s)

t+1, which may be termed the Stochastic Extended

Path. This solution is numerically very demanding and therefore rarely applied. Ex-

tended perturbation may be viewed as an efficient approximation to the Stochastic

Extended Path, as it captures effects of uncertainty by derivatives with respect to the

perturbation parameter σ instead of computing the Extended Path S times in every

time period.2

1.3 Stability Properties of the Extended Perturbation Method

We next analyze the stability properties of the process for xt as implied by extended

perturbation. Given that the control variables are functions of the states, the stability

properties of yt follow from those of xt . We proceed by first describing sufficient

2 Another way to reduce the computational burden of the Stochastic Extended Path is providedin Adjemian and Julliard (2013) by only accounting for uncertainty in the first few time periods in theStochastic Extended Path after which all innovations are set to zero.

1.3. STABILITY PROPERTIES OF THE EXTENDED PERTURBATION METHOD 9

conditions for stability in first-order nonlinear Markov systems in Section 1.3.1,

which we apply in Section 1.3.2 to analyze the stability properties of the extended

perturbation approximation. Given that stability can not be guaranteed when using

extended perturbation beyond a second order approximation, we finally present a

numerical test for stability in Section 1.3.3.

1.3.1 Stability in First-order Nonlinear Markov Systems

Stability of a first-order nonlinear Markov system as in (1.4) is typically ensured by

requiring h(xt ,σ

)to be contracting. However, Potscher and Prucha (1997) argue that

this contraction condition is too strong and often violated for stable systems. The

prime example is the companion representation of a stable VAR(p) model, where

the contraction condition never holds. One way to obtain a less restrictive stability

condition is to iterate the system in (1.4) forward in time and instead impose the

contraction restriction on the iterated system. To formally present the condition,

iterate (1.4) forward by k time periods to obtain

xt+k = h(k) (xt ,εt+1,εt+2, ...εt+k−1,σ

)+σηεt+k ,

where h(2)(xt ,εt+1,σ

) ≡ h(h

(xt ,σ

)+σηεt+1,σ)

and so forth. Potscher and Prucha

(1997) then show that the system in (1.4) is stable if only h(k) is contracting. Two

sufficient conditions ensure that this contraction property holds. The first states that

there must exist an integer k ≥ 1 at which

sup

∣∣∣∣∣∣stacnx

j=1

[i′j∂h(k)

∂x′

(x j ,

ε

jd

k−1

d=1,σ

)]∣∣∣∣∣∣< 1, (1.13)

given x j ∈Xx and ε j ∈Xε. Here, ∂h(k)∂x′

(x j ,

ε

jd

k−1

d=1,σ

)is an nx ×nx Jacobian matrix

evaluated at

(x j ,

ε

jd

k−1

d=1

), and |A| denotes the norm given by the square root of

the largest eigenvalue of the matrix product A′A. The vector i j is the j ’th column in

the nx ×nx identity matrix, and the stac-operator creates a matrix using the rows

shown as arguments to the operator.3 Hence, the condition in (1.13) states that for

a sufficiently large integer k, the largest norm of ∂h(k)/∂x′ must be strictly smaller

than one for all values of xt and εt in their feasible domains. The second condition

for h(k) to display the contraction property is much weaker than (1.13) and given by

sup

∣∣∣∣∣∂h(k)

∂ε′l

(x,

εd

k−1d=1 ,σ

)∣∣∣∣∣<∞, (1.14)

3For instance, let a j denote the j ’th row of an m×n matrix A, then stacmj=1a j = A. The stac-operator is

used in (1.13) to allow rows in ∂h(k)/∂x′ to be evaluated at different points, as indicated by the superindex

j on the arguments at which ∂h(k)/∂x′ is evaluated.


where x ∈ Xx and εd ∈ Xε for l = 1,2, ...,k −1. It is clear that this second condition

holds for basically all smooth approximations to DSGE models if xt is finite, meaning

that the essential stability condition is given by (1.13). We therefore focus on (1.13) in

our subsequent discussion and leave (1.14) as a technical regularity condition.

1.3.2 Stability of the Extended Perturbation Approximation

Before analyzing the extended perturbation approximation, it is useful to study the

stability properties of the perfect foresight solution. As emphasized by Boucekkine

(1995), the perfect foresight solution can only be obtained for DSGE models that

are stable under perfect foresight. An assumption which may be tested using the

procedure in Boucekkine (1995). This stability requirement means that the state

process under perfect foresight xPFt is stable, where xPF

t evolves as xPFt+1 = hPF

(xPF

t

)+

σηεt+1. In other words, hPF satisfies condition (1.13) and explosive sample paths for

xPFt do not appear.

We next analyze the stability of extended perturbation when gradually increasing

the approximation order, i.e. the Taylor expansion of hstoch . For this analysis, it is

useful to write the extended perturbation approximation as xt+1 = hE xPer(xt ,σ

)+σηεt+1, where hE xPer

(xt ,σ

)≡ hPF(xt

)+ hstoch(xt ,σ

). In a first-order approximation,

there is no correction for uncertainty because hstoch = 0, meaning that extended

perturbation reduces to the stable perfect foresight solution.

In a second-order approximation, there is a constant correction for uncertainty

as hstoch = 12 hssσ

2. This means that partial derivatives of hE xPer with respect to the

state variables are equal to those of hPF for all values of xt , implying that the sta-

bility condition (1.13) also holds for hE xPer . Accordingly, the extended perturbation

method at second order guarantees a stable approximation because the uncertainty

correction only re-centers the stable perfect foresight solution.

In a third-order approximation, the uncertainty correction is a linear function

of the state variables as seen in (1.11). This implies that partial derivatives of hE xPer

differ from those of hPF and the stability condition (1.13) can not be guaranteed to

hold for hE xPer , although it is satisfied for hPF . In other words, the process for xt does

not necessarily inherit stability from the perfect foresight solution, because hσσx may

generate instability if the linear approximation of hstoch is insufficiently accurate.

Given that the uncertainty correction typically is small in most DSGE models, we

expect that most approximations by extended perturbation will be stable. In general,

any instability in the extended perturbation method and explosive sample paths only

occur if the approximation of the stochastic part of the policy function is insufficiently

accurate.

Going beyond third order, the stochastic component of the policy function is

approximated more accurately and this reduces the risk of getting unstable state

1.3. STABILITY PROPERTIES OF THE EXTENDED PERTURBATION METHOD 11

dynamics with the extended perturbation method. We can not guarantee a stable ap-

proximation for the same reason as provided at third order, because partial derivatives

of hstoch may violate the stability condition in (1.13) for hE xPer although satisfied for

hPF .

1.3.3 Testing for Stability

Given that extended perturbation does not necessarily provide a stable approxima-

tion, it seems useful to have a test to determine if a given approximation is stable or

not. The test we propose applies two simplifying assumptions to get an operational

version of the stability condition in (1.13). We first propose to only evaluate (1.13) on

a sparse grid containing extreme state values since unstable state dynamics are most

likely to appear at such points. To construct the grid, let Si =

l xi ,ux

i

for i = 1,2, ...,nx

contain the lower bound l xi and the upper bound ux

i of the i th state variable. The

values of

l xi ,ux

i

nx

i=1should cover the region where the approximation is used. Guid-

ance on how to set these bounds may be obtained from unconditional moments or

extreme values in a simulated sample path for the extended perturbation approxima-

tion. We then form the Cartesian set Sx≡ S1 ×S2 × ...×Snx having 2nx elements. Our

second simplifying assumption is only to consider the stability condition in (1.13)

when the rows in ∂h(k)/∂x′ are evaluated at the same point.4

Given these simplifying assumptions, the stability condition in (1.13) reduces to

the testable requirement that h(k) is contracting if there exists an integer k ≥ 1 such

that

max

∣∣∣∣∣∂h(k)

∂x′

(x,

ε(v)

d

k−1

d=1,σ

)∣∣∣∣∣ , for all x ∈ Sx and v = 1,2, ...,M

< 1. (1.15)

Here, each point in Sx is evaluated usingM sample paths of the structural innovationsε(v)

d

k−1

d=1to avoid that a non-stable system may satisfy the contraction condition

given a fortunate sample path for the innovations. The test may be carried out for dif-

ferent values of k and M. Some guidance on a reasonable value of k may be obtained

by implementing the test on a stable linear solution.5 We generally recommend using

a fairly large value of k, say 100 or 500, because it is easier for h(xt ,σ

)to display the

contraction property when iterated many periods forward in time. It should finally

be emphasized that this stability test is not limited to the extended perturbation

4At the expense of increasing the computational cost of the test, it is obvious that a finer grid for the

state variables may be considered and that rows in ∂h(k)/∂x′ could be evaluated at different points.5Given that the Jabocian ∂h(k)/∂x′ is computed by numerical differentiation, the most efficient

implementation of the test is to evaluate∣∣∣∂h

(jx

)/∂x′

∣∣∣ by gradually increasing jx and then stop for a

given x ∈ Sx when the condition is met, even though jx may be less than a pre-determined value of k. If

max

jx

x∈Sx

< k, then the stability condition in (1.15) is satisfied.


approximation, but may also be used for other approximations, including standard

perturbation as shown in Section 1.4 and 1.5.

1.4 The Neoclassical Growth Model

This section studies accuracy and stability of standard and extended perturbation us-

ing the neoclassical growth model, which constitutes the core of many DSGE models.

Throughout this section, a log-transformation is adopted to obtain approximations in

percentage deviation from steady state, as typically done in the literature. Our focus

is devoted to the performance of standard and extended perturbation at third order,

because they capture a time-varying uncertainty correction and remain computa-

tionally tractable for most DSGE models. We proceed by describing the neoclassical

growth model in Section 1.4.1 before analyzing accuracy in Section 1.4.2 and stability

in Section 1.4.3.

1.4.1 Model Description

A representative agent obtains utility from consumption ct and optimizes

Et

[∑∞l=0

βl

1−γc1−γt+l

], where β ∈ (

0,1)

is the subjective discount factor and γ controls

risk aversion. The optimization is subject to the real budget constraint ct + it = at kαt ,

where it denotes investment and α ∈ [0,1

]describes the production function. The

capital stock kt evolves as kt+1 = (1−δ)kt +it , where δ ∈ [0,1

]is the depreciation rate.

Total factor productivity at is exogenous and given by log at+1 = ρa log at +σaεa,t+1,

where εa,t+1 ∼NID(0,1). The optimality condition for the representative agent is

given by Et

[β

(ct+1/ct

)−γ (αat+1kαt+1 +1−δ

)]= 1.

1.4.2 Accuracy Analysis

Table 1.1 provides parameters for the neoclassical model.6 The left column of Figure

1.1 considers the benchmark specification and plots the approximated policy func-

tions for consumption when capital kt ≡ log(kt /kss

)ranges from −5 to +5 standard

deviations in a log-linearized solution. When technology is at its steady state level

(the middle row), all perturbation methods perform very well compared to a highly

accurate 10th order projection approximation, considered as a stand-in for the true

6In contrast to the paper by den Haan and de Wind (2012), we do not consider the specificationwith δ= 1 and γ= 1, where the closed-form solution is certainty equivalent and given by kt+1 =αβat kαtand ct = (1−αβ)at kαt . The reason being that the Extended Path and also extended perturbation for this

specification recover the true solution with gstoch (xt ,σ

)= 0 and hstoch (xt ,σ

)= 0.

1.4. THE NEOCLASSICAL GROWTH MODEL 13

solution (see Appendix B for further details).7 The top and bottom row of Figure 1.1

show that we draw the same conclusion, when the technology level is altered to −3

and +3 standard deviations, respectively. A more detailed inspection of the approxi-

mation errors in the right column of Figure 1.1 reveals two interesting findings. Firstly,

extended perturbation and standard perturbation at third order are highly accurate

and perform equally well. Secondly, extended perturbation improves substantially on

the accuracy provided by a first-order approximation and a perfect foresight solution,

where the latter case documents the benefit of correcting for uncertainty in the policy

function.

Our second specification labeled ’High risk aversion’ increases γ from 5 to 25 to

more clearly illustrate the benefit of removing approximation errors in the perfect

foresight component of the policy function. The middle chart in Figure 1.2 to the left

shows that standard perturbation at third order is unable to maintain monotonicity

and convexity of consumption for this specification - even when the technology level

is at the steady state. Note also how rapidly the accuracy of standard perturbation

deteriorates when moving away from steady state. In contrast, extended perturbation

is very close to the true solution and preserves both monotonicity and convexity of

consumption. The satisfying performance of extended perturbation is also evident

from the reported errors in the policy function which remain low even far from the

steady state. These findings suggest that the unsatisfying performance of standard

perturbation at third order is related to large errors in the perfect foresight component

of the policy function, whereas these errors are eliminated by extended perturbation.

To see where the perfect foresight solution struggles, we finally consider a speci-

fication labeled ’High variance’ with an ordinary degree of risk aversion(γ= 5

)but

with highly volatile technology shocks(σa = 0.1

). Charts to the left in Figure 1.3 show

that ignoring the uncertainty correction in the perfect foresight solution pushes con-

sumption above its true level. On the other hand, third order approximations using

either standard or extended perturbation perform much better due to the included

uncertainty correction.

We summarize the performance of standard and extended perturbation in Table

1.2 by reporting root mean squared errors (RMSEs) on a grid with 20 points uniformly

spaced along each dimension of the state variables. The RMSEs are computed using

log(ct /c tr ue

t

)= ct − c tr ue

t , where c tr uet denotes consumption in percentage deviation

from steady state in the true solution (i.e. in the projection approximation).

Extended and standard perturbation deliver broadly the same performance in

7See also Aruoba et al. (2006) who document the high accuracy of the projection method for theneoclassical growth model.


the benchmark specification, whereas extended perturbation dominates in the two

other specifications, as indicated by the bold figures in Table 1.2. The table also shows

that extended perturbation outperforms the perturbation +1 and +2 approximations

by den Haan and de Wind (2012) and a modified version of their algorithm named

perturbation +13r d and +23r d , where the system is closed using a third-order pertur-

bation approximation instead of the linearized solution as in den Haan and de Wind

(2012).8

The RMSEs from a simulated sample path of 20,000 observations (with a burn-in

of 1,000 observations) using the same set of innovations for technologyεa,t

20,000t=1 in

the approximations are reported in Table 1.3. Observations of capital and technol-

ogy close to the steady state appear frequently in this simulation and are therefore

assigned a higher weight than points far from the steady state. This is contrary to

the accuracy results computed in Table 1.2, where all points in the grid are weighted

equally. For our benchmark specification, standard perturbation at third order deliv-

ers the best performance with a RMSE of 7.57×10−6 and is marginally more accurate

than extended perturbation having a RMSE of 1.63×10−5. For our high risk aversion

specification, extended perturbation delivers the best performance, whereas the

standard perturbation approximation explodes. Extended perturbation also delivers

the lowest RMSEs in the high variance specification. We also note that extended per-

turbation is more accurate than the pruned third-order perturbation approximation

of Andreasen et al. (2013), which always ensures stability in contrast to extended

perturbation at third order. The second part of Table 1.3 reports the standard de-

viation of quarterly consumption ct = log(ct /css

)in the simulated samples. In our

benchmark specification we find that standard perturbation, pruning, and extended

perturbation all generate the same variability as the true solution (i.e. the projec-

tion approximation). In the two other specifications with stronger non-linearities,

extended perturbation displays the best performance with variability in consumption

closest to the true solution, as indicated by the bold figures in Table 1.3.

1.4.3 Stability Analysis

To analyze the stability properties of standard and extended perturbation beyond

simple inspection of sample paths, Figure 1.4 plots future capital in deviation from

the diagonal line dt , i.e. kt+1 −dt , as a function of kt .9 For the benchmark speci-

fication in the first row of Figure 1.4, the two perturbation methods induce stable

approximations for all values of technology, as decreasing lines correspond to kt+1

8We are in debt to Joris de Wind for suggesting this modifcation of their solution method to us.9The approximated functions of kt+1 are very close to the diagonal line for all values of kt , and

simply plotting kt+1 as a function of kt would therefore make it nearly impossible to analyze the stabilityproperties of the approximated functions.

1.4. THE NEOCLASSICAL GROWTH MODEL 15

first being above the diagonal line and then below this line when kt+1 −dt turns

negative. Hence, the equilibrium level of capital for a given technology level is pro-

vided by the intersection with the horizontal axis. We draw the same conclusion from

running our stability test, where standard and extended perturbation at third order

pass the test with k = 150, M = 50, and Sx constructed by setting the bounds for

capital and technology equal to ±4 and ±3 standard deviations in a log-linearized

solution, respectively. Note that we use wider bounds for capital compared to the

technology process, because non-linearities in the approximated law of motion for

capital may generate a wider distribution for capital than found in the log-linear

solution.

The corresponding stability plots with high risk aversion are given in the second

row of Figure 1.4. Extended perturbation once again returns a stable approximation,

but the inability of standard perturbation to maintain monotonicity and convexity of

consumption in this specification affects the law of motion for capital with kt+1 =at kαt +(1−δ)kt −ct and generates an unstable approximation. To realize this, consider

the case where technology is at the steady state. Although the function for kt+1 attains

positive values of kt+1 −dt when kt < 0 and intersects the horizontal axis around

0.5, the problematic feature relates to the second crossing with the horizontal axis

around 1.7. If kt exceeds this value, kt+1 is incorrectly increasing in kt because the

approximated consumption function does not maintain monotonicity and decreases

for higher values of kt , as shown in Figure 1.2. Accordingly, the law of motion for

capital diverts if kt exceeds this second crossing with the horizontal axis. The situation

is even more problematic when the technology level is either high or low, because

the function for kt+1 does not even intersect the horizontal axis and hence diverts.

Given that the process for technology moves between the high and low level in

Figure 1.4, this explains why we experienced an explosive sample path for standard

perturbation in Section 1.4.2. Indeed, using our test from Section 1.3.3, we reject

stability of standard perturbation at third order, because the system explodes when

we attempt to iterate it sufficiently many periods forward in time as required to

verify the contraction property. On the other hand, extended perturbation passes the

stability test with k = 350, M= 50, and Sx constructed using the same procedure as

for our benchmark specification.

The last row in Figure 1.4 shows that standard and extended perturbation generate

stable approximations in our specification with high volatility. The same conclusion

follows from our stability test with k = 150, M = 50, and Sx constructed using the

same procedure as in the two previous specifications.


1.5 A New Keynesian Model

We next explore accuracy and stability of standard and extended perturbation using a

New Keynesian model with price stickiness as in Calvo (1983). Two reasons motivate

our choice of model. Firstly, the New Keynesian model with Calvo pricing is one

of the most popular DSGE models. Secondly, and perhaps somewhat surprisingly,

some dimensions of this New Keynesian model are highly non-linear even for a

standard calibration. The strong non-linearities in the model also imply that standard

perturbation at third order easily generate explosive sample paths, suggesting that

unstable approximations are not only obtained at extreme calibrations, as found in

the neoclassical growth model.

To make the global projection method of our New Keynesian model computation-

ally feasible, we initially only consider technology shocks and defer the inclusion of

additional disturbances to Section 1.6. As in our analysis of the neoclassical growth

model, we adopt a log-transformation and study the performance of standard and

extended perturbation at third order. We proceed by describing the New Keynesian

model in Section 1.5.1, before studying accuracy in Section 1.5.2 and stability in

Section 1.5.3.

1.5.1 Model Description

A representative household maximizes

Ut = Et

∞∑l=0

βl

c1−φ2

t+l1−φ2

+φ0

(1−ht+l

)1−φ1

1−φ1

, (1.16)

where ct is consumption and ht is labor supply. In addition to a no-Ponzi-game

condition, the optimization is subject to the real budget constraint

ct +bt + it = ht wt + r kt kt + Rt−1bt−1

πt+di vt , (1.17)

where resources are allocated to consumption, one-period nominal bonds bt , and

investment it . Letting wt denote the real wage and r kt the real price of capital kt ,

the household receives i) labor income wt ht , ii) income from capital services sold

to firms r kt kt , iii) payoffs from bonds purchased in the previous period Rt−1bt−1/πt ,

and iv) dividends from firms di vt . Here, πt ≡ Pt /Pt−1 is gross inflation and Rt is the

gross nominal interest rate. The optimization of (1.16) is also subject to the law of

motion for capital kt+1 =(1−δ)

kt +it − κ2

(itkt

−ψ)2

kt , where κ≥ 0 introduces capital

adjustment costs based on ii /kt as in Jermann (1998). The constant ψ ensures that

these adjustment costs are zero in the steady state.

1.5. A NEW KEYNESIAN MODEL 17

We consider a perfectly competitive representative firm that produces final output

using yi ,t and the production function yt =(∫ 1

0 y(η−1)/ηi ,t di

)η/(η−1)with η > 1. This

generates the demand function yi ,t =(

Pi ,tPt

)−ηyt , with aggregate price level Pt =[∫ 1

0 P 1−ηi ,t di

]1/(1−η).

The intermediate goods are produced by monopolistic competitors using the

production function yi ,t = at kθi ,t h1−θi ,t , where technology at evolves according to

log at+1 = ρa log at +σaεa,t+1 with εa,t+1 ∼NID(0,1

). The i th firm sets Pi ,t , hi ,t , and

ki ,t by maximizing the present value of dividends, i.e.

Et

∞∑l=0

D t ,t+l Pt+l

(Pi ,t+l

Pt+l

)yi ,t+l − r k

t+l ki ,t+l −wt+l hi ,t+l

,

where D t ,t+l is the nominal stochastic discount factor for payments between time

period t and t + l . Beyond a no-Ponzi-game condition, the firm must satisfy demand

for the i th good. When setting prices, we follow Calvo (1983) and assume that only a

fraction α ∈ [0,1

)of firms set their prices optimally, with the remaining firms letting

Pi ,t = Pi ,t−1.

Finally, monetary policy is determined by the Taylor-rule

log

(Rt

Rss

)= ρR log

(Rt−1

Rss

)+ (

1−ρR)κπ log

(πt

πss

)+κy log

(yt

yss

) . (1.18)

The model’s equilibrium conditions are provided in Appendix C.10 We adopt a

relative standard parametrization for a quarterly model, where households have an

intertemporal elasticity of substitution of 0.5 (φ2 = 2), a Frisch labor supply elasticity

of 0.5(φ1 = 2

), and allocate one third of their time endowment to labor in steady

state (hss = 0.33). Firms optimally reset prices once a year on average (α= 0.75) and

impose a mark-up of 20% (η= 6). The main objective of the central bank is to stabilize

inflation (κπ = 1.5, κy = 0.125), subject to a desire to smooth changes in the policy

rate(ρR = 0.8

). Our parametrization is summarized in Table 1.4.

1.5.2 Accuracy Analysis

Our New Keynesian model can be summarized by the four control variables(ct , it ,πt , x2

t

), where x2

t denotes an auxiliary variable for the recursive representation

of firms’ first-order condition with respect to the optimal price. These control vari-

ables are a function of the states(Rt−1,kt , st , at

), with st denoting the price dispersion

10These conditions are derived in a technical appendix accompanying the paper. The technical ap-pendix is available from the homepage of the corresponding author.


index linked to the Calvo pricing. One way to display these policy functions is to con-

dition on representative values for the first three state variables, and plot the control

variables as a function of the remaining state variable, i.e. technology. This exercise

reveals that some of the largest differences between the approximation methods

considered appear for a low nominal interest rate, a low capital stock, and a high

value of the price dispersion index as the model here displays strong non-linearities.

To conserve space, we therefore focus on this state configuration in Figure 1.5, before

studying accuracy on a grid covering the entire state space.

The first chart of Figure 1.5 shows that extended perturbation captures most

of the non-linear pattern in the policy function for consumption and outperforms

standard perturbation for low levels of technology. Extended perturbation therefore

displays smaller errors with low values of technology (charts to the right in Figure

1.5), whereas the two methods display roughly similar performance for higher levels

of technology. The accuracy of the two perturbation solutions is evaluated using a

highly accurate 12th-order projection approximation, considered as a stand-in for

the true solution (see Appendix C for further details). The plots for the remaining

control variables also reveal that there generally is a significant gain in accuracy from

using extended instead of standard perturbation, in particular for investment and

the auxiliary control variable x2t .

To analyze accuracy on the entire state space, Table 1.5 reports RMSEs for the con-

sidered approximation methods on a grid with 20 points uniformly spaced along

each dimension of the state variables, giving a total of 204 = 160,000 points. As

in Section 1.4.2, we compute the RMSEs using log(zt /z tr ue

t

)= zt − z tr ue

t , where

zt ≡

ct , it ,πt , x2t

and the true solution is given by the projection approximation. We

find that standard perturbation at third order is more accurate than a log-linearized

approximation, but is generally outperformed by the perfect foresight solution.

Adding an uncertainty correction to the perfect foresight solution further improves ac-

curacy, and extended perturbation therefore delivers the best overall approximation

to the four control variables. Notable improvements in RMSEs from using extended

instead of standard perturbation appear for investment (0.0088 vs. 0.0172) and the

auxiliary control variable (0.0387 vs. 0.0558).

We also study accuracy on a simulated sample path of 20,000 observations (with

a burn-in of 1,000 observations) using the same set of innovations for technologyεa,t

20,000t=1 in all of the approximations. Table 1.6 shows that extended perturbation at

third order also in this setting is more accurate than standard perturbation. To explain

where some of this gain in accuracy comes from, Table 1.6 also reports the RMSEs for

a modified version of the standard perturbation solution, where the approximated

1.5. A NEW KEYNESIAN MODEL 19

transition function for st in the simulation is replaced by the exact expression, i.e.

st+1 = (1−α)1

1−η[

1−απη−1t

] ηη−1 +απηt st . (1.19)

That is, the price dispersion index is computed using (1.19) with πt =π3r dt , i.e. infla-

tion from the third-order perturbation approximation. Under the label "Perturbation:

3r d order, exact st " in Table 1.6, we find that using the exact transition function for

st lowers the RMSEs for all control variables, because we more accurately track the

non-linear evolution in st .11 Extended perturbation also includes the exact transition

function for st and additional non-linearities for the control variables (although only

under perfect foresight), and this explains why extended perturbation outperforms

this modified perturbation approximation. We also note that extended perturbation

is more accurate than the pruned third-order perturbation approximation which

always ensures stability. The second part of Table 1.6 reports quarterly standard devi-

ations in the simulated samples. We find minor negative biases for all approximations

compared to the projection method with extended perturbation having a lower bias

than standard perturbation.

1.5.3 Stability Analysis

The graphical stability analysis used in Section 1.4.3 is not applicable to our New

Keynesian model with three endogenous state variables. Instead, we use our stability

test from Section 1.3.3 to study the dynamic properties of standard and extended

perturbation and to understand why a given approximation may be unstable.

To run the stability test for the New Keynesian model, we first construct the set Sx .

The bounds for at are given by ±3 standard deviations of technology, while bounds

for the two endogenous state variables(Rt−1, kt

)are set to ±4 standard deviations in

a log-linearized solution, and hence slightly wider than for technology to account for

effects of non-linearities. The price dispersion index st is constant in a log-linearized

solution without steady state inflation, and we therefore use a simulated sample

path of extended perturbation to guide our bounds of −0.005 and 0.05. Using this

specification of Sx , we find that standard perturbation at third order passes our

stability test with k = 150 and M= 50. We also find that the extended perturbation

approximation is stable with k = 100 and M= 50.

Although standard perturbation at third order displays stable dynamics for the

considered specification, the approximation is fragile to even minor modifications.

We illustrate this point by increasing πss from 1.00 to 1.0015, giving an annual steady

11Unreported results show that using the exact transition function for all endogenous state variables incombination with the standard third-order perturbation approximation of the control variables do notdeliver a further improvement in accuracy.


state inflation rate of 0.6%. The bounds for(Rt−1, kt , at

)in the set Sx are determined

using the same procedure as in our benchmark specification, and we increase the

upper bound of st to 0.10. Our stability test reveals that extended perturbation re-

mains stable, whereas standard perturbation now induces unstable dynamics.12

Accordingly, the explosive behavior of standard perturbation is due to inaccuracies in

the perfect foresight component of the policy function. An inspection of the 24 = 16

state configurations in Sx reveals that it is only when we simultaneously have a low

nominal interest rate, a low capital stock, a high price dispersion index, and a low

technology level that the approximation explodes when iterated forward in time to

evaluate the contraction condition.13

To understand why standard perturbation displays unstable dynamics with πss =1.0015, consider Figure 1.6 plotting the policy function for inflation and the transition

equation for st at this state configuration. The bottom chart to the left shows that

inflation in a standard perturbation approximation increases sharply for higher

values of the price dispersion index st . This in turn leads to even higher values of

the price dispersion index in the next period st+1, as seen in the middle chart. This

then increases πt+1, which in turn increases st+2 and so on. In contrast, inflation

increases only slowly in st with extended perturbation, because this approximation

accounts for the upper bound(1/α

)1/(η−1) on inflation. This bound follows from

(1.19), as the term[

1−απη−1t

] ηη−1

implies complex numbers for inflation beyond(1/α

)1/(η−1). Accounting for the upper bound on inflation ensures broadly the same

moderate increase in st+1 for higher values of st as in the true solution (i.e. the

projection approximation) and explains why extended perturbation generates stable

dynamics.14

A careful inspection of Figure 1.6 reveals that the policy functions for πt and st+1

in standard perturbation are nearly identical for πss = 1.00 and πss = 1.0015 when

considered at a given value of st . The same applies for extended perturbation. Hence,

the main effect from introducing steady state inflation is that the distribution of st

moves to the right and attains an even longer right tail, as shown in the third column

of Figure 1.6. It is these extreme values of st that start a price-inflation spiral in the

standard perturbation approximation and generates explosive dynamics.

12This result is confirmed by simulating repeated sample paths using a standard third order perturba-tion approximation.

13At this critical state configuration, we interestingly find for extended perturbation both withπss = 1.00and πss = 1.0015 that h is contracting without iterating this function forward in time. This means thatthe number of considered sample paths M in the stability test is irrelevant at this state configuration forextended perturbation.

14We conjecture that the presence of the upper bound on inflation explains the relative high approxi-mation order needed in the projection method to obtain a sufficiently accurate approximation.

1.6. EFFICIENT IMPLEMENTATION OF EXTENDED PERTURBATION 21

1.6 Efficient Implementation of Extended Perturbation

Having documented the gain in accuracy and stability from extended perturbation,

we next address its computational costs. The first step of extended perturbation

involves computing derivatives of the model at the steady state, as outlined in Sec-

tion 1.2.2. This can be done within a few seconds for third order approximations

when using the optimized MATLAB codes of Binning (2013).15 A computationally

more demanding aspect of extended perturbation is to obtain the perfect foresight

component of the policy function by the Extended Path, as it requires solving a large

fixed-point problem. Although this fixed-point problem typically is solved within a

few iterations using the Newton-Raphson algorithm, the computational burden may

nevertheless be substantial if the perfect foresight solution is called repeatedly, for

instance when simulating a long sample path.

This section therefore analyzes the computational costs of the Extended Path

and presents several ways to reduce the time spent computing the perfect foresight

component of the policy function. That is, the focus of this section is entirely devoted

to the perfect foresight solution. We proceed as follows. Section 1.6.1 extends our New

Keynesian model to get a medium-sized DSGE model. Section 1.6.2 derives efficient

starting values for the fixed-point problem in the Extended Path, and Section 1.6.3 and

1.6.4 discuss how to efficiently set the terminal condition in the Extended Path. We

finally explore the possibility of occasionally using the perturbation approximation

of the perfect foresight component in the policy function in Section 1.6.5.

1.6.1 Expanding the New Keynesian Model

Given that many non-linear solution methods either perform poorly or become

infeasible in large models, we first extend our model from Section 1.5 to study the

computational complexity of extended perturbation on a fairly large model. We

therefore replace the utility function in (1.16) by

Ut = Et

∞∑l=0

βl dt+l

(ct+l −bct−1+l

)1−φ2

1−φ2+φ0

(1−ht+l

)1−φ1

1−φ1

,

where b introduces external habit formation and dt are preference shocks evolving

according to logdt+1 = ρd logdt +σdεd ,t with εd ,t ∼ NID(0,1

). We also augment

(1.18) with monetary policy shocks σRεR,t where εR,t ∼ NID(0,1

), and introduce

15Obtaining a third-order perturbation approximation to the model described below in Section 1.6.1with nine state variables takes only one second on a standard desktop in MATLAB 2014a using an Intel(R)Core(TM) i5-4200 CPU with 2.50 GHz.


investment specific shocks et by replacing (1.17) with

ct +bt + it /et = ht wt + r kt kt + Rt−1bt−1

πt+di vt ,

where loget+1 = ρe loget +σeεe,t+1 and εe,t+1 ∼NID(0,1

). Finally, trends in technol-

ogy are introduced through zt in the production function, i.e. yi ,t = at kθi ,t

(zt hi ,t

)1−θ

where log zt+1 = logµz,ss + log zt +σzεz,t+1 and εz,t+1 ∼NID(0,1

). As a result, this

New Keynesian model has five shocks and nine state variables, making its size com-

parable to many of the DSGE models typically used in the literature when studying

the business cycle (see for instance Christiano et al. (2005), Fernandez-Villaverde and

Rubio-Ramirez (2007), among others).16

For the simulation experiments below, we let b = 0.3, ρd = 0.98, ρe = 0.90, σd =0.015,σe = 0.01,σR = 0.0025, µz = 1.005, and πss = 1.005. Given the additional shocks

and steady state inflation of 2% per year, we initially eliminate the price-inflation spi-

ral in the standard perturbation approximation by letting κπ = 2.0 to make the central

bank more aggressive to deviations in the inflation gap. With the remaining parame-

ters as in Table 1.4, we obtain a calibration where standard third-order perturbation

generates stable dynamics. This specification is therefore referred to as the ’stable

perturbation calibration’. For our simulation experiments, it is also useful to consider

a setting where standard third-order perturbation explodes. We therefore lower κπ to

1.95 in our second calibration (which otherwise is identical to the first calibration)

and refer to this second specification as the ’explosive perturbation calibration’.

1.6.2 Efficient Starting Values for the Extended Path

It is essential to have good starting values to obtain fast convergence when solving the

fixed-point problem implied by the Extended Path. These starting values are typically

derived based on a first-order approximation. However, when the Extended Path is

used in the extended perturbation method, higher-order derivatives of the functions

g and h are already available as they are required to compute the uncertainty correc-

tions. It therefore seems natural to use these higher-order derivatives to improve the

starting values from a linearized solution. To minimize the computational burden

of derivingEt

[xt+i

],Et

[yt+i−1

]N

i=1, we use the perturbation method of Andreasen

and Zabczyk (2015).17 The method is outlined in Appendix D and illustrated for a

16The computational cost of running the Extended Path is largely unaffected by the number of exoge-nous shocks because they are concentrated out when solving the fixed-point problem, as explained inAppendix A. Hence, the execution time reported below for the perfect foresight component of the policyfunction should be representative of the computational costs implied by models with more than fivestructural shocks.

17Given a third order perturbation approximation and N = 200, it takes only 0.04 and 0.40 seconds tocompute the required loadings for the conditional expectations up to second and third order, respectively,in our New Keynesian model.


third-order approximation but generalizes easily to any orders.

We evaluate the effect of different starting values in the Extended Path by simu-

lating a sample path of 5,000 observations. For the stable perturbation calibration it

takes 2,465 seconds to generate this sample path when using starting values from

a linearized solution, but only 1,926 seconds when the starting values are obtained

from a third-order approximation. This is equivalent to an efficiency gain of 21.9%.18

The corresponding execution times for the explosive perturbation calibration are

2,878 and 2,478 seconds, respectively, and imply an efficiency gain of 13.9%. The

execution times are slightly higher in this second calibration, because it is time-

consuming to obtain convergence in the Extended Path for state values far from

the steady state where standard perturbation explodes.19 Based on this finding, we

therefore use starting values from a third-order approximation in the remaining part

of this section when testing the performance of the Extended Path.

1.6.3 The Terminal Condition in the Extended Path

The execution time of the Extended Path is also affected by the length of the horizon

N considered before closing the DSGE model with a terminal condition for the

control variables yt+N . As the size of the fixed-point problem in the Extended Path

grows linearly in N , it is desirable to use a relatively low value of N (see Appendix

A). Adjemian and Julliard (2010) note that N can be lowered if the standard terminal

condition of yt+N = yss is replaced by yt+N = Et

[y1st

t+N

], where y1st

t+N denotes the

control variables at time t+N in a first-order approximation. When using this terminal

condition, N should be set such that Et

[y1st

t+N

]is within the radius of convergence for

the linearized solution, and this clearly is a weaker requirement on N than imposing

yt+N = yss . However, higher-order approximations to the conditional expectation of

yt+N are easily obtained from the process of computing efficient starting values in

Section 1.6.2, and it therefore seems natural to consider the terminal condition yt+N =Et

[y3r d

t+N

], with y3r d

t+N denoting the control variables at time t + N in a third-order

approximation. Compared to using Et

[y1st

t+N

], this alternative terminal condition

imposes even weaker requirements on N , because the radius of convergence is larger

for a third-order approximation than a linearized solution.

Table 1.7 analyzes the accuracy of the Extended Path with respect to the hori-

zon N and the choice of terminal condition on a simulated sample path of 5,000

observations. For both calibrations, all terminal conditions imply the same level of

18The computations are carried out in MATLAB 2014a using an Intel(R) Core(TM) i5-4200 CPU with2.50 GHz and a horizon of N = 200 in the Extended Path.

19For state values where Extended Path struggles to converge using starting values from a third-orderperturbation approximation, we re-run the Extended Path using the solution in the previous period asstarting values to obtain convergence. These alternative starting values are used for 0.24% and 0.36% of theobservations when using initial starting values from a first- and third-order approximation, respectively.


accuracy with a long horizon of N = 250, which serve as the benchmark for com-

puting the RMSEs. When we gradually reduce the horizon, the standard terminal

condition of yt+N = yss is clearly less accurate compared to using the terminal condi-

tion from a linearized solution. The final column in Table 1.7 shows that a third-order

approximation for the terminal condition is even more accurate than the linearized

solution, except when N = 50 in the explosive perturbation calibration. Hence, we

generally obtain the highest level of accuracy in the Extended Path by using a third-

order approximation to compute the terminal condition, and we therefore adopt this

specification in the remainder of the section.

1.6.4 A State-dependent Horizon in the Extended Path

We have so far assumed that the same horizon applies to all state values in the

Extended Path. But if xt is close to the steady state, a relative short horizon should

be sufficient to ensure that Et

[y3r d

t+N

]is within the radius of convergence. We next

exploit this observation to introduce a state-dependent horizon N∗, where the aim is

to dynamically adjust the horizon to reduce the computational cost of the Extended

Path. We implement this idea by the rule

N∗ (Dss ,xt

)= min

N ∈N : max

∣∣∣∣Et

[y3r d

t+N

]∣∣∣∣≤ Dss st. N ∈ [Nmin, Nmax

],

(1.20)

where Dss denotes the tolerated distance of Et

[y3r d

t+N

]from the steady state. That

is, we use the shortest horizon where the largest element in Et

[y3r d

t+N

]is within the

distance Dss from the steady state, subject to N ∈ [Nmin, Nmax

]. This implies that N∗

depends on Dss and the current state xt through Et

[y3r d

t+N

], as indicated in (1.20).

Table 1.8 evaluates the performance of N∗ with Nmin = 20 and Nmax = 200 on

a simulated sample path of 5,000 observations. The execution time with a fixed

horizon of N = 200 corresponds to Dss = 0 and requires 385 seconds per 1,000 draws.

Introducing the state-dependent horizon with Dss = 0.01 implies hardly any loss of

accuracy (RMSE = 7.91×10−5), but reduces the execution time to just 196 seconds

per 1,000 draws due to an average horizon of just 87 time periods. For larger values of

Dss , the average horizon and the execution time fall even further but it also affects

the precision of the Extended Path, which only outperforms the standard third-order

perturbation approximation when Dss ≤ 0.02.

Turning to the explosive perturbation calibration in the lower part of Table 1.8, we

once again find that a state-dependent horizon lowers the execution time of the Ex-

tended Path with only a small loss in accuracy. For instance, with Dss = 0.02 we have

a RMSE of 2.00×10−4, and it only takes 263 seconds per 1,000 draws compared to

496 seconds with a fixed horizon of N = 200. Table 1.8 also shows that Dss should not


exceed 0.05 for the Extended Path to outperform the pruned third-order perturbation

approximation, which is a natural benchmark given that the unpruned perturbation

approximation explodes for this calibration. Compared to the Extended Path, we

also note that the pruned approximation in this case displays an impressive high

precision with a RMSE of 0.0072, given that it only uses 0.12 second per 1,000 draws.

Based on these findings we conclude that a state-dependent horizon in the Extended

Path may substantially reduce execution time with only a minimal loss of accuracy

for sufficiently low values of Dss .

1.6.5 Using the Perturbation Approximation of the Perfect Foresight

Component

We have so far used the Extended Path to compute the perfect foresight component of

the policy function for all state values. But a third-order perturbation approximation

of this component may for some state values be sufficiently accurate. This may be

the case if xt is close to the steady state or if the model is nearly linear along some

dimensions of the state space. Exploiting this observation should substantially reduce

the execution time for extended perturbation as it no longer relies on the Extended

Path for all state values.

To formalize this idea, let y3r dt denote the third-order perturbation approximation

of the control variables at xt . The perturbation approximation of Et

[y3r d

t+1

]is derived

in Section 1.6.2, and x3r dt+1 follows directly from the perturbation approximation of the

state equations. We then use (1.1) to compute the Euler-equation residuals under

perfect foresight of the standard perturbation approximation at time t , i.e. Ψt ≡f(

xt ,x3r dt+1,y3r d

t ,Et

[y3r d

t+1

]), where f is expressed in unit-free terms. Next, let EE denote

the tolerated Euler-equations errors, and consider the approximation

yt = 1max|Ψt |≤EE

g3r d (xt

)+ (1−1

max|Ψt |≤EE)gPF (

xt)

xt+1 = 1max|Ψt |≤EE

h3r d (xt

)+ (1−1

max|Ψt |≤EE)hPF (

xt)+σηεt+1

where 1max|Ψt |≤EE

is the indicator function. Here, g3r d(xt

)denotes the standard

perturbation approximation of the perfect foresight component of the policy function,

and similarly for h3r d(xt

). That is, we use standard perturbation when it is sufficiently

accurate, i.e. max∣∣Ψt

∣∣≤ EE , otherwise the Extended Path is used. When using the

Extended Path in this context, we also apply EE to determine convergence of the fixed-

point solver and stop iterating the Newton-Rahpson algorithm when max∣∣Ψt

∣∣≤ EE .

This is a weaker convergence criteria than previously used where all Euler-equation

residuals until horizon N should be less than 10−10. We adopt this modification of the


Extended Path to make its convergence criteria comparable to the accuracy condition

used for the perturbation approximation.20

The performance of this combined approximation to the perfect foresight compo-

nent of the policy function is shown in Table 1.9 for the stable perturbation calibration

using a simulated sample path of 5,000 observations. The first part of the table im-

poses Dss = 0 and uses a fixed horizon of N = 200 in the Extended Path. Letting

EE = 5×10−5, the Extended Path is only used for 27% of the observations, and this

allows us to generate 1,000 draws in just 58 seconds compared to 385 seconds with

EE = 0. This sizable reduction in execution time is achieved with nearly no loss

in accuracy as seen from the RMSEs of 3.28× 10−5. For larger values of EE , even

more observations are computed by the perturbation approximation and this further

reduces the computational cost at the expense of a small loss in accuracy.21 The

remaining part of Table 1.9 adds a state-dependent horizon to the Extended Path and

this further lowers the execution time. For instance, using Dss = 0.01 and EE = 0.0001,

we are able to generate 1,000 draws in just 18 seconds with nearly no loss in accuracy

(RMSEs= 6.24×10−5).

The results for the explosive perturbation calibration in Table 1.10 also document

a large reduction in execution costs with only a small loss in accuracy by occasion-

ally using the perturbation approximation. Contrary to the previous calibration, the

highest level of EE = 0.01 does not imply that all observations are computed by the

perturbation approximation, because the Extended Path is used for roughly 1% of

the observations, where standard perturbation would generate explosive dynamics.

Hence, in a parametrization of extended perturbation with a relatively high value of

EE , we only use the Extended Path to avoid explosive sample paths and otherwise rely

on the standard perturbation approximation. This is illustrated in Figure 1.7, where

we plot the distance of the state variables from steady state∥∥xt

∥∥=√∑nx

i=1 x2i ,t for the

part of our simulated sample where standard perturbation explodes. With a high

value of EE as considered in the bottom chart, extended perturbation only relies on

the Extended Path (marked by gray areas) from observation 4125 to 4250 where stan-

dard perturbation explodes, as shown by the diverting blue line. Hence, the suggested

rule for combining the perfect foresight component of standard perturbation and

the Extended Path may also be interpreted as a way to ensure stability of a standard

perturbation approximation. The case with a relatively low value of EE is considered

in the top chart of Figure 1.7, where the Extended Path is used to avoid explosive

20The exception is when convergence can not be obtained by the standard or extended Newton-Raphson algorithm and we instead use the Levenberg-Marquardt routine to minimizeΣt+N−1

i=t Ψ′i Ψi using

standard convergence criteria.21When EE = 0.01 all observations are computed by the perturbation approximation. Its execution

time is here higher than reported in Table 1.8, mainly due to the computational costs of obtainingΨt .

1.7. CONCLUSION 27

dynamics but also to improve accuracy when standard perturbation does not explode.

Based on these findings we conclude that the computational costs of obtaining

the perfect foresight component of the policy function can be greatly reduced by

occasionally using the perturbation approximation. Adopting a state-dependent hori-

zon in the Extended Path reduces the execution time further and makes extended

perturbation sufficiently fast to incorporate the approximation in estimation routines.

The most obvious estimators are probably simulated method of moments following

Duffie and Singleton (1993) or Indirect Inference with moments obtained from an

auxiliary model (Smith (1993)). Another appealling alternative is to consider the

quasi-maximum likelihood approach suggested by Andreasen (2013), as it requires

relatively few evaluations of the policy function.

1.7 Conclusion

This paper introduces the extended perturbation method which improves accuracy

and stability of standard perturbation by using a better approximation to the per-

fect foresight component of the policy function. For the neoclassical growth model

and a New Keynesian model with Calvo pricing, we find that extended perturbation

achieves higher accuracy than standard perturbation. We also find that the gain in

accuracy is sufficient to generate stable approximations by extended perturbation

when standard perturbation explodes. Our results therefore suggest that the explo-

sive behavior of standard perturbation reported in the literature may be related to

inaccuracies in the perfect foresight component of the policy function and may be

eliminated with the extended perturbation method. To reduce the computational

costs of implementing extended perturbation, we also introduce several improve-

ments of the Extended Path which substantially lowers execution costs and makes

our method feasible for estimation purposes.

Acknowledgments

We thank Rhys Bidder, Lawrence Christiano, Vasco Curdia, Anders B. Kock, Eric Swan-

son, Joris de Wind, Konstantinos Theodoridis, and participants at the econometrics

workshop at University of Pennsylvania for useful comments and discussions. M.

Andreasen greatly acknowledges financial support from the Danish e-Infrastructure

Coorporation (DeIC). We also appreciate financial support to CREATES - Center for

Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish

National Research Foundation.


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1.7. CONCLUSION 31

Appendix A: The Extended Path

Obtaining the Perfect Foresight Solution

Uncertainty about future shocks is absent under perfect foresight, implying that (1.1)

reduces to

f(xt ,xt+1,yt ,yt+1

)= 0 for all t = 1,2, ... (1.21)

The system contains an infinite number of equations and can not be solved without

simplifying assumptions. The approach in the Extended Path of Fair and Taylor (1983)

is to truncate the problem at some finite horizon N , after which the variables are

assumed to be constant, for instance at their steady state values. Hence, the approxi-

mation errors from the truncation decrease in N and can be made arbitrary small

for an appropriately chosen horizon N (see Fair and Taylor (1983) and Boucekkine

(1995)). Thus, the Extended Path closes the infinite system in (1.21) by a terminal

value for yt+N , giving rise to the finite dimensional system

f(xt ,xt+1,yt ,yt+1

)= 0n×1 (1.22)

f(xt+1,xt+2,yt+1,yt+2

)= 0n×1

f(xt+2,xt+3,yt+2,yt+3

)= 0n×1

...

f(xt+N−1,xt+N ,yt+N−1,yt+N

)= 0n×1.

The initial state xt and yt+N are known by assumption, whereas we solve for the n∗N

unknowns(yt ,xt+1,yt+1,xt+2,yt+2, ...,xt+N−1,yt+N−1,xt+N

)using the same number

of equations in (1.22). Hence, the perfect foresight solution is obtained by the fixed-

point problem in (1.22), provided the solution is unaffected when increasing N (see

Fair and Taylor (1983)).

To reduce the computational burden when solving (1.22), we concentrate outx2,t+ j

N

j=1as they can be computed directly by iterating on (1.2). The concentrated

fixed-point problem is given by

f1(x1,t ,x1,t+1,yt ,yt+1

)= 0n1×1 (1.23)

f1(x1,t+1,x1,t+2,yt+1,yt+2

)= 0n1×1

f1(x1,t+2,x1,t+3,yt+2,yt+3

)= 0n1×1

...

f1(x1,t+N−1,x1,t+N ,yt+N−1,yt+N

)= 0n1×1,

where we introduce the partitioning

f(xt ,xt+1,yt ,yt+1

)≡ [f1

(x1,t ,x1,t+1,yt ,yt+1

)f2

(x2,t ,x2,t+1

) ].


The function f2 (·) has dimensions n2 ×1, where n2 ≡ nx2 , and contains the law of

motions for the exogenous variables, whereas the function f1 (·) has dimensions n1×1

with n1 ≡ ny +nx1 and contains all the remaining equilibrium conditions. Note that

we suppress the dependence of x2,t in f1 (·) to reduce the notational burden. The

concentrated system in (1.23) has only n1 ∗N equations with the same number of

unknowns. For numerical stability of the proposed routines below, we recommend

expressing the residuals of f in terms of unit-free errors.

Computing the Perfect Foresight Solution Numerically

To solve the fixed-point problem in (1.23), let

Z ≡[

yt yt+1 ... yt+N−2 yt+N−1

x1,t+1 x1,t+2 ... x1,t+N−1 x1,t+N

],

meaning that (1.23) can be condensely expressed as

F (Z)≡

f1

(x1,t ,x1,t+1,yt ,yt+1

)f1

(x1,t+1,x1,t+2,yt+1,yt+2

)f1

(x1,t+2,x1,t+3,yt+2,yt+3

)...

f1(x1,t+N−1,x1,t+N ,yt+N−1,yt+N

)

= 0.

Linearizing this system around the point Z∗ gives

F (Z) ≈ F(Z∗)+ J

(Z∗)(

vec (Z)− vec(Z∗))

, (1.24)

where J(Z∗)

denotes the Jacobian evaluated at Z∗. That is, J(Z∗)≡ ∂F(Z)

∂vec(Z)′∣∣∣

Z=Z∗ , hav-

ing dimensions(n1 ∗N

)× (n1 ∗N

), and vec (Z) is the stacked vector, i.e.

vec (Z) =

yt

x1,t+1

yt+1

x1,t+2

...

yt+N−1

x1,t+N

,

with dimension(n1 ∗N

)×1.

The Standard Newton-Raphson Routine

The most efficient way to solve (1.23) is the Newton-Raphson routine, i.e. to iterate

on

vec(Zi+1

)= vec

(Zi

)− J

(Zi

)−1F

(Zi

)

1.7. CONCLUSION 33

until convergence, where Zi denotes the value of Z at iteration i . This routine is

sometimes referred to as a Newton-Raphson relaxation algorithm (see Boucekkine

(1995)), and we compute J(Zi

)−1F

(Zi

)using the efficient method of Boucekkine

(1995).

The Extended Newton-Raphson Routine

The standard Newton-Raphson routine may not converge if the problem is very

non-linear. In this case an extended Newton-Raphson routine is considered

vec(Zi+1 (

δ))= vec

(Zi

)−δ×

(J(Zi

))−1

F(Zi

), (1.25)

where the scaling parameter δ ∈ R is determined by minδ F(Zi+1

(δ))′

F(Zi+1

(δ))

,

solved by a rough grid search. That is, δ accounts for the possibility that a linear

approximation of F (Z) may be insufficiently accurate. We refer to this algorithm as

an extended Newton-Raphson relaxation routine.

Minimizing the Squared Model-Residuals by the LM Optimizer

If the standard and extended Newton-Raphson routine are unsuccessful in solving

the fixed-point problem in (1.23), then the Levenberg-Marquardt (LM) optimizer is

used to minimize F (Z)′ F (Z) across Z.

Appendix B: The Neoclassical Growth Model: A ProjectionApproximation

We approximate consumption using a 10th order Chebyshev polynomial constructed

by the tensor of 10th order polynomials for capital and technology. Capital in period

t +1 is obtained by using the law of motion for capital. The conditional expectation

in the consumption Euler-equation is evaluated using Gauss-Hermite quadratures

with five points. The grid for the state variables is determined from the unconditional

standard deviations in a log-linearized approximation, multiplied by ±5 for capital

and by ±3 for technology. We determine the loadings in the Chebyshev polynomial for

consumption by the collocation method (see Judd (1992)). To evaluate the accuracy

of the approximation, we consider a grid using 20 uniformly spaced points along

each of the state dimensions, i.e. a total of 400 points. The largest absolute errors are:

i) 4.86×10−15 in the benchmark specification, ii) 1.87×10−11 with high risk aversion,

and iii) 4.99×10−6 with high variance where the residuals are expressed in unit-free

errors.


Appendix C: The New Keynesian Model

Equilibrium Conditions

The Households

Eq. 1 λt = c−φ2t

Eq. 2 qtλt =βλt+1[r kt+1 +qt+1

(1−δ)−qt+1

κ2

(it+1kt+1

− Isskss

)2 +qt+1κ(

it+1kt+1

− Isskss

)it+1kt+1

]

Eq. 3 φ0(1−ht

)−φ1 =λt wt

Eq. 4 1 = qt

(1−κ

(iikt

− Isskss

))Eq. 5 λt =βRt

[λt+1πt+1

]The Firms

Eq. 6 mct at(1−θ)( ht

kt

)−θ = wt

Eq. 7 at mctθ(

htkt

)1−θ = r kt

Eq. 8 (η−1)x2t

η = yt mct p−η−1t +

[αβλt+1

λt

(pt

pt+1

)−η−1πηt+1

(η−1)x2t+1

η

]Eq. 9 x2

t = yt p−ηt +

[αβλt+1

λt

(pt

pt+1

)−ηπη−1t+1 x2

t+1

]Eq. 10 1 = (1−α) p1−η

t +α(

1πt

)1−η

The Central Bank

Eq. 11 logRt − logRss = ρR(logRt−1 − logRss

)+ (1−ρR

)(κπ log

(πtπss

)+κy log

(ytyss

))Other relations

Eq. 12 at kθt h1−θt = yt st+1

Eq. 13 st+1 = (1−α) p−ηt +απηt st

Eq. 14 kt+1 =(1−δ)

kt + it − κ2

(iikt

− Isskss

)2kt

Eq. 15 yt = ct + it

Exogenous processesEq. 16 log at+1 = ρa log at +σaεa,t+1

Here, the expectation operator has been omitted for notational simplicity. In

addition to the variables introduced in Section 1.5.1, λt is the household’s lagrange

multiplier for the budget constraint, qt is the lagrange multiplier for the law of motion

for capital, mct is firms’ marginal costs, and x2t is an auxiliary control variable needed

to obtain an exact recursive representation of the first-order condition for the optimal

real price pt of firms that adjust their prices in a given time period.

A Projection Approximation

When implementing the projection approximation, it is convenient to reduce the

number of control variables to minimize the number of unknown coefficients in

1.7. CONCLUSION 35

the Chebyshev polynomials. We therefore observe that the number of control vari-

ables can be greatly reduced by substituting the following expressions into the key

equations given below:

• Eq 1: λt = c−φ2t

• Eq 15: yt = ct + it

• Eq 10: pt = 1−α

(1πt

)1−η

1−α

11−η

• Eq 12: at kθt(ht

)1−θ = yt

((1−α) p−η

t +απηt st

)m

at kθt(ht

)1−θ = yt st+1

m

ht =[

yt

((1−α)p

−ηt +απηt st

)at kθt

] 11−θ

• Eq 3: wt =φ0(1−ht

)−φ1 /λt

• Eq 6: mct = wt

at (1−θ)(

htkt

)−θ

• Eq 7: r kt = at mctθ

(htkt

)1−θ

• Eq 8: qt = 1

1−κ2

(iikt

− Isskss

)

• Eq 11: Rt = explogRss +ρR

(logRt−1 − logRss

)+(

1−ρR)(κπ log

(πtπss

)+κy log

(ytyss

))

Thus we can write the system condensely as:


The Households

Eq. 2 qtλt =βλt+1[r kt+1 +qt+1

(1−δ)−qt+1

κ22

(it+1kt+1

− Isskss

)2

+qt+1κ2

(it+1kt+1

− Isskss

)it+1kt+1

]

Eq. 5 λt =βRt

[λt+1πt+1

]The Firms

Eq. 8 (η−1)x2t

η = yt mct p−η−1t +

[αβλt+1

λt

(pt

pt+1

)−η−1 (1

πt+1

)−η (η−1)x2t+1

η

]Eq. 9 x2

t = yt p−ηt +

[αβλt+1

λt

(pt

pt+1

)−η (1

πt+1

)1−ηx2

t+1

]Other relations

Eq. 13 st+1 = (1−α) p−ηt +απηt st

Eq. 14 kt+1 =(1−δ)

kt + it − κ22

(iikt

− Isskss

)2kt

Exogenous processesEq. 16 log at+1 = ρa log at +εa,t+1

The reduced system has four states(kt , st ,Rt−1, at

)and only four control variables

(ct , it ,πt , x2t ). We approximate the controls by Chebyshev polynomials using complete

polynomials, and obtain the control variables in period t +1 from Eq 13, 14, and

16 to get states in period t +1. The conditional expectations in Eq 2, 5, 8, and 9 are

evaluated using Gauss-Hermite quadratures with five points. The grid for the state

variables is constructed using 11 points along each dimension, where the points

are determined as the Chebyshev nodes. That is, we use a total of 114 = 14,641 grid

points. The upper and lower bounds along each dimension is determined from the

unconditional standard deviation of the states in a log-linearized approximation,

multiplied by ±3 for Rt−1, kt , and at . For the price dispersion index, st , the range is

set to −0.005 to 0.05. Given that we have more nodes than unknown coefficients in

the Chebyshev polynomials, we determine these coefficients by least squares (see

Judd (1992)). The optimization was implemented in FORTRAN 90 and executed on a

computer cluster using about 80 to 100 CPUs. Multiprocessing was exploited when

computing the Jacobian numerically in the Levenberg-Marquardt optimizer. In the

optimization, the solution at approximation order m was used when starting the

optimization at order m + 1. On the grid used for the approximation, the largest

residual is 1.39×10−5 when measured in terms of unit-free errors for the 12th order

approximation.

Appendix D: Conditional Expectations in DSGE Models

Consider the case where the DSGE model reports the endogenous variable rt and

we want to compute conditional expectations of this variable, i.e. r1,t ≡ Et[rt+1

],

1.7. CONCLUSION 37

r2,t ≡ Et[rt+2

], r3,t ≡ Et

[rt+3

], etc.22 The law of iterated expectations implies r2,t ≡

Et[rt+2

]= Et

[Et+1

[rt+2

]]= Et[r1,t+1

]and so on. Hence, only a formula for comput-

ing pt ≡ Et[rt+1

]is needed because all other expectations can be found be iterating

on this formula. We therefore consider the problem

p(xt

)= Et

[r(xt+1

)],

where we omit the perturbation parameter σ, given our focus on the perfect foresight

solution.23 We then observe that

F(xt

)≡ Et

[−p

(xt

)+ r(h

(xt

)+σηεt+1

)]= 0, (1.26)

because xt+1 = h(xt

)+σηεt+1. Note then that (1.26) must hold for all values of xt . This

allow us to compute all derivatives of p with respect to xt around the deterministic

steady state, i.e. xt = xss and σ= 0, given derivatives of h(xt

)and r

(xt+1

)around the

same point. For the indices we adopt the convention that the subscript indicates the

order of differentiation. I.e. a subscript 1 is for the first time we take derivatives and

so on. Thus,

α1,α2,α3 = 1,2, ...,nx γ1,γ2,γ3 = 1,2, ...,nx .

To compute the first-order terms, straightforward differentiation of (1.26) implies

[px

]α1

= [rx

]γ1

[hx

]γ1α1

,

or in standard matrix notation

px(1; :

)= rx(1, :

)hx.

The second-order terms are given by

[pxx

]α1α2

= [rxx

]γ1γ2

[hx

]γ2α2

[hx

]γ1α1

+ [rx

]γ1

[hxx

]γ1α1α2

,

or in the standard matrix notation

pxx = h′xrxxhx +

nx∑γ1=1

rx(1,γ1

)hxx

(γ1, :, :

),

22If the variable of interest is a control variable, then the function r(xt+1

)follows from the function

g (·). If the variable of interest is a state variable, then we let rt ≡ i′xt to obtain moments for the i ’th statevariable with i

(k,1

)= 1 for k = i , otherwise i(k,1

)= 0.23Derivatives of the conditional expectation with respect to the perturbation parameter are derived in

a technical appendix accompanying Andreasen (2012a).


where hxx has dimensions nx ×nx ×nx and contains all second order derivatives of

h (·) with respect to (xt ,xt ). Finally, the third-order terms are given by[pxxx

]α1α2α3

= [rxxx

]γ1γ2γ3

[hx

]γ3α3

[hx

]γ2α2

[hx

]γ1α1

+[rxx

]γ1γ2

[hxx

]γ2α2α3

[hx

]γ1α1

+[rxx

]γ1γ2

[hx

]γ2α2

[hxx

]γ1α1α3

+[rxx

]γ1γ3

[hx

]γ3α3

[hxx

]γ1α1α2

+[rx

]γ1

[hxxx

]γ1α1α2α3

,

or in the standard matrix notation

pxxx(α1,α2,α3

) =nx∑γ3=1

hx(:,α1

)′ rxxx(:, :,γ3

)hx

(:,α2

)hx

(γ3,α3

)+hx

(:,α1

)′ rxxhxx(:,α2,α3

)+

nx∑γ1=1

rxx(γ1, :

)hx

(:,α2

)hxx

(γ1,α1,α3

)+

nx∑γ1=1

rxx(γ1, :

)hx

(:,α3

)hxx

(γ1,α1,α2

)+rx

(1, :

)hxxx

(:,α1,α2,α3

).

Here, hxxx has dimensions nx ×nx ×nx ×nx and contains all third order derivatives of

h (·) with respect to (xt ,xt ,xt ). Similarly, rxxx and pxxx have dimensions nx ×nx ×nx

and contain all third order derivatives of the r - and p-functions, respectively.

1.7. CONCLUSION 39

Figure 1.1. The Neoclassical model: Accuracy Plots using the Benchmark Specification

Charts in the left column plot consumption in percentage deviation from steady state, i.e.

ct , as a function of kt = log(kt /kss

), ranging from −5 to +5 standard deviations in a log-

linearized solution. Charts in the right column report the log10 errors in consumption using

the projection approximation as the true solution. The conditioning level of technology in

the top and bottom charts equal −3 and +3 standard deviations in a log-linearized solution,

respectively.

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.3

−0.2

−0.1

0

0.1

Capital, kt

Con

sumption

,c t

Policy function: Low technology level

Perturbation: 1st-order Perturbation: 3rd-order Perfect foresight Extended Perturbation: 3rd-order Projection: 10th-order

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−5

−4

−3

−2

Capital, kt

Log10-errors

Errors: Low technology level

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.2

−0.1

0

0.1

0.2

Capital, kt

Con

sumption

,c t

Policy function: Technology at steady state

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−7

−6

−5

−4

−3

−2

Capital, kt

Log10-errors

Errors: Technology at steady state

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.1

0

0.1

0.2

0.3

Capital, kt

Con

sumption

,c t

Policy function: High technology level

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−5

−4

−3

−2

Capital, kt

Log10-errors

Errors: High technology level


Table 1.1. The Neoclassical Model: The Structural Parameters

Benchmark High risk aversion High varianceβ 0.99 0.99 0.99δ 0.025 0.025 0.025α 0.36 0.36 0.36γ 5.00 25 5.00ρa 0.98 0.98 0.98σa 0.01 0.01 0.10

Table 1.2. The Neoclassical Model: Accuracy of Consumption on Grid

The RMSEs are computed based on log(ct /c tr ue

t

)= ct − c tr ue

t , where c tr uet denotes

consumption in percentage deviation from the steady state in the true solution (i.e. theprojection approximation). The grid is constructed using 20 points uniformly spaced alongeach dimension of the state space, giving a total of 400 grid points. The bounds for capital andtechnology in the grid range from -3 to +3 standard deviations in a log-linearized solution.Bold figures highlight the approximation with the lowest RMSEs.

Benchmark High risk aversion High varianceRMSEsPerturbation: 1st order 0.0044 0.0393 0.3959Perturbation: 3r d order 1.19×10−5 0.0183 0.1242Perfect foresight 0.0033 0.0262 0.3141Perturbation +1 0.0125 0.0304 N aNPerturbation +2 0.0117 0.0296 N aNPerturbation +13r d 8.75×10−5 0.0251 N aNPerturbation +23r d 7.97×10−5 0.0251 N aNExtended Perturbation: 3r d order 2.46×10−5 0.0024 0.0822

1.7. CONCLUSION 41

Table 1.3. The Neoclassical Model: Accuracy of Consumption in Simulation

The RMSEs are computed based on log(ct /c tr ue

t

)= ct − c tr ue

t , where c tr uet denotes

consumption in percentage deviation from the steady state in the true solution (i.e. theprojection approximation). The reported standard deviations are for consumption inpercentage deviation from the steady state. The RMSEs and the standard deviations arecomputed using simulated paths based of 20,000 observations with a burn-in of 1,000observations. The symbol N aN indicates that consumption explodes using thisapproximation. Bold figures highlight the best performing approximation method(s).

Benchmark High risk aversion High varianceRMSEsPerturbation: 1st order 0.0033 0.0329 0.2865Perturbation: 3r d order 7.57×10−6 N aN 0.1290Perturbation pruned: 3r d order 6.46×10−5 0.0353 0.3624Perfect foresight 0.0024 0.0309 0.1936Perturbation +1 0.0029 0.0311 N aNPerturbation +2 0.0028 0.0303 N aNPerturbation +13r d 1.09×10−5 N aN N aNPerturbation +23r d 1.04×10−5 N aN N aNExtended Perturbation: 3r d order 1.63×10−5 0.0022 0.0921

Standard deviations for ct

Perturbation: 1st order 0.0635 0.0744 0.6351Perturbation: 3r d order 0.0625 N aN 0.3339Perturbation pruned: 3r d order 0.0625 0.0404 0.1838Perfect foresight 0.0628 0.0654 0.5311Perturbation +1 0.0630 0.0643 N aNPerturbation +2 0.0629 0.0639 N aNPerturbation +13r d 0.0630 N aN N aNPerturbation +23r d 0.0629 N aN N aNExtended Perturbation: 3r d order 0.0625 0.0497 0.3484Projection: 10th order 0.0625 0.0487 0.3698

Table 1.4. The New Keynesian Model: The Structural Parameters

β 0.99 α 0.75hss 0.33 ρR 0.80φ1 2.00 κπ 1.50φ2 2.00 κy 0.125κ 2.00 πss 1.00δ 0.025 ρa 0.95θ 0.36 σa 0.006η 6.00


Table

1.5.Th

eN

ewK

eynesian

Mo

del:A

ccuracy

on

Grid

Th

eR

MSE

sare

com

pu

tedb

asedo

nlo

g (zt /z

true

t

)=z

t −z

true

t,w

here

zt ≡ c

t ,it ,πt ,x

2t an

dth

etru

eso

lutio

nis

givenb

yth

ep

rojectio

n

app

roximation

.Th

egrid

iscon

structed

usin

g20

poin

tsu

niform

lysp

acedalon

geach

dim

ension

ofthe

statesp

ace,giving

atotalof20

4=160,000

gridp

oin

ts.Th

eb

ou

nd

sfo

rth

ein

terestrate,capital,an

dtech

no

logy

inth

egrid

range

from

-3to

+3stan

dard

deviatio

ns

ina

log-lin

earizedso

lutio

n.Fo

rth

ep

riced

ispertio

nin

dex

the

gridran

gesfro

m−

0.005to

0.05.Bo

ldfi

gures

for

eacho

fthe

variables

high

lightth

eap

proxim

ation

with

the

lowestR

MSE

.

Stand

ardPertu

rbatio

n:

Perfectfo

resight

Exten

ded

Perturb

ation

:1

sto

rder

3rd

ord

er3

rdo

rder

RM

SEs

Co

nsu

mp

tion

:ct

0.002530.00170

0.001440.00124

Investm

ent:it

0.068130.01718

0.009570.00884

Infl

ation

:πt

0.001900.00058

0.000600.00049

Au

xiliaryco

ntro

lvariable:x

2t0.18741

0.055780.03863

0.03866

1.7. CONCLUSION 43

Table 1.6. The New Keynesian Model: Accuracy in Simulation

The RMSEs are computed based on log(zt /z tr ue

t

)= zt − z tr ue

t , where zt ≡

ct , it ,πt , x2t

and

the true solution is given by the projection approximation. The reported standard deviationsare in percentage deviation from the steady state. The RMSEs and the standard deviations arecomputed using simulated paths based of 20,000 observations with a burn-in of 1,000observations. Bold figures highlight the best performing approximation method(s).

Consumption Investment Inflation Aux. controlct it πt x2

tRMSEsPerturbation: 1st order 0.00263 0.00811 0.00091 0.01284Perturbation: 3r d order 0.00116 0.00431 0.00043 0.00499Perturbation: 3r d order, exact st 0.00103 0.00415 0.00038 0.00401Perturbation pruned: 3r d order 0.00134 0.00457 0.00049 0.00579Perfect foresight 0.00137 0.00371 0.00045 0.00234Extended Perturbation: 3r d order 0.00094 0.00362 0.00038 0.00313

Standard deviationsPerturbation: 1st order 0.01517 0.05083 0.00579 0.05874Perturbation: 3r d order 0.01570 0.05202 0.00595 0.06114Perturbation: 3r d order, exact st 0.01577 0.05222 0.00598 0.06158Perturbation pruned: 3r d order 0.01564 0.05182 0.00593 0.06088Perfect foresight 0.01562 0.05185 0.00594 0.06237Extended Perturbation: 3r d order 0.01579 0.05215 0.00599 0.06210Projection: 12th order 0.01617 0.05328 0.00618 0.06368


Table 1.7. Extended Path: The Terminal Condition

All three terminal conditions imply the same RMSEs for the control variables (denoted RMSEy)using N = 250, where N denotes the horizon in the Extended Path. The RMSEs are computedin a simulated sample path of 5,000 observations. Starting values for the Extended Path arecomputed from a third-order perturbation approximation. Bold figures highlight the bestperforming method for a given value of N .

RMSEs for terminal conditions

yt+N = yss yt+N = Et

[y1st

t+N

]yt+N = Et

[y3r d

t+N

]Stable perturbation calibrationN = 200 6.41×10−9 1.32×10−10 1.29×10−10

N = 175 6.06×10−8 1.25×10−9 1.21×10−9

N = 150 5.63×10−7 1.18×10−8 1.11×10−8

N = 125 5.22×10−6 1.10×10−7 9.91×10−8

N = 100 7.01×10−5 1.04×10−6 8.59×10−7

N = 75 5.63×10−4 1.02×10−5 7.08×10−6

N = 50 0.0035 1.32×10−4 5.54×10−5

Explosive perturbation calibrationN = 200 2.77×10−8 2.45×10−10 2.16×10−10

N = 175 2.71×10−7 6.37×10−9 3.62×10−9

N = 150 2.67×10−6 1.44×10−7 9.49×10−8

N = 125 2.67×10−5 2.91×10−6 2.08×10−6

N = 100 0.0074 6.00×10−5 4.46×10−5

N = 75 0.0165 0.0040 4.83×10−4

N = 50 0.0120 0.0183 0.0197

1.7. CONCLUSION 45

Table 1.8. State-Dependent Horizon in the Extended Path

The RMSEs for the control variables are computed using the horizon N = 200 as thebenchmark in a simulated sample path of 5,000 observations. The optimal horizon N∗ isrestricted to the interval from 20 to 200. Starting values and the terminal condition areobtained from a third-order perturbation approximation. The perturbation approximation atthird order and the pruned version are computed without the uncertainty correction to get theperfect foresight approximation. The computations are carried out in MATLAB 2014a using anIntel(R) Core(TM) i5-4200 CPU with 2.50 GHz.

RMSEs Mean(seconds) Mean(N∗)per 1,000 draws

Stable perturbation calibrationExtended Path: Dss = 0 0 385 200Extended Path: Dss = 0.01 7.91×10−5 196 87Extended Path: Dss = 0.02 2.50×10−4 136 58Extended Path: Dss = 0.03 3.92×10−4 101 43Extended Path: Dss = 0.05 5.92×10−4 74 29Extended Path: Dss = 0.08 7.43×10−4 62 21Extended Path: Dss = 0.10 7.89×10−4 51 20Perturbation: 1st order 0.0071 0.004 −Perturbation: 3r d order 2.87×10−4 0.05 −Perturbation pruned: 3r d order 5.05×10−4 0.12 −

Explosive perturbation calibrationExtended Path: Dss = 0 0 496 200Extended Path: Dss = 0.01 6.31×10−5 361 100Extended Path: Dss = 0.02 2.00×10−4 263 69Extended Path: Dss = 0.03 3.40×10−4 208 52Extended Path: Dss = 0.05 7.84×10−4 137 35Extended Path: Dss = 0.08 0.0127 75 25Extended Path: Dss = 0.10 0.0176 66 22Perturbation: 1st order 0.0134 0.004 −Perturbation: 3r d order N aN − −Perturbation pruned: 3r d order 0.0072 0.12 −


Table 1.9. Combining Perturbation and Extended Path: Stable Perturbation Calibration

The RMSEs for the control variables are computed using N = 200 and EE = 0 as thebenchmark in a simulated sample path of 5,000 observations. The optimal horizon N∗ isrestricted to the interval from 20 to 200. Starting values and the terminal condition areobtained from a third-order perturbation approximation. The perturbation approximation atthird order is computed without the uncertainty correction to get the perfect foresightapproximation. The computations are carried out in MATLAB 2014a using an Intel(R)Core(TM) i5-4200 CPU with 2.50 GHz.

RMSEs Mean(seconds) Pct of times Extendedper 1,000 draws Path is used

Dss = 0 EE = 0 0 385 100EE = 0.00005 3.28×10−5 57 27EE = 0.0001 6.20×10−5 35 15.3EE = 0.001 2.38×10−4 1.4 0.36EE = 0.01 2.87×10−4 0.5 0

Dss = 0.01 EE = 0 7.91×10−5 196 100EE = 0.00005 3.65×10−5 33 27EE = 0.0001 6.24×10−5 18 15.3EE = 0.001 2.38×10−4 1.0 0.36EE = 0.01 2.87×10−4 0.5 0

Dss = 0.03 EE = 0 3.92×10−4 101 100EE = 0.00005 1.61×10−4 16 27EE = 0.0001 1.11×10−4 12 15EE = 0.001 2.38×10−4 1.0 0.36EE = 0.01 2.87×10−4 0.5 0

Dss = 0.05 EE = 0 5.92×10−4 74 100EE = 0.00005 3.33×10−4 11 27EE = 0.0001 2.38×10−4 9 15EE = 0.001 2.38×10−4 1.0 0.36EE = 0.01 2.87×10−4 0.5 0

Dss = 0.08 EE = 0 7.43×10−4 62 100EE = 0.00005 4.63×10−4 7 27EE = 0.0001 3.49×10−4 5 15EE = 0.001 2.38×10−4 0.7 0.36EE = 0.01 2.87×10−4 0.5 0

1.7. CONCLUSION 47

Table 1.10. Combining Perturbation and Extended Path: Explosive Perturbation Calibra-tion

The RMSEs for the control variables are computed using N = 200 and EE = 0 as thebenchmark in a simulated sample path of 5,000 observations. The optimal horizon N∗ isrestricted to the interval from 20 to 200. Starting values and the terminal condition areobtained from a third-order perturbation approximation. The perturbation approximation atthird order is computed without the uncertainty correction to get the perfect foresightapproximation. The computations are carried out in MATLAB 2014a using an Intel(R)Core(TM) i5-4200 CPU with 2.50 GHz.

RMSEs Mean(seconds) Pct of times Extendedper 1,000 draws Path is used

Dss = 0 EE = 0 0 498 100EE = 0.00005 0.0052 151 45EE = 0.0001 0.0058 119 34EE = 0.001 0.0068 42 5EE = 0.01 0.0079 27 0.9

Dss = 0.01 EE = 0 6.31×10−5 361 100EE = 0.00005 0.0052 101 45EE = 0.0001 0.0058 82 34EE = 0.001 0.0068 29 5EE = 0.01 0.0079 18 0.9

Dss = 0.03 EE = 0 3.40×10−4 208 100EE = 0.00005 0.0044 78 45EE = 0.0001 0.0050 62 34EE = 0.001 0.0069 18 5EE = 0.01 0.0080 11 0.9

Dss = 0.05 EE = 0 7.84×10−4 137 100EE = 0.00005 0.0050 55 45EE = 0.0001 0.0061 39 34EE = 0.001 0.0073 13 5EE = 0.01 0.0079 8 0.9

Dss = 0.08 EE = 0 0.0127 75 100EE = 0.00005 0.0106 31 45EE = 0.0001 0.0106 28 34EE = 0.001 0.0127 10 5EE = 0.01 0.0145 7 1.2


Figure 1.2. The Neoclassical Model: Accuracy Plots with High Risk Aversion

Charts in the left column plot consumption in percentage deviation from steady state, i.e.

ct , as a function of kt = log(kt /kss

), ranging from −5 to +5 standard deviations in a log-

linearized solution. Charts in the right column report the log10 errors in consumption using

the projection approximation as the true solution. The conditioning level of technology in

the top and bottom charts equal −3 and +3 standard deviations in a log-linearized solution,

respectively.

−1 −0.5 0 0.5 1

−0.4

−0.2

0

0.2

Capital, kt

Con

sumption

,c t



−1 −0.5 0 0.5 1

−4

−3

−2

−1

Capital, kt

Log10-errors


−1 −0.5 0 0.5 1

−0.3

−0.2

−0.1

0

0.1

0.2

Capital, kt

Con

sumption

,c t


−1 −0.5 0 0.5 1

−5

−4

−3

−2

−1

Capital, kt

Log10-errors


−1 −0.5 0 0.5 1

−0.2

−0.1

0

0.1

0.2

0.3

Capital, kt

Con

sumption

,c t


−1 −0.5 0 0.5 1

−3.5−3

−2.5−2

−1.5−1

−0.5

Capital, kt

Log10-errors


1.7. CONCLUSION 49

Figure 1.3. The Neoclassical Model: Accuracy Plots with High Variance

Charts in the left column plot consumption in percentage deviation from steady state, i.e. ct ,as a function of kt = log

(kt /kss

), ranging from −5 to +5 standard deviations in a

log-linearized solution. Charts in the right column report the log10 errors in consumptionusing the projection approximation as the true solution. The conditioning level of technologyin the top and bottom charts equal −3 and +3 standard deviations in a log-linearized solution,respectively.

−3 −2 −1 0 1 2 3−2.5−2

−1.5−1

−0.50

0.5

Capital, kt

Con

sumption

,c t



−3 −2 −1 0 1 2 3

−3

−2.5

−2

−1.5

−1

−0.5

0

Capital, kt

Log10-errors


−3 −2 −1 0 1 2 3

−1

−0.5

0

0.5

1

Capital, kt

Con

sumption

,c t


−3 −2 −1 0 1 2 3

−2.5

−2

−1.5

−1

−0.5

0

0.5

Capital, kt

Log10-errors


−3 −2 −1 0 1 2 3

0

0.5

1

1.5

2

Capital, kt

Con

sumption

,c t


−3 −2 −1 0 1 2 3

−2

−1

0

1

Capital, kt

Log10-errors



Figure 1.4. The Neoclassical Model: Stability Plots

The low and high conditioning level of technology equal −3 and +3 standard deviations in alog-linearized solution, respectively.

−1 −0.5 0 0.5 1−0.02

−0.01

0

0.01

0.02

0.03

kt+

1−

dt

Capital, kt

Benchmark: Standard 3rd order perturbation

Low technology level Steady state technology level High technology level

−1 −0.5 0 0.5 1−0.02

−0.01

0

0.01

0.02

0.03

kt+

1−

dt

Capital, kt

Benchmark: Extended 3rd order perturbation

−2 −1 0 1 2−0.02

−0.01

0

0.01

0.02

0.03

kt+

1−

dt

Capital, kt

High risk aversion: Standard 3rd order perturbation

−2 −1 0 1 2−0.02

−0.01

0

0.01

0.02

0.03

kt+

1−

dt

Capital, kt

High risk aversion: Extended 3rd order perturbation

−10 −5 0 5 10−0.5

0

0.5

1

kt+

1−

dt

Capital, kt

High variance: Standard 3rd order perturbation

−10 −5 0 5 10−0.5

0

0.5

1

kt+

1−

dt

Capital, kt

High variance: Extended 3rd order perturbation

1.7. CONCLUSION 51

Figure 1.5. New Keynesian Model: Accuracy Plots

Charts in the left column plot the control variables as a function of at , ranging from −3 to +3standard deviations. Charts in the right column report the log10 errors in the control variablesusing the projection approximation as the true solution. The conditioning level of the nominalinterest rate and the capital stock equals −3 standard deviations in a log-linearized solution.The conditional level of st is 0.04.

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05

−0.05

−0.04

−0.03

−0.02

−0.01

0

Technology, at

Deviation

from

SS

Consumption, ct


−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

Errors in Consumption, ct

Log10-errors

Technology, at

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05

0.4

0.5

0.6

0.7

0.8

0.9

1

Technology, at

Deviation

from

SS

Investment, it

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

Errors in Investment, it

Log10-errors

Technology, at

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05

0.02

0.03

0.04

0.05

0.06

Technology, at

Deviation

from

SS

Inflation, πt

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

Errors in Inflation, πt

Log10-errors

Technology, at

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

Technology, at

Deviation

from

SS

Auxiliary control variable, x2t

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05

−3

−2.5

−2

−1.5

−1

−0.5

0

Errors in Auxiliary control variable, x2t

Log10-errors

Technology, at


Figure 1.6. The New Keynesian Model: Stability Plots

The policy function for inflation and the transition function for the price dispertion index areplotted as a function of st . The conditioning level of the nominal interest rate and the capitalstock equals −4 standard deviations in a log-linearized solution. The conditioning level oftechnology equals −3 standard deviations. The histogram for the distribution of st iscomputed from a simulated sample path of 20.000 observations using the projectionapproximation.

0 0.01 0.02 0.03 0.04 0.050.054

0.056

0.058

0.06

0.062

0.064

0.066

0.068

0.07

Inflation (πss = 1.00)

πt

Price dispersion index, st

Perturbation: 3rd-order Extended Perturbation: 3rd-order Projection: 12th-order

0 0.01 0.02 0.03 0.04 0.05

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

Price dispersion index (πss = 1.00):

s t+1

Price dispersion index, st0.01 0.02 0.03 0.04

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Distribution of st (πss = 1.00)


0 0.02 0.04 0.06 0.08 0.1

0.055

0.06

0.065

0.07

0.075

0.08

0.085Inflation (πss = 1.0015)

Price dispersion index: st

πt

0 0.02 0.04 0.06 0.08 0.1

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Price dispersion index (πss = 1.0015)

Price dispersion index: st

s t+1

0 0.02 0.04 0.06 0.080

0.05

0.1

0.15

0.2

Distribution of st (πss = 1.0015)


1.7. CONCLUSION 53

Figure 1.7. Combining Perturbation and Extended Path: Plot of Sample Path

This figure shows part of the simulated sample where standard third-order perturbationexplodes. That is, the simulation is for the explosive perturbation calibration and with Dss = 0.The extended perturbation approximation is computed by standard perturbation, except atthe areas shaded gray where the Extended Path is used. The y-axis reports the distance of the

state variables from the steady state, i.e. xt from the steady state, i.e.∥∥xt

∥∥=√∑nx

i=1 x2i ,t .

4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 45000

0.1

0.2

0.3

0.4

Tolerated Euler Error, EE = 0.0001

∥xt∥

Extended Path Perturbation 3rd-order Extended Perturbation: 3rd-order

4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 45000

0.1

0.2

0.3

0.4

Tolerated Euler Error, EE = 0.01

∥xt∥

C H A P T E R 2NEW EVIDENCE ON DOWNWARD NOMINAL WAGE

RIGIDITY AND THE IMPLICATIONS FOR

MONETARY POLICY

Anders Kronborg


Abstract

This paper examines the degree of downward nominal wage rigidity and its macroe-

conomic implications. For this purpose, a simple dynamic stochastic general equilib-

rium model is estimated where the nominal wage rigidity is allowed to be asymmetric.

As a novelty, the nonlinear model equilibrium is approximated using the extended

perturbation method in Andreasen and Kronborg (2016) which improves the accu-

racy relative to standard perturbation when the model is characterized by strong

nonlinearities. The estimates show that wages are more downwardly than upwardly

rigid, which generates asymmetric responses to shocks. However, the asymmetries

generated are generally less pronounced than what is found in the literature. The

estimated model is subsequently used to compute the welfare implications of differ-

ent inflation targets. For the U.S., the optimal inflation target is approximately 0.25

percent per year which provides support for low but positive inflation.

55

56

CHAPTER 2. NEW EVIDENCE ON DOWNWARD NOMINAL WAGE RIGIDITY AND THE

IMPLICATIONS FOR MONETARY POLICY

2.1 Introduction

Downward nominal wage rigidity (henceforth DNWR), a phenomenon where wages

are more downwardly than upwardly rigid, has received increased attention during

the recent Great Recession. Schmitt-Grohe and Uribe (2013) show that during the

recent economic downturn, the average hourly nominal wage has stayed largely the

same as before the crisis in several European economies even as unemployment rates

have risen dramatically. In a speech at the Federal Reserve Bank of Kansas City in

2014, the Chair of the Federal Reserve Janet Yellen suggested that the sluggish wage

growth during the recovery of the U.S. economy might be a result of ’pent-up wage

deflation’ because wages did not adjust sufficiently during the crisis.1

This paper looks at DNWR and its macroeconomic implications for the busi-

ness cycle and monetary policy. This is done by estimating a simple New Keynesian

dynamic stochastic general equilibrium (DSGE) model similar to that in Kim and

Ruge-Murcia (2009). A key feature in the economy is that the wage rigidity is allowed

to be asymmetric, in the sense that it is more costly for the households to cut their

nominal wages than to increase them. This differs from most of the DSGE literature,

in which sticky prices and wages are usually modeled in a symmetric fashion (see

Blanchard (2009) for a discussion), partly reflecting the use of linearization as the

preferred solution technique. Thus, to capture the asymmetric features of DNWR in

the DSGE model it is necessary to apply a nonlinear approximation.

As a novelty, the nonlinear model solution is approximated using the extended

perturbation method of Andreasen and Kronborg (2016). This approach combines the

perfect foresight solution obtained from the Extended Path (Fair and Taylor (1983))

with higher-order risk corrections from standard perturbation. This approach gener-

ally improves the accuracy relative to a higher-order perturbation approximation by

removing error terms in the model solution under perfect foresight. Further, it is likely

to preserve characteristics of the true solution, such as monotonicity or convexity,

something that standard perturbation may struggle to obtain, as emphasized by Den

Haan and De Wind (2012). These improvements can be substantial if the model is

characterized by strong nonlinearities. As shown in this paper, this is in fact the case

for the applied model, even for modest levels of DNWR.

Using U.S. data from 1964Q2-2015Q1 the structural parameters of the model

are estimated by the simulated method of moments (SMM) as described in Duffie

and Singleton (1993). The choice of econometric methodology reflects the nonlinear

nature of the approximated solution, which does not allow for the moments to be

computed in closed-form, making GMM infeasible. Instead, as shown in Ruge-Murcia

(2012), SMM can be a feasible way of estimating DSGE models by minimizing the

1The remarks, titled "Labor Market Dynamics and Monetary Policy", can be found at the webpage ofthe Board of Govenors of the Federal Reserve System.

2.1. INTRODUCTION 57

weighted difference between the data moments and those from model generated

sample paths.

The main findings in this paper are as follows: First, and in line with previous

studies, both prices and wages are found to be nominally rigid. For wages, the data

suggest that the adjustment to shocks is asymmetric, meaning DNWR is present in

the U.S. economy. Hence, the estimated model is characterized by asymmetric propa-

gation of shocks, which is especially pronounced for nominal wages and inflation but

feed through to the real economic variables where contractions are generally larger

than expansions. While the results confirm the qualitative findings in the literature

(see below), the degree of asymmetry found in this paper is generally less pronounced.

Specifically, the parameter estimate relating to the asymmetry in wage adjustments

is orders of magnitude smaller than what is found in previous studies, which can

attributed to the change in solution methodology. Second, as argued by Tobin (1972),

the presence of DNWR can help bridge the apparent gap between much of the classic

monetary literature prescribing price stability or deflation (examples include Fried-

man (1969) and Woodford (2003)) and the fact that this is not the monetary policy

pursued by central banks. In the DSGE model in this paper, the central bank faces

a trade-off: A positive steady state inflation reduces the need for nominal wages to

decline in face of adverse shocks, so it may be prudent to operate with a positive infla-

tion target in lieu of price stability. On the other hand, price rigidity implies that this is

associated with systematic costs. Based on the estimates, the optimal inflation target

is found to be approximately 0.25 percent per year when implemented as a strict

inflation target in a Taylor rule which provides support for low but positive inflation

in the presence of DNWR. The welfare improvements of a higher inflation target are

decreasing in the level of price rigidity and increasing in the level of macroeconomic

volatility.

The finding that wages are more downwardly than upwardly rigid is well docu-

mented at the micro level (see for example Dickens et al. (2007)). Some researchers

suggest that this may be caused by the tendency of firm managers to cut wage only

in periods of severe financial distress, as they worry about adverse effects on worker

morale (Kahneman et al. (1986) and Akerlof et al. (1996)). Others point to institu-

tional or legal factors such as minimum wages, wage indexation, restrictions on firing

workers and collective bargaining (Babecky et al. (2010)). It is not clear however, to

which extent DNWR at the micro level translates into the aggregated level, e.g. due

to composition or business cycle effects (Holden and Wulfsberg (2009)). Further, in

order to examine how nominal rigidities affect the business cycle and what the impli-

cations are for monetary policy it is necessary to use a macroeconomic framework.

An agnostic approach to DNWR is taken in this paper, implying that the reason(s)

behind this friction is not modeled explicitly. This is done to ensure tractability and

to keep the analysis as simple as possible.

58



This work is related to a series papers in the DSGE literature which use a similar

framework. In Kim and Ruge-Murcia (2009) and Kim and Ruge-Murcia (2011) the

authors estimate a model with asymmetric wage costs similar to the one in this paper

using a second-order pruned perturbation approximation. They find that DNWR

implies that the optimal inflation target is between 0.75 and 1 percent per year. Fahr

and Smets (2010) extend this framework to calibrate a model with two countries in

a monetary union and show that the region with DNWR adjusts slowly to adverse

shocks due to a persistent loss of competitiveness. Abbritti and Fahr (2013) calibrate a

version of this model to match the positive skewness of annual growth rates of wages

and the negative skewness of employment, investment, and output found for most

economies. Other studies find the optimal inflation target in economies with DNWR

to be higher than the above. For example, Akerlof et al. (1996) consider a model in

which wages can never fall and find the optimum to be 3 percent per year.

The methodological choice in this paper has several implications for the analysis:

First, given that the approximated solution is more accurate, the estimates are likely

to be closer to the "true" values as well, provided that the model is correct. Second,

the interpretation of the parameter estimates of the approximated equilibrium is

closer to that of the underlying model. For example, since the optimality conditions

includes taking the first-order derivative of a linex function, second-order perturba-

tion implies that the approximated equilibrium relies on its third-order derivative. As

a result, when using second- and third-order perturbation the resulting approximated

adjustment costs are no longer convex and become negative when wages increase

beyond a certain point, even for moderate amounts of DNWR. With the extended

perturbation method this does not happen as it includes higher-order terms of the

perfect foresight component of the policy functions. Third, since the extended pertur-

bation does not rely on a local approximation of the perfect foresight component its

precision is relatively high even far from the steady state. This allows us to study the

nonlinearities of the model far from the non-stochastic steady state more accurately,

for example when the economy is hit by large adverse shocks.

The remainder of the paper is structured as follows. Section 2.2 presents the

DSGE model. Section 2.3 introduces the extended perturbation method used to ap-

proximate the model. The estimation methodology used is described in Section 2.4.

Section 2.5 briefly lists the data and moments used in the estimation. The model

properties are analyzed in Section 2.6. Section 2.7 provides an estimate of the op-

timal inflation target and discusses the robustness of the results while Section 2.8

concludes.

2.2. THE DSGE MODEL 59

2.2 The DSGE Model

This section presents the DSGE model to be estimated and used for the analysis. The

model is identical to the one found in Kim and Ruge-Murcia (2009). A key feature

of the model is the labor market. The representative household is a monopolistic

supplier of differentiated labor and hence has some market power through the setting

of wages. Nominal wage rigidity is allowed to be asymmetric in the model by assuming

a more general cost function than the usual quadratic costs in Rotemberg (1982). The

economy is assumed to feature shocks to preferences and productivity.

2.2.1 Households

Preferences

The households are assumed to be infinitely lived and indexed by n ∈ [0,1]. They

maximize the expected discounted utility

Ut = Et

∞∑s=0

βs dt+s

c1−ρn,t+s −1

1−ρ −χhn,t+s

, (2.1)

where Et is the conditional expectation at time t . Consumption of household n at

time t is cn,t and hn,t is the labor supply. The discount factor is denoted β ∈ (0,1),

ρ > 0 controls the utility curvature of consumption andχ> 0 is a parameter governing

the disutility from hours worked. The preference shock, dt , shifts the intertemporal

utility of the household and is assumed to evolve as

log dt+1 = ρd log dt +σdεdt+1, εd

t+1 ∼N.i .d .(0,1),

where ρd ∈ (0,1). Consumption is a basket of differentiated goods indexed by i ∈ [0,1],

and aggregated by

cn,t =[∫ 1

0

(cn,i ,t

) η−1η di

] ηη−1

,

where η> 1 is the elasticity of substitution between consumption goods, implying a

steady state markup of ηη−1 . Cost minimization gives the following demand for good i

by household n

cn,i ,t =(

Pi ,t

Pt

)−ηcn,t , (2.2)

where Pi ,t is the nominal price of good i and Pt =[∫ 1

0

(Pi ,t

)1−ηdi

] 11−η

is the aggregate

price index at time t . Hence, total consumption expenditure is Pt cn,t .

60



Wage Setting

The household enters in a market with monopolistic competition for its labor ser-

vices. Thus, the household has some market power through the determination of its

nominal wage, Wn,t , subject to the firm demand for labor. The labor market is subject

to frictions in the form of nominal rigidity in the wage adjustment so that changing

the nominal wage is associated with the following adjustment costs

Φ

(Wn,t

Wn,t−1

)=φ

[exp

−ψ

(Wn,t

Wn,t−1−1

)+ψ

(Wn,t

Wn,t−1−1

)−1

ψ2

]. (2.3)

The parameter φ≥ 0 determines the general degree of convexity while ψ controls the

asymmetry in wage adjustments. When ψ> 0 the exponential term will dominate

the linear term whenWn,t

Wn,t−1< 1 and hence nominal wage decreases are more costly

than increases. Thusψ> 0 corresponds to the case of DNWR. Asψ→∞, the function

obtains an "L-shape" implying that nominal wages can never fall. The function is

convex and differentiable. The latter is necessary to apply perturbation and the

extended perturbation method.

Finally, note that the linex function nests quadratic costs in the limiting case,

l i mψ→0

Φt = φ

2

(Wn,t

Wn,t−1−1

)2

.

This implies that we can test for DNWR econometrically by testing the significance of

ψ in a one-side test.

Budget Constraint and Optimality Conditions

In every period t , the budget constraint for household n is given by (in consumption

units)

cn,t +Et

[Qt ,t+1 An,t

Pt

]+ Bn,t

Pt= Rt−1Bn,t−1

Pt+ An,t−1

Pt+ Wn,t

Pt

(1−Φn,t

)hn,t +

Dn,t

Pt,

(2.4)

where An,t is a portfolio of state-contingent Arrow-Debreu assets, i.e. An,t (ω j ) pays

out one dollar if state ω j is reached in period t +1. The price of this portfolio is deter-

mined using the nominal stochastic discount factor, Qt ,t+1. Further, the household

can buy one-period nominal bonds, Bn,t , which carry the gross nominal interest rate

Rt from period t to t+1. The household uses resources for consumption and acquires

assets for next period. Expenditures are financed by the payoff from assets carried


from the previous period, wages net of adjustment costs2, and dividends received

from firms, Dn,t .

After imposing a symmetric equilibrium, the optimality conditions for the repre-

sentative household for consumption and nominal wages are, respectively3

0 =βRtEt

[λt+1

λtΠ−1

t+1

]−1, (2.5)

0 = λt ht

Pt

[(ν−1)

(1−Φt

)+ ∂Φt

∂ωtωt

]−dtχν

ht

Wt−βEt

[∂Φt+1

∂ωt+1

λt+1

Pt+1ω2

t+1ht+1

]. (2.6)

Here the gross nominal wage growth is denoted ωt ≡ WtWt−1

and gross inflation is

Πt = PtPt−1

. The stochastic discount factor is given as Qt ,t+1 = βλt+1λtΠ−1

t+1 and λt =c−ρt dt is the marginal utility of consumption. The parameter ν> 1 is the elasticity of

demand for labor, implying a steady state markup of νν−1 . Equation (2.5) is the Euler

equation which balances the expected marginal rate of substitution for intertemporal

consumption with the relative price of consumption between periods, given by the

nominal interest rate and the inflation rate. Equation (2.6) is the wage Phillips curve.

The household equals the marginal cost of increasing the nominal wage with its

marginal benefit. The costs include lower demand for its labor when firms substitute

towards cheaper labor input as well as the adjustment costs. The benefits consist

of a higher hourly wage, lower disutility as the hours worked are reduced, and next

period’s expected net utility gain from lower adjustment costs.

2.2.2 Firms

Production Technology and Labor Demand

The production side of the economy is populated by a continuum of firms, indexed by

i ∈ [0,1]. Each firm i produces a differentiated good by using labor as input, according

to the following production function

yi ,t = at h1−θi ,t , (2.7)

where θ ∈ (0,1) is a production parameter. The variable at refers to an economy wide

exogenous productivity level and is assumed to evolve as follows

log at+1 = ρa log at +σaεat+1, εa

t+1 ∼N.i .d .(0,1),

2Φn,t gives the fraction of gross real wage income,Wn,t hn,t

Pt, wasted due to adjustment costs. This in

fact implies that the asymmetry in total wage adjustment costs depends on both ν and ψ.3These optimality conditions can be derived by combining the first-order conditions for consumption,

hours worked, demand for state-contingent claims and bonds, and the nominal wage level. Detailedderivations used in this paper can be found in a technical appendix, available upon request. A modeloverview is given in Appendix A.

62



where ρa ∈ (0,1). The labor used in the production by firm i is a composite of the

differentiated labor types

hi ,t =[∫ 1

0

(hn,i ,t

) ν−1ν dn

] νν−1

.

The demand by firm i for each labor unit n is derived from the cost minimization

problem

mi nhn,i ,t

∫ 1

0Wn,t hn,i ,t dn, s.t . hi ,t =

[∫ 1

0

(hn,i ,t

) ν−1ν di

] νν−1 ≥ hi ,t ,

where the nominal wage index paid for the labor composite at time t is given as Wi ,t =[∫ 10

(Wn,t

)1−νdn

] 11−ν

and the aggregated quantities are defined as hn,t =∫ 1

0 hn,i ,t di

and ht =∫ 1

0 hi ,t di . The resulting demand for type n labor by firm i is given as

hn,i ,t =(

Wn,t

Wt

)−νhi ,t . (2.8)

The aggregate demand for labor faced by household n is then given by aggregating

(2.8) over all of the firms

hn,t =∫ 1

0

(Wn,t

Wt

)−νhi ,t di =

(Wn,t

Wt

)−νht . (2.9)

Price Setting

Output prices are nominally sticky in the Rotemberg sense, i.e. adjusting prices

implies a cost for firm i at time t given by4

Γ

(Pi ,t

Pi ,t−1

)= γ

2

(Pi ,t

Pi ,t−1−1

)2

. (2.10)

The parameter γ≥ 0 controls the degree of convexity in adjustment costs.

At time t , firm i maximizes the expected discounted profits (in real terms)

maxPi ,t+s ,hi ,t+s

∞s=0

Et

∞∑s=0

βs λt+s

λt

1

Pt+s

[(1−Γi ,t+s

)Pi ,t+s yi ,t+s −Wt+s hi ,t+s

],

subject to the production technology in (2.7) and the demand yi ,t = ci ,t . Using (2.2)

and aggregating over the households we can obtain the total demand faced by firm i

ci ,t =∫ 1

0

(Pi ,t

Pt

)−ηcn,t dn =

(Pi ,t

Pt

)−ηct . (2.11)

4Like in Kim and Ruge-Murcia (2009), price adjustment costs are assumed to be quadratic to econo-mize with the number of parameters in the estimation. Preliminary estimates using pruning suggest thatthe asymmetry parameter in the generalized function is in fact insignificant.


After imposing a symmetric equilibrium, the optimality conditions for the represen-

tative firm for labor demand and price are, respectively

Wt

Pt= mct (1−θ)at h−θ

t , (2.12)

0 = (η−1)(1−Γt

)+ ∂Γt

∂PtΠt −ηmct −βEt

[λt+1

λt

ct+1

ct

∂Γt+1

∂Pt+1Πt+1

], (2.13)

where mct is the real marginal cost of production (i.e. the Lagrangian multiplier

for the optimization problem). Equation (2.12) gives the labor demand by equating

the marginal product of labor with the cost. The optimal price in (2.13) is the price

Phillips-curve and can be interpreted in similar fashion as (2.6), where the firm bal-

ances the marginal costs and benefits of a price increase. The costs include reduced

demand for the firm’s goods as consumers substitute towards cheaper ones and the

adjustment costs. The benefits include a higher unit price, a reduction in marginal

production costs (a result of lower demand), and next period’s expected net profit

gain from lower adjustment costs.

2.2.3 Aggregation

In equilibrium, all households supply the same amount of labor and set the same

nominal wage. Further, the net supply of all financial assets is zero. Using this, the

budget constraint in (2.4) simplifies to

ct = Wt

Pt(1−Φt )ht + D t

Pt. (2.14)

As all firms set the same price and produce the same quantity in equilibrium, aggre-

gate dividends are given by D t = (1−Γt )Pt yt −Wt ht . Combining this with (2.14), the

economy resource constraint simplifies to

yt =ct + Wt

PtΦt ht

1−Γt. (2.15)

Hence, nominal price and wage rigidity lowers the aggregate consumption level

relative to its frictionless counterpart. Further, if the economy is characterized by

DNWR, this gap increases when nominal wages decrease.

2.2.4 Monetary Policy

The model is closed by a monetary policy rule. The central bank is assumed to set the

interest rate to stabilize inflation and hours worked in accordance with the following

Taylor rule

l og

(Rt

Rss

)= ρR l og

(Rt−1

Rss

)+κπlog

(Πt

Πss

)+κh l og

(ht

hss

), (2.16)

64



where ρR ∈ (0,1) and κπ,κh ≥ 0 are constant policy parameters.Πss is the (quarterly)

gross inflation target and Rss and hss are the steady state values of the interest rate

and labor, respectively.

2.3 Solution Methodology

This section presents the extended perturbation method used to approximate the

solution of the model in Section 2.2. For further details, the reader is referred to

Andreasen and Kronborg (2016). Let the ny ×1 vector yt denote the control variables

and let the state variables be denoted by the nx ×1 vector xt =[

x′1,t x′2,t

]′, where

x1,t and x2,t are the endogenous and exogenous state variables with dimensions

nx1 ×1 and nx2 ×1, respectively. The vectors yt and xt belong to the sets χy ⊂Rny and

χx ⊂Rnx , respectively.

Consider a broad class of DSGE models, which can be represented as

Et[f(yt ,yt+1,xt ,xt+1

)]= 0, (2.17)

where f :χy ×χy ×χx ×χx 7→Rn and n = ny +nx . It is assumed that this mapping is

at least m times differentiable, where m will denote the order of approximation. Now,

let εt be an nε×1 vector of i.i.d. shocks to the exogenous state variables. Provided

that the DSGE model in (2.17) has a unique solution, this can be expressed as (see

Schmitt-Grohe and Uribe (2004))5

yt = g(xt ,σ), (2.18)

xt+1 = h(xt ,σ)+σηεt+1, (2.19)

η≡[

0nx1×nε

η

].

The perturbation parameter σ≥ 0 scales the square root of the covariance matrix, η,

which has dimensions nx ×nε. This enables us to capture the effects of uncertainty

in the policy functions g and h. For most DSGE models, however, we do not know the

true policy functions.

Now, let (2.18) and (2.19) be decomposed into

g(xt ,σ) = gPF (xt )+gstoch(xt ,σ), (2.20)

h(xt ,σ) = hPF (xt )+hstoch(xt ,σ), (2.21)

where gstoch(xt ,σ) and hstoch(xt ,σ) contain the effects of uncertainty when the per-

fect foresight component is removed from the policy functions. Thus, gstoch(xt ,σ=5In the following estimation, the parameter space will be restricted to the determinacy region.

2.3. SOLUTION METHODOLOGY 65

0) = 0 and hstoch(xt ,σ = 0) = 0. Note trivially that all derivatives of g and gPF with

respect to the state variables xt are identical atσ= 0. Further, all derivatives involving

σ are identical for g and gstoch (similarly for h).

In standard perturbation, a Taylor series expansion around the non-stochastic

steady state is applied to both components in (2.20) and (2.21). However, this may

not be necessary as gPF and hPF can be approximated to arbitrary precision, using

the Extended Path (see Fair and Taylor (1983)). In Andreasen and Kronborg (2016)

we therefore suggest that the perfect foresight components are approximated in this

fashion whereas the stochastic parts of the policy functions gstoch and hstoch , remain

approximated by perturbation. The order of approximation for the extended pertur-

bation is defined by the order of the Taylor series expansion used to approximate the

stochastic components.

Note that the extended perturbation can be thought of as adding the higher-order

terms,∑∞

i=m+1g(xss ,0)xi

i !

(xt −xss

)⊗i and∑∞

i=m+1h(xss ,0)xi

i !

(xt −xss

)⊗i , to an m’th-order

perturbation approximation. To the extent that the approximation errors of gPF

and hPF from perturbation are large this is likely to be a significant improvement

for several reasons. First, given that the approximated solution is more accurate,

the estimates will be closer to the "true" values as well, provided that the model is

correct, as discussed in Fernandez-Villaverde and Rubio-Ramirez (2005) and An and

Schorfheide (2007). Thus, the choice of solution method will feed through to the

estimation of the parameters. Second, the improved accuracy is likely to mitigate the

tendency of higher-order perturbation to generate exploding sample paths.6 Third,

the interpretation of the parameter estimates is closer to that of the underlying model,

e.g. by preserving convexity of the adjustment costs.

In the model equilibrium both the function and its first-order derivative enter the

equilibrium dynamics, implying that a second-order perturbation approximation

depends on a third-order Taylor expansion hereof. Figure 2.1 shows the function in

(2.3) when the degree of asymmetry is low (left graph) as well as when it is high (right

graph). Both the actual function and the third-order Taylor expansion are shown. As

seen from the figure, the approximated cost function is no longer convex for positive

wage inflation. Further, as wages increase beyond a certain point the adjustment

costs become negative under the standard perturbation approximation. This point is

found to lie well within the ergodic distribution for the estimates by a second-order

perturbation approximation, which questions whether this is sufficiently accurate. Fi-

nally, it is clear that the approximation struggles more when the model nonlinearities

are strong, i.e. the above problems are exaggerated in the right graph of Figure 2.1. As

the extended perturbation method does not rely on a Taylor expansion of finite order

6A tractable alternative that preserves stability is pruning (see Kim et al. (2008) and Andreasen et al.(2013)). This approximation however may still struggle to preserve monotonicity and may not be suffi-ciently accurate.

66



for the perfect foresight components in (2.20) and (2.21) these approximation errors

should be significantly smaller than under standard perturbation.

In this application, the model will be estimated using the extended perturbation

method for both second and third order. Using a second-order approximation im-

plies there will be a constant correction for risk, given by the terms gσσ and hσσ,

respectively. As shown in Andreasen and Kronborg (2016) this ensures that the ex-

tended perturbation is stable for any state xt , provided the same holds for the model

under perfect foresight. Under a third-order approximation, the risk correction is

a linear function of the state variables. In addition to the risk correction under the

second-order approximation this implies that the terms gσσσ, gσσx and hσσσ, hσσx

are added to the respective approximated policy functions. Although stability can

no longer be guaranteed, these terms are generally quite small and do not seem to

induce instability in the resulting approximation.

A simple description of the extended perturbation algorithm used in the simula-

tions is the following: For a given set of parameters, θ, and a given order m: First, run

the standard perturbation method to compute all relevant derivatives of gstoch(xt ,σ)

and hstoch(xt ,σ). Second, for each period, t , use the Extended Path to compute

gPF (xt ) and hPF (xt ) and approximate the policy functions by adding the stochastic

terms. This approach can be relatively demanding in terms of computational costs,

especially if the algorithm is called repeatedly, e.g. through long simulated sample

paths since the Extended Path requires solving a large fixed-point problem for every

period. To improve computational efficiency, a state-dependent truncation as well

as a combination with perturbation, dependent on some tolerance parameters is

applied.7 Intuitively, this means that standard perturbation is used when xt is suffi-

ciently close to the steady state, where the Taylor expansion of the perfect foresight

component is accurate.

2.4 Econometric Methodology

Since a nonlinear approximation is used to solve the model in Section 2.2, the like-

lihood function can not be obtained from the Kalman filter. Instead, Ruge-Murcia

(2012) shows that the Simulated Method of Moments (SMM), first described in Duffie

and Singleton (1993), is a feasible way of estimating the structural parameters in

nonlinear DSGE models. This estimation method is well suited for the purpose of this

7The model is solved by standard perturbation if, for xt , the unit free Euler equation errors are lessthan a specified tolerance level, EE . The truncation of the perfect foresight problem is determined by aspecified radius of convergence, Dss , based on a third-order approximation of the solution. Specifically,EE = 0.001 and Dss = 0.005 are set. The boundaries of the truncation length is set as N ∈

20,200

. For theestimated parameter set, this implies that the fraction of periods in which the Extended Path is applied isapproximately 8 percent.

2.4. ECONOMETRIC METHODOLOGY 67

paper since closed-form expressions of the model moments are not obtained from

the extended perturbation method. Instead, the model moments are approximated

by simulating sample paths, Y1:τT , minimizing the weighted distance of these to the

moments in the data, Y1:T .

Let θ ∈Θbe the q×1 vector of variables to be estimated, whereΘ⊂Rq is a compact

set. Let gt be a p ×1 vector of data transformations at time t , where p ≥ q , for which

we are interested in the unconditional expectation (the moment conditions), and let

gt (θ) be the corresponding series generated by using the DSGE model in simulation.

The size of this simulated sample path is given as τT , where τ≥ 0 is an integer. Finally,

let WT be some positive-definite weighting matrix with dimension p ×p. The SMM

estimate, θ, is found by minimizing the weighted distance between the data moments

and those of the model. Formally, the estimator is given by

θ = ar g . mi n.θ∈Θ

QT (θ) = GT (θ,τ)′WT GT (θ,τ), (2.22)

where

GT (θ,τ) = 1

T

T∑t=1

gt − 1

τT

τT∑t=1

gt (θ), (2.23)

denotes the difference in sample moments. Local identification of θ requires that

the matrix J = E[∂Gt (θ)∂θ

]has rank q . While it is hard to prove global identification, the

parameters can be confirmed to be locally identified around the estimated values.

Further, when simulating a series of artificial data from the model, this rank condition

is found to be satisfied for all samples.

The optimal weighting matrix is found from the two-step procedure: In the first

step, the parameter estimates are found using the inverse standard errors of the

moments. In the second step, based on these estimates, the weighting matrix is set as

WT = S−1, where S is the sample estimate of

S0 =∞∑

j=−∞E

([g t −E

(g t (θ)

)][g t+ j −E

(g t+ j (θ)

)]′). (2.24)

The matrix S is found non-parametrically, using the Newey-West estimator (see

Appendix C). This weighting matrix implies that the moments with the smallest

variance in the simulated sample are given a relatively higher weight in the objective

function in (2.23) and hence, choosing the optimal WT improves the efficiency of the

SMM estimator as is also a well-known result from the GMM literature. Further, as

shown in Ruge-Murcia (2012), these efficiency gains are likely to be increasing in the

degree of nonlinearity in the DSGE model.

68



Under the regularity conditions given in Duffie and Singleton (1993), the asymp-

totic distribution of the SMM estimator in (2.22) is

pT

(θ−θ0

)→N

(0,(1+1/τ)(J′S−1J)−1

).

In the application τ= 10 is set, which is similar to or slightly larger than what is used

in the literature. This choice obviously reflects a balance between computational

and statistical efficiency. Since the extended perturbation can be computationally

demanding, and since a new sample path is generated for every function evaluation

in the numerical optimization, one would like to keep τ as small as possible. Further,

for both the pruning and extended perturbation approximation, the estimates are

found to be fairly stable when increasing τ beyond 5.

2.5 Data and Moments

The model is estimated using quarterly seasonally adjusted U.S. data series from

1964Q2 to 2015Q1 giving a total of 204 observations. The variables used are real con-

sumption per capita (Personal Consumption Expenditures divided by the quarterly

average of monthly Civilian Noninstitutional Population), hours worked (Aggregate

Weekly Hours Index: Total Private Industries), quarterly CPI inflation (Consumer

Price Index for All Urban Consumers: All items), quarterly wage inflation (Average

Hourly Earnings: Total Private Industries), and the nominal interest rate (Effective

Federal Funds Rate). All series can be downloaded from the FRED database at the

Federal Reserve Bank of St. Louis web page. Prior to estimation, the log was taken to

all variables and they have been detrended by a linear deterministic trend to conform

to the stationarity of the DSGE model.

Similar to Kim and Ruge-Murcia (2009), the moments used in the estimation are

the 15 variances and covariances as well as the five first-order autocovariances of the

data series.

2.6 Model Properties

This section presents the parameter estimates and examines the resulting model

properties. I follow the literature and calibrate a number of parameters prior to the

estimation, using fairly standard values. An overview of the calibrated parameters is

given in Table 2.1.

This leaves 11 structural parameters for the estimation. Table 2.2 reports the es-

timated values, the standard errors, and the value of the objective function for both

second and third-order pruning and extended perturbation.

2.6. MODEL PROPERTIES 69

Consider first the extended perturbation estimates. The sets of parameter esti-

mates are similar for most parameters which indicates that the third-order terms

for the risk corrections are small. Consequently, the quantitative properties of the

two approximations will be quite similar as well and thus, for ease of exposition,

only the third-order extended perturbation approximation will be considered in the

remainder of the paper. For the price and wage rigidity, both γ and φ are found to be

positively significant at the 5 percent level with estimates of 44.13 and 20.94, respec-

tively. Thus, the data clearly reject the hypothesis that the economy is characterized

by fully flexible goods and factor prices. Further, in the case of wages, the rigidity is

asymmetric since ψ is estimated to be 51.04. Again, the hypothesis that ψ = 0 can

be rejected at the 5 percent level which means that the data support the presence of

DNWR in the U.S. economy.

It is instructive to compare these estimates to those obtained by a pruned per-

turbation approximation. Table 2.2 shows that both second and third-order pruning

estimates imply a higher utility curvature of consumption since ρ is found to be

more than twice as large. Further, with perturbation the price rigidity is found to

be less pronounced especially for the second-order approximation. The parameters

relating to the Taylor rule and exogenous shocks are remarkably similar for stan-

dard perturbation and extended perturbation. However, there are large differences

found in the estimates for the parameters in the wage adjustment cost function in

(2.3). First, the φ has an estimated value of 1,314.03 and 4,330.10 for second and

third-order perturbation, respectively, while it is only 20.94 for extended perturbation.

Second, the ψ estimates are 6,654.26 and 6,395.19 for perturbation but only 51.04 for

extended perturbation. While the estimated value of ψ using perturbation is compa-

rable to what is found or used in previous studies (for example Kim and Ruge-Murcia

(2009), Fahr and Smets (2010), Kim and Ruge-Murcia (2011), and Abbritti and Fahr

(2013)), where this parameter ranges from 3,844 to 26,000, the estimate is orders of

magnitude smaller when using extended perturbation. Hence, while the results in

this paper qualitatively confirm the findings in previous studies, it can be seen from

Table 2.2 that the model is able to match the data moments with a smaller degree of

asymmetry in the wage rigidity when using this solution method.

2.6.1 Impulse Response Functions

It has become common in the DSGE literature to study the endogenous propagation

of exogenous shocks by considering impulse response functions. However, when the

model is solved using the extended perturbation method, closed-form expressions

for the IRFs are not available. Below, let the vector ut consist of the control and state

variables of interest. Now let the i ’th exogenous shock be hit by a disturbance of

size vi in period t +1. The generalized impulse response function (GIRF) for ut+l

70



proposed by Koop et al. (1996) is then defined as

G I RFu(l , vi ) = E[

ut+l

∣∣∣∣εi ,t+1 = vi ,xt

]−E

[ut+l

∣∣∣∣xt

], (2.25)

for l = 1, ...,L. Note that the GIRF is only conditioned on the i ’th shock and only for l =1. The impulse responses generated from (2.25) are not scaleable (i.e. G I RFu(l ,c vi ) 6=c G I RFu(l , vi )) nor symmetric (i.e. G I RFu(l , vi ) 6= −G I RFu(l ,−vi )), and they are de-

pendent on the current state of the economy. The expectations in (2.25) are found by

simulation, i.e. by drawing ε j ,t+1 for j 6= i and ε j ,t+l for all j shocks when l = 2, ...,L.

5,000 simulations are used for both shock types. The state vector xt from which the

GIRF is computed is set to its unconditional expectation.

Figure 2.2 shows the generalized impulse responses of consumption, hours

worked, inflation, nominal wage inflation, real wages, and the nominal interest rate to

productivity shocks of difference sizes. Both positive and negative shocks (solid and

dashed lines, respectively) of 2 and 3 standard errors (smaller and larger linewidth,

respectively) are shown. An increase in the productivity level increases output and

thus should be considered as an expansionary shock. Following (2.25), the figure

depicts the expected percentage deviations of the variables from a path without

conditioning on the shocks as a function of the horizon l .

The impulse responses show asymmetries for most variables considered. The

obvious case is nominal wage inflation where initial response is for nominal wages to

increase for both expansionary and contractionary shocks. However, as the marginal

product of labor decreases following an adverse productivity shock this implies that

real wages must fall in accordance with (2.12). This is instead obtained through higher

inflation which responds asymmetrically since nominal wage adjust more flexibly

for expansionary shocks. Thus, when DNWR is present we note how an increase

in inflation can serve as a mean of restoring equilibrium after adverse productivity

shocks. The asymmetric inflation response in Fig. 2.2 is an example of how DNWR

effectively acts as an additional cost push shock when productivity decreases. This

in turn affects the response of the nominal interest rate: Since the central bank is

targeting inflation, the interest rate decreases following an expansionary shock and

vice versa. However, the monetary policy response is asymmetric, reflecting the in-

flation impulse response. Hence, the initial response of the nominal interest rate

following contractionary productivity shocks is larger than expansionary ones. In

accordance with the Euler equation in (2.5), aggregate consumption responds to the

change in interest rate and wealth effect by deviating persistently from the uncondi-

tional expectation. The changes in consumption for contractionary shocks are larger

than for their positive counterpart, partly reflecting the asymmetric monetary policy.

Thus, the accumulated difference over the depicted horizon is approximately 0.8 and

2.2 percent relative to the unconditional expectation for shocks of 2 and 3 standard

2.7. OPTIMAL MONETARY POLICY 71

errors, respectively.

Figure 2.3 shows the impulse responses to preference shocks. A positive preference

shock increases marginal utility of consumption today relative to future periods.

Hence, a positive shock should be considered as an expansionary demand shock and

it temporarily increases aggregate consumption, in accordance to the Euler equation

in (2.5). Following an expansionary shock, final goods equilibrium implies that hours

worked increase to satisfy the increased demand. Increased employment lowers the

marginal product of labor and thus increases the real marginal cost of production

in (2.12). As a result, inflation increases for expansionary shocks and decrease for

contractionary. Again, nominal wages are clearly more restricted downwardly than

upwardly. The larger change in nominal wages following an expansionary shock

means that, in equilibrium, inflation increases more than it decreases following a

contractionary shock. The asymmetric inflation response implies an asymmetric

interest rate response by the inflation targeting central bank which increases the

interest rate more following expansionary preference shocks than it decreases it

following contractionary ones. As opposed to the productivity shocks this coun-

tercyclical response actually reduces the real economic effects of DNWR. Overall,

DNWR implies that consumption falls more following contractionary shocks than it

increases in the opposite case. The accumulated difference over the depicted horizon

is approximately 1.5 and 4.1 percent for shocks of 2 and 3 standard errors, respectively.

The impulse responses in Figure 2.2 and 2.3 show how nominal rigidities cause

real economic effects as consumers and firms adapt to exogenous shocks. In the

presence of DNWR, changes in nominal wages are particularly restricted when con-

tractionary shocks hit the economy. When the nominal wage adjustment is more

sluggish, optimal behavior implies that other variables must compensate to obtain

output and labor market equilibrium. Intuitively, this results in a flattening of the

wage Phillips curve as employment decreases. As a results, the model with DNWR

implies asymmetric business cycles in line with the stylized facts in macroeconomic

time series. Finally, the graphs show that the asymmetry is more pronounced for

larger shocks. Hence, the economic effects of DNWR might be small in tranquil peri-

ods but on the other hand have important implications in states far from the steady

state such as in severe recessions.

2.7 Optimal Monetary Policy

This section examines the optimal inflation target when the economy is characterized

by DNWR. As in Kim and Ruge-Murcia (2009), this is done by allowing the central

bank to conduct its monetary policy in (2.16) as a strict inflation target. Thus, the

72



nominal interest rate is set to stabilize inflation only, i.e. irrespective of the state of

the economy and with no preference for smoothing or stabilizing employment. The

optimal policy is then defined as the inflation target that maximizes the expected

discounted utility of the representative household in (2.1).

When choosing its inflation target the central bank faces a trade-off: On one

hand, inflation generates systematic costs since both output prices and wages are

rigid. Hence, with little or no macroeconomic volatility it would be suboptimal to

operate with an inflation target much above price stability. On the other hand, higher

steady state inflation reduces the probability that the economy ends up in a state

where adverse shocks create a large downward pressure on nominal wages. As dis-

cussed earlier, inflation will serve as a mean of adjustment to a new equilibrium

when nominal wages can not adjust. This will also increase the changes in the real

variables such as consumption and employment, something which is disliked by

the households. Hence, it might be prudent for the central bank to operate with a

positive inflation target and to incur small systemic costs so as to reduce overall

macroeconomic volatility.

Figure 2.4 depicts the change in unconditional welfare of different inflation targets

relative to price stability (solid black line). The expectation is found by simulation,

using 5,000 sample paths of 500 observations with a burn-in of 100 observations.8

The welfare is measured as percentage changes in consumption equivalents, i.e. the

change in (2.1) scaled with the inverse of marginal utility of consumption. Given the

estimates in Table 2.2, the optimal inflation target is found to be approximately 0.25

percent per year. The results show that, under uncertainty and with an economy

characterized by DNWR, it is optimal to reduce the probability of a high-cost event

by incurring small but systematic costs. As the inflation target is increased, further

lowering the probability of downward pressure on nominal wages is associated with

diminishing welfare gains. Furthermore, higher steady state inflation is increasingly

associated with systematic costs due to price rigidity and this latter effect will eventu-

ally dominate. The costs are found to outweigh the benefits once the inflation target

is raised beyond 0.50 percent per year in the benchmark case. Hence, overall the

results in this paper lend support to the notion of a small but positive inflation target

in lieu of price stability but is not able to fully justify the typically observed inflation

targets of most central banks, solely based on DNWR. The optimal inflation target

is lower than the 3 percent per year suggested by Akerlof et al. (1996) who assume

that nominal wages can never fall as well as the estimates of 0.75 to 1 percent per

year found in Kim and Ruge-Murcia (2009) and Kim and Ruge-Murcia (2011) where

DNWR is more pronounced than found in this paper. This underlines the importance

of an accurate assessment of the degree of DNWR for monetary policy. To the extent

8The same sets of innovations,εd

t ,εat

500

t=1, are used to compare across different inflation targets.

2.7. OPTIMAL MONETARY POLICY 73

that this paper underestimates the potential asymmetry in the wage rigidity so will it

underestimate the optimal inflation target.

To examine the robustness of the results, the welfare gains are computed for

different levels of volatility (by scaling σd and σa , see the left column of Figure 2.4)

and price rigidity (by scaling γ, see the right column of Figure 2.4). As seen from the

graph, increasing volatility generally shifts the curve up, implying that the optimal

inflation target increases. The reason is fairly intuitive: When volatility increases,

the benefits of a precautionary buffer against nominal wage deflation also increases

since, for any level of inflation, it is now more likely that the economy will be in a

state where DNWR is binding. Thus, in the case where the standard deviations of

both shocks are increased by 50 percent, the optimal inflation target increases to

approximately 0.40 percent per year. Lowering the volatility by 50 percent implies

that the optimal target falls to approximately 0.1 percent per year. Increasing the price

rigidity generally shifts the curve down. As γ increases this means that the systematic

costs of higher steady state inflation increase, counteracting the benefits of the buffer.

As a result, when price rigidity is increased by 50 percent this lowers the optimal

target to 0.15 percent per year whereas it increases to 0.35 percent per year when

price rigidity is 50 percent higher. Further, a higher value of γ has a significant effect

on the rate at which the costs will eventually outweigh the benefits of inflation.

How general are the results? Obviously, the results will depend on the model

specification. For example, Fagan and Messina (2009) show that if heterogeneity

across workers is assumed, this raises the optimal inflation level to avoid resource

misallocation in the presence of DNWR when real wages vary cross-sectionally. Simi-

larly, heterogeneity across firms or sectors may also increase the optimal inflation

target. On the other hand, the cost of inflation would be raised further if households

needed to hold money for economic transactions (this could be obtained e.g. through

a money-in-utility specification). It might also be beneficial to consider the effects

shocks to the disutility of labor, something which is not considered in this paper. Fur-

ther, the central bank can obtain a smoother and more symmetric adjustment of real

variables to supply shocks by targeting inflation less strictly than what is assumed in

the above. This is likely to lower the optimal inflation target below the estimate found

in this paper. Other factors, such as the zero lower bound restriction on monetary

policy, will tend to increase the optimal inflation rate, as discussed in Blanchard et al.

(2010). Previous micro studies (see for example Dickens et al. (2007)) suggest that

the degree of DNWR may vary a lot from country to country. In that case, so will the

real economic costs of deflationary wage pressure and the inflation buffer needed.

Finally, the model (like most New Keynesian DSGE models) is exposed to the Lucas

(1976) critique. While the nominal frictions might be a fairly good approximation of

the aggregate economy, the implicit assumption that the parameters are completely

policy invariant is, of course, questionable. Specifically, the in-sample inflation level

74



used to estimate the model is significantly higher than what is found as the optimal

level. If the mechanism by which monetary policy is transmitted to the economy

changes as a result this might result in misleading conclusions.

2.8 Conclusion

This paper examines the extent of downward nominal wage rigidity in the U.S. econ-

omy and its implications for monetary policy. This is done by specifying a simple

dynamic stochastic general equilibrium model in which wage rigidity is allowed to

be asymmetric. The model estimates show that nominal rigidities are important for

both prices and wages. For wages, it is found that nominal rigidities are asymmetric,

i.e. that DNWR is present in the U.S. economy.

The model equilibrium is approximated using the extended perturbation method

in Andreasen and Kronborg (2016) instead of a standard perturbation-based approxi-

mation. This solution method generally improves the accuracy of the approximation

and is more likely to preserve characteristics of the underlying model such as con-

vexity and monotonicity. The change in the solution methodology is found to have a

substantial impact on the parameter estimates associated with the wage rigidity. In

particular, the parameter governing the asymmetry in wage adjustments is estimated

to be orders of magnitude smaller than in previous studies. While the estimated

model is characterized by asymmetric wage rigidity and thus confirms the qualitative

findings in the literature, the asymmetric propagation of shocks is generally less

pronounced than what has been found previously.

Based on the estimated model, the optimal inflation target is computed when

implemented as a strict inflation target in a Taylor rule. This choice is governed by a

trade-off between systematic inflation costs and the benefits of reducing the probabil-

ity of ending in a state with deflationary pressure on nominal wages. I find the optimal

net inflation target to be approximately 0.25 percent per year. Hence, the findings in

this paper lend support to the notion of a small but positive inflation target albeit

less than what is implemented by most central banks. Increasing macroeconomic

volatility increases the optimal inflation target whereas higher price rigidity lowers it.

Acknowledgments

The author gratefully acknowledges support from Aarhus University, Department

of Economics and Business Economics and from CREATES - Center for Research in

Econometric Analysis of Time Series (DNRF78), funded by the Danish National Re-

search Foundation. Comments from participants at the Danish Graduate Programme

in Economics Workshop and CREATES seminars are also gratefully acknowledged.

2.8. CONCLUSION 75

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2.8. CONCLUSION 77

Appendix A: Model Overview

1. λt = dt c−ρt ,

2. 1 =βRtEt

[λt+1λt

],

3. 0 =λt ht

[(ν−1)

(1−Φt

)+ ∂Φt∂ωt

wtwt−1

Πt

]−dtχν

htwt

−βEt

[∂Φt+1∂ωt+1

λt+1Πt+1

(wt+1

wtΠt+1

)ht+1

],

4. mct (1−θ)at h−θt = wt ,

5. 0 = (η−1)(1−Γt

)+ ∂Γt∂PtΠt −ηmct −βEt

[λt+1λt

ct+1ct

∂Γt+1∂Pt+1

Πt+1

],

6. yt(1−Γt

)= ct +wtΦt ht ,

7. yt = at h1−θt ,

8. l og(

RtRss

)= ρR log

(Rt−1Rss

)+ (1−ρR )

[κπlog

(ΠtΠss

)+κh log

(hthss

)],

9. l og dt+1 = ρd log dt +σdεdt+1,

10. l og at+1 = ρa l og at +σaεat+1.

The model has 10 equations + 2 link equations (for Rt−1 and wt−1). The 4 state

variables are Rt−1, wt−1, dt , and at . The 8 control variables are λt , ct , Rt ,Πt , ht , mct ,

yt , and wt (where wt ≡ WtPt

).

Appendix B: Model Steady State

LetΠ be the steady state inflation and use the normalization for the shocks: d = a = 1.

R = Πβ

,

mc = 1

η

[(η−1)(1−Γ)+ (1−β)π

∂Γ

∂P

],

h =[

χν

(ν−1)(1−Φ)+ (1−β)Π ∂Φ∂P

1[1−Γ−mc(1−θ)Φ

]−ρmc(1−θ)

] 1−ρ(1−θ)−θ

,

w = mc(1−θ)h−θ,

c = h1−θ[1−Γ−mc(1−θ)Φ],

y = h1−θ,

λ= c−ρ .

78



Appendix C: Sample Estimate of (2.24)

Let θ1 be the step 1 estimates:

θ1 = ar g . mi n.θ∈Θ

QT (θ) = GT (θ,τ)′Ip GT (θ,τ).

Denote the sample mean of the outer product of the moment distance as

Γ j = 1

T −1

T− j∑t=1

[gt − 1

τT

τT∑s=1

gs (θ1)

][gt+ j − 1

τT

τT∑s=1

gs (θ1)

]′, j = 0, ...,T −1.

Note that Γ j = Γ− j . The non-parametric estimate of the covariance matrix is then

S =T−1∑

j=−T+1κ

(j

CT

)Γ j , κ (x) =

1−|x|, |x| ≤ 1

0 other wi se,

where CT is a bandwidth parameter.

2.8. CONCLUSION 79

Table 2.1. Calibrated parameters

Parameter Description Valueα (1−α) is the wage share of income 0.3333β Discount factor 0.9900η Elasticity of substitution, intermediate goods 11.0000ν Elasticity of substitution, labor input 4.3333χ Disutility of labor 1.5000Πss Average gross quarterly inflation 1.0112

Nominal wage growth0.96 0.98 1 1.02 1.04

Adju

stm

ent c

osts

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5φ = 100, ψ = 100

Quadratic costsDNWRDNWR, 3rd-order approximation

Nominal wage growth0.96 0.98 1 1.02 1.04

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5φ = 100, ψ = 1,000

Figure 2.1. Wage adjustment costsThe figure depicts the adjustment cost function in (2.3). Both plots show the function for

quadratic costs (ψ= 0) and with DNWR (ψ> 0). In the latter, case both the true function and

the third-order Taylor expansion are shown. The left and right plots show a case of relatively

low and high asymmetry, respectively.

80



Table 2.2. Parameter estimates

The estimates are shown for both pruned perturbation (first two columns) and extended

perturbation (last two columns). The figures in parenthesis denote the standard errors of the

parameter estimates.

Pruning Extended perturbation2nd order 3r d order 2nd order 3r d order

ρ 2.883 2.467 1.161 1.159(0.418) (0.541) (0.112) (0.084)

φ 1,314.03 4,330.10 20.91 20.94(431.24) (454.79) (5.40) (4.53)

ψ 6,654.26 6,395.19 51.15 51.04(249.08) (1,440.75) (8.82) (16.28)

γ 23.292 38.289 44.238 44.134(6.835) (11.301) (11.900) (10.892)

ρR 0.809 0.861 0.824 0.822(0.031) (0.024) (0.043) (0.039)

κπ 1.584 1.968 1.563 1.563(0.203) (0.214) (0.346) (0.405)

κh 0.073 0.049 0.125 0.124(0.014) (0.010) (0.107) (0.062)

ρa 0.962 0.957 0.957 0.955(0.007) (0.007) (0.007) (0.008)

σa 0.0107 0.0128 0.0120 0.0121(0.0012) (0.0013) (0.0013) (0.0013)

ρd 0.8725 0.8960 0.8186 0.7837(0.0264) (0.0311) (0.0510) (0.0445)

σd 0.0428 0.0412 0.0265 0.0258(0.0057) (0.0056) (0.0024) (0.0016)

2.8. CONCLUSION 81

0 5 10 15 20−4

−3

−2

−1

0

1

2

3

4Consumption

+2 σ−2 σ+3 σ−3 σ

0 5 10 15 20−1

−0.5

0

0.5

1

1.5Hours worked

0 5 10 15 20−1

−0.5

0

0.5

1Inflation

0 5 10 15 20−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3Nominal wage inflation

0 5 10 15 20−4

−3

−2

−1

0

1

2

3

4Real wages

0 5 10 15 20−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4Nominal interest rate

Figure 2.2. Generalized impulse responses: Productivity shockThe GIRFs are shown in percentage deviations from the expected value. The expectations are

approximated numerically from 5,000 simulated paths. Extended perturbation of third order

is shown.

82



0 5 10 15 20−4

−3

−2

−1

0

1

2

3

4Consumption

+2 σ−2 σ+3 σ−3 σ

0 5 10 15 20−6

−4

−2

0

2

4

6

8Hours worked

0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

1.5Inflation

0 5 10 15 20−3

−2

−1

0

1

2

3

4Nominal wage inflation

0 5 10 15 20−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5Real wages

0 5 10 15 20−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Nominal interest rate

Figure 2.3. Generalized impulse responses: Preference shockThe GIRFs are shown in percentage deviations from the expected value. The expectations are

approximated numerically from 5,000 simulated paths. Extended perturbation of third order

is shown.

2.8. CONCLUSION 83

0 0.1 0.2 0.3 0.4 0.5 0.6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Inflation target

Con

sum

ptio

n eq

uiva

lent

s (in

%)

0.5σ1.0σ1.5σ

0 0.1 0.2 0.3 0.4 0.5 0.6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

Inflation target

0.5γ1.0γ1.5γ

Figure 2.4. Optimal inflation targetThe figure depicts the welfare change for various yearly (net) inflation targets compared to

the case of price stability. Welfare is measured as the percentage change in consumption

equivalents. Each line shows the welfare changes for different levels of volatility (left column)

or price rigidity (right column). Extended perturbation of third order is shown.

C H A P T E R 3FORECASTING USING A DSGE MODEL WITH A

FIXED EXCHANGE RATE

Anders Kronborg


Abstract

Dynamic stochastic general equilibrium (DSGE) models are increasingly being used

when conducting forecasting of macroeconomic time series. This paper examines the

forecasting accuracy of a small open economy DSGE model in which the exchange

rate is fixed. As shown, the fixed exchange rate has implications for the relative

importance of the structural shocks in the model. Using Danish data the model is

estimated recursively to assess the out-of-sample forecast accuracy of several time

series from one to eight quarters ahead. The DSGE model is generally comparable to

an AR(1) model in terms of root mean square error while it outperforms the random

walk. Consistent with previous literature, the DSGE model largely underestimates the

severity of the Great Recession. However, the model correctly predicts a continued

fall in GDP growth and a subsequent slow recovery when forecasting from 2009Q1.

85

86 CHAPTER 3. FORECASTING USING A DSGE MODEL WITH A FIXED EXCHANGE RATE

3.1 Introduction

Macroeconomic forecasters can choose between a wide range of models to generate

predictive distributions. Increasingly, dynamic stochastic general equilibrium (DSGE)

models are applied by economic institutions and policy makers when conducting

such quantitative forecasts at business cycle frequencies. The popularity of DSGE

models can at least partially be credited to research that has shown that New Keyne-

sian models with nominal and real frictions are comparable to VAR models in terms

of marginal likelihood and forecast ability, for example as shown in the well-known

paper by Smets and Wouters (2007).

Using Danish data, this paper investigates the forecasting performance of an open

economy DSGE model in which the exchange rate is fixed, as to reflect the currency

peg of the Danish Krone against the Euro. The model builds on the open economy

model by Adolfsen et al. (2007b) in which the exchange rate is flexible. However, it

is not clear a priori that the fixed exchange rate DSGE model will produce similar

results as its flexible exchange rate counterpart in terms of forecasting performance.

As discussed in Pedersen and Ravn (2013), the choice of exchange rate regime is likely

to have significant effects on the DSGE model, since both the relative importance of

shocks and their transmission are likely to differ. First, as monetary policy in a fixed

exchange rate regime is devoted to maintain the currency peg it must abandon other

objectives such as targeting domestic output and inflation. Second, shocks to the

foreign economy to which the small open economy has fixed its currency transmit

more forcefully through domestic variables since the nominal exchange rate can

not serve as a buffer, e.g. by allowing a nominal depreciation improve the domestic

competitiveness following an adverse foreign demand shock.

The structural parameters of the model are estimated using a Bayesian econo-

metric approach on a dataset from 1990Q2-2010Q4. Following an initial estimation

sample, the model is estimated and evaluated recursively to assess its predictive

performance for out-of-sample forecasting horizons between 1 and 8 quarters. The

evaluation is based on actual forecasts and root mean square errors (RMSE) of point

forecasts, using the series for GDP, consumption, investment, real wages, imports,

exports, the output deflator and the consumer price index. The forecasting accuracy

of the DSGE model is compared to an AR(1) model and the random walk.

The main findings of this paper are as follows: First, the estimated DSGE model

is able to produce unconditional second moments that are generally in line with

the data. Second, the accuracy of point forecasts generated by the DSGE model are

generally comparable to those of the AR(1) model and better than the random walk

when measured by RMSEs. Third, special attention is given to the DSGE model’s

prediction prior to and during the Great Recession which for Denmark I define as

starting in 2008Q3. In line with the findings in Del Negro and Schorfheide (2013), the

3.1. INTRODUCTION 87

DSGE model underestimates the severity of the downturn at the outset of the crisis.

However, when increasing the information set to include 2009Q1 the DSGE model

correctly predicts a further decline in GDP growth as well as a relatively slow recovery.

By performing a historical decomposition of the filtered output gap in the model the

causes of the crisis are analyzed. The main drivers are found to be domestic demand

and foreign variables at the outset of the crisis while domestic demand and markup

shocks explain the continued suppressed output.

Why choose a DSGE model for forecasting? Because DSGE models deliver a set of

dynamic equations that are based on equilibrium conditions and the optimizing be-

havior of forward-looking agents they deliver forecasts that have a strong theoretical

coherence. This model class allows the researcher to give a structural interpretation

of the state of the economy as well as attribute business cycle fluctuations to under-

lying structural shocks. However, as discussed in Pagan (2003), there might exist a

trade-off between theoretical and empirical coherence. For DSGE models, parameter

restrictions on the resulting state space representation of the model might lead to a

poor empirical fit if they are not a good description of the data. To the practitioner,

examining the ability of a fixed exchange rate DSGE model to predict future paths of

macroeconomic time series is in itself an interesting topic of research. Further, if the

model is to be trusted to deliver quantitative credible answers to more theoretically

based questions like effects of policy initiatives, counterfactuals, etc., it must first be

able to adequately explain and predict the comovements in the data.

This work contributes to an expanding body of literature on DSGE models and

their empirical performances. Much of the recent work is based on the New Keynesian

models with frictions such as habit formation, investment adjustment costs, price

and wage rigidities as well as various exogenous shocks proposed by Christiano et al.

(2005) and Smets and Wouters (2003, 2007). The forecasting performance of these

closed economy DSGE models have been examined extensively (see for example Del

Negro and Schorfheide (2013) and Amisano and Geweke (2013)). By extending the

model framework to include open economy aspects, Adolfsen et al. (2007a) examine

the predictive abilities of a DSGE model with a flexible exchange rate. The general

finding in this literature is that the DSGE models can compete with reduced-form

statistical models such as AR or V AR models in terms of out-of-sample forecasting

but that it is possible to generate better forecasts through more sophisticated models

such as dynamic factor models. Pedersen and Ravn (2013) suggest a model that

incorporates some of the characteristics of the Danish economy, including the fixed

exchange rate regime. However, little research has been conducted to examine the

forecasting performance of open economy DSGE models with fixed exchange rates.

Hence, the main contribution of this paper is to take a first look at the forecasting

ability of such a model. As a result, the model presented below will be kept as simple

as possible, laying the groundwork for future extensions. This implies that interesting


aspects such as modeling financial frictions or a more detailed fiscal policy framework

is left out.

The remainder of the paper is structured as follows. Section 3.2 presents the DSGE

model. Section 3.3 briefly presents the estimation methodology while Section 3.4

outlines the data series used. Section 3.5 gives the prior distributions and calibrated

parameters. Some posterior properties of the estimated model are examined in

Section 3.6. Section 3.7 performs the forecasting exercise while Section 3.8 concludes.

3.2 The DSGE Model

This section presents the open economy DSGE model used in the forecasting exercise.

The model contains several nominal and real frictions in the New Keynesian tradition

with open economy elements similar to those found in Adolfsen et al. (2007b). Impor-

tantly, as monetary policy is assumed to be characterized by a credible fixed exchange

rate regime this implies that - up to a risk premium - the central bank will set the

domestic interest rate equal to that of the economy to which the exchange rate is fixed.

The domestic economy is assumed to be small compared to its foreign counterpart

which implies that the latter can be perceived as approximately exogenous.

To model the trends in the macroeconomic time series, a unit root is introduced

in the model through a non-stationary productivity shock. Together with steady state

inflation, this introduces a real and a nominal trend in the model which is subse-

quently stationarized. An overview of the log-linearized model is given in Appendix

A.

3.2.1 Firms

There are three types of firms in the model. Domestic goods are produced by inter-

mediate firms in a differentiated fashion and subsequently aggregated into a final

homogenous good. The domestic firm demand labor and capital for production,

which is subject to a stochastic productivity level. The importing firms buy the ho-

mogenous goods at the world market and differentiate them before selling them

to domestic households. The imported good enters both aggregate consumption

and investment. This helps the model in explaining the growth in imports which are

more volatile than domestic consumption. Similarly, the exporting firms buy the final

domestic goods and differentiate them, before selling them at the world market. All

firm types set prices subject to nominal rigidities according to a Calvo model with

indexation.


Domestic Goods Firms

At time t , the final domestic output, Yt , consists of an aggregation of the intermediate

goods, Yi ,t , indexed by i ∈ [0,1], by the following production function

Yt =[∫ 1

0Y

1

λdt

i ,t di

]λdt

,

where 1 ≤λdt <∞ is the the stochastic time-varying markup over marginal cost in the

domestic intermediate goods market, whereλd

t

λdt −1

is the elasticity of demand. Taking

the prices, Pi ,t and Pt , as given, costs minimization leads to the demand for each

intermediate good i

Yi ,t =(

Pi ,t

Pt

)− λdt

λdt −1

Yt .

Each intermediate goods producer i operates subject to the following production

function

Yi ,t = εt Kαi ,t

(zt Hi ,t

)1−α−φzt , (3.1)

where Ki ,t is the capital stock and Hi ,t is the labor input in production. A covariance-

stationary productivity shock is denoted εt , whereas zt is a permanent labor-augmenting

productivity shock. Introducing productivity growth induces a common stochastic

trend in the model, implying that the real variables will be cointegrated with zt . The

parameter φ captures the fixed cost in production and set to ensure zero profits in

the steady state. It is assumed to be proportional to the permanent productivity level

to ensure that it grows with the real variables of the economy. The capital share in

production is given by α.

The optimal capital and labor demand of firm i solves the following costs mini-

mization problem

mi nKi ,t ,Hi ,t

Rt−1Wt Ht +Rkt Kt , s.t . εt Kα

i ,t

(zt Hi ,t

)1−α−φzt ≥ Yi ,t .

Nominal wages are denoted Wt , Rt is the gross nominal interest rate on one-period

zero-coupon bonds, and Rkt is the gross nominal rental rate of capital. To allow for

working capital, wages are assumed to be payed one period ahead, which implies

that the wage bill has to be financed at the risk free rate. After imposing a symmetric

equilibrium, the optimality conditions for capital and labor demand are given as,

respectively

r kt = α

1−αwt Rt−1Ht

ktµz,t , (3.2)

mcdt =

(1

1−α)1−α (

1

α

)α (r k

t

)α (wt Rt−1

)1−α 1

εt, (3.3)


where µz,t = ztzt−1

is the growth rate of zt and real marginal cost is denoted mcdt . The

optimality conditions balance the marginal cost and value of marginal product for

factor demands, where each firm takes the factor prices as given. Here and for the

remainder of the paper, small letters are used to indicate that the variables have been

appropriately detrended to account for the nominal and real trends in the model as

to obtain a stable equilibrium. Here, r kt ≡ Rk

tPt

, wt ≡ WtPt

, mcdt ≡ MC d

tPt

, and kt+1 ≡ Kt+1zt

.

The domestic firms face Calvo-style nominal rigidities. In each period t , firm i is

allowed to set a new price, P d ,newt , with probability 1−ξd . Further, prices are allowed

to be partly indexed to previous periods’ inflation, captured by the parameter κd .

This implies that the price put forth by non-reoptimizing firms will evolve according

to P dt+1 = (πd

t )κdπ1−κd P dt , where π denotes the steady state level of inflation. The

resulting profit optimizing price-setting today is characterized by the expected future

aggregated price movements.

maxP d ,new

t

= Et

∞∑s=0

(βξd )s D t+s

[(πd

t ...πdt+s−1

)κd(π...π)1−κd Pt

d ,new

−MC di ,t+s

]Yi ,t+s −MC d

i ,t+sφzt

,

where Et is the conditional expectation at time t , D t+s is the marginal utility of

household derived from consumption between period t and t + s. The parameter β

is the household discount factor. Note that the effective stochastic discount factor

applied between period t and t + s is (βξd )s D t+s . High levels of price stickiness imply

that the forward-looking element in the optimal price behavior gets more pronounced

as more weight is attached to future cash flows and less on setting the intra period

optimal markup over marginal cost. The log-linearized first-order condition yields

the familiar New Keynesian Phillips-curve

πdt = β

1+κdβEt π

dt+1 +

κd

1+κdβπd

t−1 +(1−ξd )(1−βξd )

ξd (1+κdβ)(mcd

t + λdt ). (3.4)

Here and for the remainder of the paper, variables denoted with a hat are in log-

deviations from their steady state, xt ≡ log xt − log xss . Three factors determine the

current level of inflation for domestic goods: First, higher future inflation raises

current inflation since this implies higher future marginal costs. As a result, the

representative firm raises prices in anticipation of these costs due to the possibility

that it might not be able to due so in future periods. Second, lagged inflation carries

over to the current period through price indexation. Third, an increase in the current

marginal cost or the mark-up will increase inflation, the latter since a lower degree of

competitiveness between domestic firms allow them to increase prices.


Importing Firms

The import sector consists of two types of producers, each indexed by i ∈ [0,1]. The

importing firm buys a homogenous good from foreign producers at price P∗t (in

local currency), differentiates and imports them to the domestic market (e.g. through

branding). The homogeneous foreign goods are imported such as to transform it into

consumption, C mi ,t , and investment, I m

i ,t , respectively. The importing consumption

and investment firm faces Calvo-style nominal price rigidities. When allowed to set a

new price, firm i solves the following maximization problem

maxP m,c,new

t

= Et

∞∑s=0

(βξm,c )s D t+s

[(πm,c

t ...πm,ct+s−1

)κm,c(π...π)1−κm,c C m

i ,t+s P m,c,newt

−St+s P∗t+s

[C m

i ,t+s +φm,c zt+s]]

,

maxP m,i ,new

t

= Et

∞∑s=0

(βξm,i )s D t+s

[(πm,i

t ...πm,it+s−1

)κm,i(π...π)1−κm,i I m

i ,t+s P m,i ,newt

−St+s P∗t+s

[I m

i ,t+s +φm,i zt+s]]

,

where the nominal exchange rate (domestic currency units per foreign currency units)

is denoted St . The differentiated import goods at time t are aggregated using the

following CES functions

C mt =

[∫ 1

0(C m

i ,t )1

λm,ct di

]λm,ct

, I mt =

[∫ 1

0(I m

i ,t )1

λm,it di

]λm,it

, (3.5)

where λm,ct and λm,i

t are time-varying markups in the import sectors. Hence, in the

flexible price equilibrium, the representative firm will set the price as a markup over

marginal costs, P m,ct =λm,c

t St P∗t and P m,i

t =λm,it St P∗

t .

Taking the aggregate prices as given, costs minimization then implies the follow-

ing demand for the imported consumption and investment good i , respectively

C mi ,t =

P m,ci ,t

P m,ct

− λm,ct

λm,ct −1

C mt , I m

i ,t =P m,i

i ,t

P m,it

− λm,it

λm,it −1

I mt .

The resulting Phillips curve is

πm, jt = β

1+κm, jβπ

m, jt+1 +

κm, j

1+κm, jβπ

m, jt−1 +

(1−ξm, j )(1−βξm, j )

ξm, j (1+κm, jβ)(mcm, j

t + λm, jt ),


where mcm, jt = P∗

t

St Pm, jt

is the real marginal cost of the importing firm in sector j ∈ c, i .

Time-varying markups and price rigidity imply that terms of trade deviations happen

in the short run. However, since the domestic economy does not affect the foreign

economy and because of the fixed exchange rate, this implies that the relative prices

must eventually return to their long-run equilibrium.

Exporting Firms

The exporting sector consists of a continuum of firms, indexed by i ∈ [0,1]. Firm i buys

the homogenous domestic good, differentiates it and sells it for P xi ,t (denominated in

local currency). Exports are aggregated according to

X t =[∫ 1

0(Xi ,t )

1λx

t di

]λxt

,

where λxt is the time-varying markup in the export sector. Firm i takes the aggregate

price as given and faces the following demand

Xi ,t =(

P xi ,t

P xt

)− λxt

λxt −1

X t .

The exporter faces Calvo-style nominal rigidities and non-reoptimized prices are

indexed in the same fashion, P xt+1 = (πx

t )κxπ1−κx P xt . When the exporting firm i is able

to reset its price, P new,xt , it solves the following optimization problem

maxP x,new

t

= Et

∞∑s=0

(βξx )s D t+s

[(πx

t ...πxt+s−1

)κx (π...π)1−κx Xi ,t+s P x,newt

− Pt+s

St+s

[Xi ,t+s +φx zt+s

]].

The resulting Phillips curve is

πxt = β

1+βκxπx

t+1 +κx

1+βκxπx

t−1 +(1−ξx )(1−βξx )

ξx (1+βκx )(mcx

t + λxt ),

where the real marginal cost of the exporter is given as mcxt = Pt

St P xt

. Note that the law

of one price does not hold in the short run when export prices are sticky (ξx > 0).

Since the domestic economy is of negligible size, the domestic price level and the

consumer prices coincide in the foreign economy, implying that P xt = P∗

t . Further,

by assuming that foreign consumption and investment are CES aggregated with the

same elasticity of substitution, the total export demand can be written as

X t =C xt + I x

t =(

P xt

P∗t

)−η f

Y ∗t , (3.6)


where Y ∗t and P∗

t denote the foreign output and price level, respectively. Hence it

is not necessary to model foreign consumption and investment separately. Instead,

exports move proportionally to aggregate foreign output.

3.2.2 Households

The household sector is characterized by infinitely lived households, indexed by

n ∈ [0,1]. The representative household maximizes the expected discounted utility,

given by

Et

∞∑s=0

βt

ζc

t+s log (Cn,t+s −bCn,t−1+s )−ζht+s AL

hσLn,t+s

1+σL

, (3.7)

where Cn,t is a consumption good composite and hn,t denotes the hours worked

by the household. The parameter b governs the level of internal habit formation in

consumption and σL is the labor supply elasticity. There are two preference shocks,

ζct and ζh

t , that shift the intertemporal margins of utility of consumption and labor.

These are common for all households and can be interpreted as aggregate demand

and labor supply shocks, respectively. Consumption is assumed to be aggregated by

a CES function that combines domestically produced consumption goods, C dt , and

imported consumption goods, C mt .

Ct =[

(1−ωc )1ηc

(C d

t

) ηc−1ηc +ω

1ηcc

(C m

t

) ηc−1ηc

] ηcηc−1

.

The parameter ηc denotes the elasticity of substitution between domestic and for-

eign goods in consumption. Hence, a high value of ηc implies a high willingness

to shift the composition of aggregated consumption which becomes smooth even

if domestic or imported consumption are more volatile. The degree of home bias

in consumption given as (1−ωc ), which affects the steady state level of domestic

consumption. Cost minimization implies the following demand function for the two

types of consumption goods

C dt = (1−ωc )

(Pt

P ct

)−ηc

Ct , C mt =ωc

(P m,c

t

P ct

)−ηc

Ct . (3.8)

Similarly, investment is assumed to be a CES aggregate of domestic and imported

investment goods, I dt and I m

t , respectively.

It =[

(1−ωi )1ηi

(I d

t

) ηi −1ηi +ω

1ηii

(I m

t

) ηi −1ηi

] ηiηi −1

,


with the demand functions

I dt = (1−ωi )

(Pt

P it

)−ηi

It , I mt =ωi

P m,it

P it

−ηi

It . (3.9)

The consumer and investment price indices are given as

P ct =

[(1−ωc )P 1−ηc

t +ωc (P m,ct )1−ηc

] 11−ηc , P i

t =[

(1−ωi )P 1−ηit +ωi (P m,i

t )1−ηi] 1

1−ηi .

It is important to note the close relationship between the substitution elasticity

between domestic and foreign goods and the markup shocks within each import

sector, given in (3.5). A higher elasticity of substitution between domestic and foreign

goods will lower the markup for all importing firms. In fact, changes in the elasticity

are observationally equivalent to changes in the exogenous markups in each sector.

Thus, in this model, the markup shocks cover both changes in the optimal pricing

behavior of firm i (for example due to changes in the degree of competition) and

the extend to which the consumers are willing to substitute between domestic and

foreign goods.

Investment is used to accumulate capital by the following law of motion

Kt+1 = (1−δ)Kt +Υt F (It , It−1)+∆t ,

where δ is the rate of which capital depreciates.Υt is a stationary investment-specific

productivity shock, which affects the intertemporal margin of the investment deci-

sion of firms. The function F (It , It−1) =(1− F

(It

It−1

))It summarizes the relationship

between current and past investment and next period’s physical capital. It is assumed

to be cost free to invest at the steady state growth rate but increasingly costly as

investment moves away from the balanced growth path1. The variable ∆t reflects

that capital can be traded frictionlessly between households at price Pk ′,t . Although

the equilibrium condition ∆t = 0 holds for all periods t , this is included in the model

to derive the market value of capital.

In each period t , household n faces the the following intertemporal budget con-

straint

Bn,t+1 +St B∗n,t+1 + (1+τc )P c

t Ci ,t +P it In,t +Pt

[Pk ′,t∆n,t

]=Rt−1Bn,t + (1−τk )Tt + (1−τy

t )Wn,t hn,t + (1−τk )Rkt Kn,t +R∗

t−1Φ

(At−1

zt−1, φt−1

)St B∗

n,t

−τk[

(Rt−1 −1)Bn,t +(

R∗t−1Φ

(At−1

zt−1, φt−1

)−1

)St B∗

n,t +B∗n,t (St −St−1)

]+Qn,t .

(3.10)

1Only the second order derivative, F ′′ is identified in the log-linearized model and will be treated as aparameter.


The household spends resources on domestic and foreign zero-coupon bond pur-

chases, Bn,t+1 and B∗n,t+1 which carry the gross interest rate, Rt and R∗

t , from period

t to t +1, respectively. Further, consumption and investment goods are purchased at

prices P ct and P i

t , respectively. Households gain resources from their bond holdings,

form profits transferred by the intermediate good producers, Tt , from labor income,

and from renting out capital to firms. The real net foreign asset position is given as

At = St B∗t+1

Pt. There is a risk premium, Φ

(At−1zt−1

, φt−1

)> 0, to holding foreign bonds if

the domestic economy as a whole is a net borrower (At < 0). Further, each household

has a negligible size, implying that increased borrowing has no aggregate effects on

the risk premium and thus does not internalize the effects on the net asset position.

Note that the households face idiosyncratic risks as suppliers of labor as the pricing

signal arrives stochastically with probability 1−ξw . To avoid ex post heterogeneity

and to preserve the representative agents framework the assumption is imposed that

the households can freely trade the entire set of Arrow-Debreu securities. This allows

the households to enter into an insurance scheme with perfect risk sharing, making

them ex post homogeneous. The net income from this portfolio is denoted Qn,t .

After imposing a symmetric equilibrium, the first-order conditions for Ct , ∆t ,

Kt+1, It , Bt+1, and B∗t+1, respectively are

0 = ζct

ct −b ct−1µz,t

−βEtbζc

t+1

µz,t+1ct+1 −bct− λt (1+τc )

P ct

Pt, (3.11)

0 =− λt

ztPk ′,t +qt , (3.12)

0 =−λt Pk ′,t +βEt

λt+1

µz,t+1

[(1−τk )r k

t+1 + (1−δ)Pk ′,t+1

], (3.13)

0 = λt

[Pk ′,tΥt F1 −

P it

Pt

]+βEt

λt+1

µz,t+1Pk ′,t+1Υt+1F2, (3.14)

0 =−λt +βEt

λt+1

µz,t+1πt+1

[Rt −τk (Rt −1)

], (3.15)

0 =−λt St +βEt

λt+1

µz,t+1πt+1

[R∗

t Φ(at , φt )St+1 −τk(R∗

t Φ(at , φt )−1)

St+1

−τk (St+1 −St )

]. (3.16)

The Lagrangian multiplier to the household budget constraint, λt , has been appro-

priately scaled as λt = Pt ztλt . Equation (3.11) is the Euler equation of consumption

which balances intertemporal marginal utility of consumption with the relative price.

The variable qt is the real Tobin’s Q, i.e. the marginal value of an extra unit of capital.

The equations (3.12)-(3.14) give the equilibrium price and optimal demand for capi-

tal, either through purchases or by investing in new capital. The relative demands of


domestic and foreign bonds in (3.15) and (3.16) are determined by the interest rate

spread and the expected capital gains from changes in the nominal exchange rate.

To model wage setting it is assumed that each household supplies monopolisti-

cally differentiated labor. The labor aggregate, ht , is constructed from the following

function

ht =[∫ 1

0h

1λw

i ,t di

]λw

, 1 ≤λw <∞,

where λw is the wage markup. Hence, the demand for labor of type n is

hn,t =(

Wi ,t

Wt

)− λw

λw −1ht . (3.17)

Wage setting is subject to Calvo-style nominal rigidities so that with probability ξw

the household is not able to set a new wage in the subsequent period which follows

an indexation rule instead, Wn,t+1 = πκwt π1−κwµz,t+1Wn,t . When the household is

able to reset its wage it solves the following optimization problem

maxW new

t

= Et

∞∑s=0

(βξw )s[−ζh

t+s ALh1+σL

n,t+s

1+σL

+ vt+s (1−τyt+s )(πt ...πt+s−1)κw (π...π)1−κw (µz,t+1...µz,t+s )W new

t hn,t+s

],

subject to (3.17).

3.2.3 UIP and Monetary Policy

The uncovered interest rate parity (UIP) relates expected changes in the nominal

exchange rate with the interest rate spread between the domestic and foreign econ-

omy. To ensure that the DSGE model is stationary, there is a risk premium on foreign

bonds which is strictly decreasing in the real net foreign asset position, at = St B∗t+1

Pt zt.2

The premium on foreign bonds is assumed to be described as

Φ(at , φt ) = exp

−φa(at −a)+ φt

, (3.18)

where the parameter φa denotes the sensitivity of net foreign asset holdings on the

risk premium and φt is a risk premium shock. As will be seen below, in an economy

with a fixed exchange rate this shock has a similar role as a monetary policy shock for

economies with a floating exchange rate.

2If this was not the case, the domestic households could borrow infinitely to finance consumption,violating the transversality condition. See Schmitt-Grohe and Uribe (2003).


By combining the household demand for domestic and foreign bonds in (3.15)

and (3.16), a no arbitrage condition can be derived. In log-linearized form the UIP

condition is given as

Rt − R∗t = Et∆St+1 −φa at + ˆφt . (3.19)

Hence, in equilibrium, if the domestic interest rate is lower than its foreign counter-

part this must be reflected in either nominal exchange rate appreciation or the risk

premium. Similarly, an increase in the foreign interest rate must be met by a similar

domestic increase as if to avoid currency depreciation, which underlines the close

link between monetary policy and the exchange rate.

Assume now that the monetary policy can be described by the following Taylor

rule

Rt = ρR Rt−1 + (1−ρR )(κππ

ct−1 +κy yt−1 +κs∆St

). (3.20)

A credible and fixed exchange rate policy implies that Et∆St+1 = 0 for every period t ,

which requires that the central bank responds forcefully to changes in the nominal

exchange rate, κs →∞. Combining this with (3.19) gives the following log-linearized

relationship for the domestic interest rate

Rt = R∗t −φa at + ˆφt , (3.21)

implying that monetary policy is now completely endogenous in the sense that

changes in the foreign interest rate will affect the domestic interest rate one-to-one.

Any interest rate spread between the domestic and foreign policy rates is due to

the risk premium that domestic consumers have to pay on foreign bonds. Hence,

a risk premium shock, φt , is similar in nature to a monetary in closed-economy

models or models where the exchange rate is flexible. However, where the latter has

the interpretation of an exogenous deviation from the policy rule, a shock to (3.21)

reflects an exogenous increase in the perceived risk of foreign investors of holding

domestic bonds, e.g. due to fear of devaluation. Note finally, that φt is not a true

structural shock, in the sense that it is derived from first principles such as utility

maximization but instead a residual that captures UIP deviations.

3.2.4 Equilibrium Conditions and Shock Processes

Final goods market equilibrium is given as

C dt +C x

t + I dt + I x

t +Gt = εt z1−αt Kα

t H 1−αt −φzt .

Inserting the demand functions (3.6), (3.8), and (3.9) and stationarizing variables

yields

(1−ωc )(γc,dt )ηc ct + (1−ωi )(γi ,d

t )ηi it + (γx,∗t )−η f y∗

t z∗t + g t = εtµ

−αz,t kαt H 1−α

t −φ.


The variable z∗t ≡ z∗t

zthas been used to detrend foreign output and allows the domestic

and foreign productivity to evolve in a stationary asymmetric fashion.

The evolution of net foreign assets is described by the balance of payments

St B∗t+1 = St P x

t (C xt + I x

t )−St P∗t (C m

t + I mt )+St R∗

t−1Φ(at−1, φt−1)B∗t .

The balance of payments gives the equilibrium dynamics between foreign debt and

the trade balance. If imports exceed exports this must be financed by an increase in

foreign debt (corresponding to a decrease in net foreign assets). This in turn increases

the interest rate that domestic households must pay on foreign bond in the next

period.

Inserting the demand functions and stationarizing variables yields

at =(mcxt )−1(γx,∗

t )−η f y∗t

z∗t

zt− (mcx

t γx,∗t )−1

[ωc

γmc,dt

γc,dt

−ηc

ct +ωi

γmi ,dt

γi ,dt

−ηi

it

]

+ St

St−1R∗

t−1Φ(at−1, φt−1)at−1

πtµz,t.

The exogenous shocks in the model are all assumed to adhere to an AR(1) process

xt+1 = ρx xt +σxεxt+1, εx

t+1 ∼N(0,1),

where x = λd ,λmc ,λmi ,λx ,µz ,ε, z∗,ζc ,ζh ,Υ, φ,τy , g .

Finally, a set of relative prices is used to define the model equilibrium. They are

given in Appendix A.

3.2.5 Foreign Economy

By assuming that the domestic economy is of negligible size compared to the foreign

economy, the latter can be modeled exogenously. This significantly simplifies the

modeling task and allows for a more flexible specification. The foreign economy is

assumed to have the following structural VAR representation

B0Xt = B1Xt−1 + ...+Bp Xt−p +εt , εt ∼N(0,Ik ). (3.22)

The vector Xt =(π∗

t , y∗t ,R∗

t

)′ consists of HP-filtered Euro area inflation and output3

and the demeaned ECB nominal interest rate. To obtain identification of the structural

3One concern of using HP-filtered data for the foreign economy is that it might make the foreignvariables "too smooth" compared the their unfiltered domestic counterparts. Preliminary results show thatusing demeaned growth rates in the foreign VAR tend to increase the relative importance of foreign shocksin the model, however the forecasting implications of this change was not investigated. This remains animportant question for further research.

3.3. ECONOMETRIC METHODOLOGY 99

shocks it is assumed that B0 is lower triangular. Thus, a Cholesky decomposition

of the estimated covariance matrix can be applied to obtain identification from

the reduced-form representation of (3.22). The parameters of the VAR model are

estimated before those of the DSGE model using a Bayesian approach with prior

distributions represented by dummy observations.4

3.3 Econometric Methodology

The solution to the DSGE model in Section 3.2 has to be approximated before it

can be estimated. In this paper, the model equilibrium will be approximated by a

log-linearization around the non-stochastic steady state. Provided that the model

has a unique and stable solution this can be expressed as

st+1 =Φ1(θ)st +Φε(θ)εt+1, (3.23)

where st is a vector of appropriately defined model variables and θ is vector contain-

ing the parameters of the DSGE model. The observables are related to the model by a

set of measurement equations

yt =Ψ0(θ)+Ψ1(θ)st +ut . (3.24)

The matrices Φ1, Φε, Ψ0, and Ψ1 of the reduced-form system depend on the un-

derlying structural model parameters, θ. Measurement errors, ut , are added to all

observables except the interest rates. This is not necessary for the applications in

this paper in the sense that the model contains enough structural shocks to avoid

stochastic singularity for the chosen number of data series. However, including mea-

surement errors can be a beneficial way of relaxing the restrictions imposed by the

model during the estimation. Further, it is well known that macroeconomic data

series are subject to revisions because their "true" values are unknown.

The equations (3.23) and (3.24) constitute the state-space representation of the

(linearized) DSGE model. It is assumed that ut ∼N(0,Σu), where Σu is a diagonal

matrix, and since εt ∼N(0,Σε) follows from the model the exact likelihood of the

model can be evaluated directly, using the Kalman filter.

In the following, let the nobs ×T matrix Y1:T matrix denote the sequence of ob-

servables

y1, ...,yT

. The structural parameters are estimated using Bayesian econo-

metrics, which is described in detail in An and Schorfheide (2007). Thus, the object

of interest is the posterior distribution

p(θ|Y1:T ) = p(Y1:T |θ)p(θ)

p(Y1:T ), (3.25)

4See Sims and Zha (1998) for a detailed description. These so-called Minnesota priors are dependenton several hyperparameters that controls the correlation structure of the dummy observations and arewell known to increase the forecasting performance, relative to a standard VAR model.


where p(Y1:T ) is the model likelihood, p(θ) is the prior distribution, and p(Y1:T ) is

the data density. For DSGE models it is generally not possible to obtain closed-form

expressions of (3.25). This is instead approximated by simulating Markov chains that

converges to this posterior ergodically. Specifically, a numerical optimizer is applied

to approximate the Hessian at the posterior mode which then guides the step size for

the Random-Walk Metropolis Hastings algorithm using a chain with 500,000 draws,

discarding the first 250,000 draws to better ensure convergence. The Hessian is scaled

such that the acceptance rate in the chain is approximately 20-30 percent, close

to what is generally recommended in the literature (see for example Roberts et al.

(1997)). Convergence of the chain was checked by using the convergence diagnostics

in Geweke (1999) which in standard in the literature (not shown).

3.4 Data

The model is estimated using quarterly seasonally adjusted Danish data from 1990Q2-

2010Q4. As the model allows for trends the raw data series can in general be used in

the estimation procedure as opposed to using pre-filtered data. However, following

Adolfsen et al. (2007b), some of the series are altered prior to estimation: First, due

to increased globalization, both imports and exports grow at a faster rate than the

overall economy in the sample period. This is at odds with the model that imposes

constant steady state ratios of imports and export to output, determined by ωc and

ωi . Instead of making these coefficients time-varying the "excess" linear trend of the

two series relative to that of GDP is removed. Second, Euro area output and inflation

are HP-filtered prior to estimation while the foreign interest rate is demeaned.

Throughout this paper, the question of data vintages is set aside. Hence, all data

series used are the latest vintage. Since the model performance is not compared with

real-time forecasts this is not likely to affect the assessment of the relative forecasting

performances.

The set of observables used in the measurement equations in (3.24) are (real) GDP,

consumption, investment, wages, exports, imports, the Danish policy rate, the GDP

deflator, the consumer price index, and Euro area output, inflation, and interest rate.

The set of measurement equations and how they relate to the model variables is given

in Appendix C. As noted in Adolfsen et al. (2007b) the foreign economy variables

can still be included in the estimation to the modeler’s advantage, even though this

part is modeled exogenously and estimated prior to the estimation of the DSGE

model. The reason is that they contain information about the transmission of foreign

shocks through the domestic economy and enables identification of the risk premium

and asymmetric productivity shocks. Further, the foreign variables contain crucial

information about the current state of the economy and hence for the forecasts.

The choice of data series in the estimation turns out to be crucial for the forecast-

3.5. PRIOR DISTRIBUTIONS AND CALIBRATED PARAMETERS 101

ing performance. On one hand, including more data series will help identifying the

structural parameters in the model. For example, including another price deflator

will sharpen the parameter estimate of that particular nominal rigidity. On the other

hand, if too many series are included this will have adverse effects on the filtering of

the unobserved state, st . This will in turn affect the mean of the forecasts through the

transition equation in (3.23). Specifically, it is found that the medium and long term

forecasts suffer if too many inflation series are included.

Finally, for the measurement equations in (3.24) a new set of model variables is

defined to consistently match the data specifications. Aggregate consumption and

investment are CES aggregated in the model but enter the national income identity

linearly. Hence, to get consistent measures, the consumption, investment, imports,

and consumer price series are transformed (see Appendix B for details).

3.5 Prior Distributions and Calibrated Parameters

The prior distributions used in the estimation are given in Table 3.2. Note that, in

the model estimation, the Kalman filter generates a sequence of probability den-

sities which form the model likelihood p(Y1:T |θ) = ∏Tt=1 p(yt |Y1:t−1,θ). Hence, the

estimation routine which uses the posterior distribution in (3.25) is based on the

1-step ahead prediction error, while we are interested in a high density at longer

horizons as well. Further, with diffuse priors DSGE models have a tendency to be

multi-modal with deep valleys of low likelihood in between. It is well known that the

Random Walk Metropolis Hastings algorithm has very low efficiency in this case. As a

result, somewhat tight priors is generally necessary to ensure a decent forecasting

performance.

The priors are broadly in line with previous literature and take into account

the a priori restrictions imposed on the parameter space. For the nominal price

and wage rigidity, the beta distribution is specified since the domain of this density

function is [0,1]. The mean is set to imply a reset probability of 25 percent, implying

an average duration of contracts of four quarters. The indexation parameters are

also beta distributed with mean 0.25, since most previous studies find relatively

low degrees of indexation. For the elasticity of foreign demand and the substitution

elasticity of domestic and foreign investment, the inverse gamma distribution with

mean 1.5 is used since this assigns probability mass only to positive parameter values.

The price markups also adhere to an inverse gamma distribution with mean 1.2,

which corresponds to a 20 pct. markup over marginal cost and a demand elasticity of

6. The prior distribution of F ′′ is a normal distribution with mean 5.0 and standard

error 1.5. The habit parameter is given a beta distribution with mean 0.5.

Since the exogenous shock processes assert great influence on the dispersion

of the endogenous variables, the magnitude of persistence and standard errors will


greatly affect the posterior predictive density of the observables. Hence, relatively

uninformative priors are generally specified for parameters governing the shocks in

the model so as mostly to let the data guide the posterior estimates. For the persis-

tence coefficients the beta distribution is used while the inverse gamma distribution

is used for the standard errors. All prior distributions are kept constant during the

sequential estimation of the model.

A subset of the parameters in the DSGE model is calibrated instead of estimated,

i.e. they are assigned a prior with infinite mass at the calibrated value. This is done

mainly because these parameters are weakly identified in the data, implying that

the model likelihood is flat in these dimensions or because they determine the so-

called "great ratios" (e.g. imports as share of total output). Examples include capital

depreciation, the discount factor, and the labor supply elasticity. Secondly, due to the

sequential estimation used in the forecasting exercise in this paper, some parameters

are calibrated rather than estimated to avoid unstable estimates with poor out of

sample properties as a result. Tihs includes the elasticity of substitution in consump-

tion is fixed at ηc = 2.5. Thirdly, the monetary policy parameters are calibrated to

reflect the fixed exchange rate regime of Denmark. Specifically, κs = 100,000 is set

while keeping the remaining parameters at standard values. Finally, the fiscal shocks

are calibrated with persistency parameter of 0.5 and a standard error of 1 percent.

The measurement errors in (3.24) are given standard errors of 10 percent of that of

the corresponding data series. Table 3.1 provides an overview of the calibration.

3.6 Posterior Model Evaluation

The estimated parameters are reported in Table 3.2 and 3.3. This section presents

some of the properties of the estimated DSGE model.

As a posterior predictive check Table 3.4 shows the model’s ability to match the

unconditional second-order moments in the data. Specifically, the standard devi-

ations, first-order autocorrelations, and the correlations with GDP growth will be

considered for the set of variables used in the estimation (all in quarterly growth

rates). The model-implied moments are the analytical moments based on the mode

of the posterior distribution.

Considering first the standard deviations. It can be seen that these are slightly

higher in the model than in the data for output, wages, and the price deflators. How-

ever, for the remaining variables the model is relatively successful in replicating the

volatility found in the data. For the first-order autocorrelations the model generally

does a good job at matching the persistency in the data. However, the model is unable

to match the low persistency found in the wage, consumption, and consumer price

series. Finally, for the cross-correlations with output growth it can be seen that wages

3.6. POSTERIOR MODEL EVALUATION 103

and consumption are somewhat too procyclical in the DSGE model. Since it is gen-

erally hard for DSGE models to generate acyclical real wages (see for example King

and Rebelo (1999)), it can be concluded that the model generally does very well in

capturing the correlation with output growth. Overall, for a DSGE model this size the

level of conformity to the moments in the data can be characterized as satisfactory.

3.6.1 Impulse Response Functions

It has become common in the DSGE literature to study the endogenous propagation

of exogenous shocks by looking at impulse response functions (IRFs). The figures

are based on the posterior mode of the estimated parameters. For ease of exposition

only a subset of the model shocks will be considered in this section: A stationary

productivity shock, a shock to the domestic price markup, and a shock to foreign

output.

Figure 3.1 shows the impulse response of output, real wage, consumption, in-

vestment, imports, exports, the output deflator, and consumer price inflation to an

expansionary shock to the stationary productivity level of one standard error. From

(3.3) it is seen that this lowers the real marginal costs of domestic producers. This

in turn feeds through to prices and reduces domestic inflation which follows from

the Phillips curve in (3.4). Aggregate consumption increases since forward looking

consumers spend parts of their increased future income today. The increased produc-

tivity raises the marginal product of capital which spurs an increase in investment.

The increase in aggregate demand leads to higher marginal production costs which

eventually cancel out the fall in inflation. Lower domestic inflation implies an im-

provement in the terms of trade which leads to an increase in demand for domestic

goods instead of imports which declines initially. Eventually, increased domestic

demand for consumption and investment increases imports above its steady state

level. Exports increase as well, due to an improvement in the terms of trade through

the real exchange rate. As pointed out by Pedersen and Ravn (2013), all shocks in

the model that affect the different inflation measures in the domestic economy will

cause overshooting. Due to the small open economy assumption (and since all goods

are tradeable), the terms of trade has to return to its long-run level. This adjustment

can only happen through the adjustment of domestic prices in a fixed exchange rate

regime.5 Note that the nominal interest rate remains constant instead of falling, as

usually implied in DSGE models where the central bank seeks to stabilize inflation.

Thus, the fixed exchange rate mitigates the real effects of productivity shocks.

Next, consider the impulse responses to an increase in the domestic price markup,

depicted in Figure 3.2. This can be interpreted as an increase in the degree of market

5From a more technical perspective, this inflation overshooting is necessary for solution determinacy.


power for the representative firm, thus allowing it to set a higher price, ceteris paribus.

As a result, domestic and consumer price inflation goes up. As markups are only

weakly correlated over time this effect only lasts a few quarters until inflation falls

again. Higher prices lower demand for domestic goods, resulting in a fall in output

as well as exports due to a real exchange rate appreciation which on the contrary

boosts imports. The reduced demand for domestic goods results in lower real wages

and hence marginal cost of production. Eventually inflation goes down to restore

competitiveness so exports return to the steady state. Again, we see the inflation

overshooting for both the output deflator and consumer prices to obtain balance of

trade equilibrium. As with the productivity shock, the exchange rate peg works coun-

tercyclical in this case as the interest rate is not increased in face of higher inflation.

Lower real interest rates support domestic aggregate consumption and investment

hence mitigates the real downturn of the shock.

Figure 3.3 shows the impulse response to an increase in the foreign output. A di-

rect effect of the increase in foreign demand is a sharp increase in exports. This

increase in aggregate demand causes output, consumption, and investment to in-

crease and remain above their steady state values for a long time. Forward looking

firms respond to the increase in current and expected future aggregate demand by

raising price so both domestic and consumer price inflation increase. Rising income

and domestic inflation cause imports to increase, the latter because of a real exchange

rate appreciation. The fixed exchange rate policy works procyclical in face of foreign

demand shocks. If instead the exchange rate was floating, a nominal depreciation

could serve as a buffer against adverse foreign shocks because it would support

exports by restoring competitiveness. Further, as monetary policy does not target

inflation the interest rate is kept constant in face of increased inflation. Hence, foreign

shocks are transmitted more forcefully and persistent through the domestic economy

because of the fixed exchange rate.

3.6.2 Forecast Error Variance Decomposition

It is useful to assess which structural shocks are important for forecasting which

variables and at different horizons. Such a forecast error variance decomposition

is closely related to the impulse response functions. LetΩhj ≡ ∂yt+h

∂ε j ,tbe the impulse

response for a given variable at horizon h to shock j with standard errorσ j , where h =1, ..., H and j = 1, ..., J . The contribution to the H-step ahead forecast error variance

from shock i can then be expressed as ΣHi =∑H

h=1

(Ωh

i

)2σ2

i /∑J

j=1

∑Hh=1

(Ωh

j

)2σ2

j .

Table 3.5 and 3.6 show the forecast error variance decomposition for the endoge-

nous variables used in estimation at the 1- and 8-step ahead horizons, respectively.

The parameters are fixed at the posterior mode for the initial estimation sample.

3.6. POSTERIOR MODEL EVALUATION 105

In line with Smets and Wouters (2003) the investment specific preference shock

is found to be a major contributor to output variations. Like the shock to consump-

tion preferences this can be interpreted as a domestic aggregate demand shock, as

both output and inflation increase. Together, these two shocks account for more than

30 percent of the variation in GDP growth for both horizons.

Productivity shocks account for approximately 5.5 and 7.5 percent of GDP growth

at the 1 and 8 quarter horizon, respectively. Shocks to labor preferences account

for between 3 and 6 percent of output variations approximately. It is characteristic

for both of these capacity shocks that their relative importance increase with the

forecasting horizon for most of the observed variables.

With approximately 40 percent in total, the largest contributors to output vari-

ations at both 1 and 8 quarters are the markup shocks, in particular the markup in

the export sector. This might be necessary for the model to explain the joint evolve-

ment of a relatively large set of observables included in the estimation. As discussed

previously, the fixed exchange rate imposes restrictions not found in models with

a flexible exchange rate, as all variations in the real exchange rate must come from

the relative prices. Thus, the somewhat large contribution to several variables by the

markup shocks might potentially reflect some model misspecification or inability to

explain comovements of both prices and quantities.

Not surprisingly, aggregate consumption growth is driven to a large extend by

shocks to consumption preferences. Similarly, for investment, almost all of the varia-

tion comes from the investment specific shock while real wage variation comes from

shocks to labor preferences and the domestic price markup.

The importance of foreign shocks6 is seen for both GDP, consumption, and ex-

ports growth, where these account for approximately 15 to 21 percent of fluctuations.

This is considerably less than what is found in Pedersen and Ravn (2013) but still

more than what is found in most of the open economy DSGE literature, for example

in Justiniano and Preston (2010). Thus, as we would expect from economic theory,

this suggests that the fixed exchange rate tends to increase the relative importance of

shocks originating from the foreign economy.

Turning to the inflation series, the tables show that the markup shocks account

for most of the variability. However, going from 1 to 8 quarters other shocks become

more important in explaining the variation in inflation, whereas the importance of

weakly correlated markup shocks diminishes (a finding that echoes that in Adolfsen

et al. (2007b)).

6"Foreign shocks" here and below are defined as shocks to the foreign VAR model, UIP deviations, andthe asymmetric productivity shock, zt .


3.7 DSGE Model-based Forecasting

This sections briefly explains how to obtain forecasts from the estimated DSGE model

(see Del Negro and Schorfheide (2013) for a more detailed discussion hereof). For

forecasting purposes we are interested in the h-step ahead predictive density of the

vector yT+h , given the past observations, Y1:T . This can be expressed as

p(YT+1:T+h |Y1:T ) =∫θ

p(YT+1:T+h |θ,Y1:T )p(θ|Y1:T )dθ

=∫

(sT ,θ)

∫ST+1:T+h

p(YT+1:T+h |ST+1:T+h)

p(ST+1:T+h |sT ,θ,Y1:T )dST+1:T+h

×p(sT |θ)p(θ|Y1:T )d(sT ,θ),

(3.26)

where ST+1:T+h denotes the sequence sT+1, ...,sT+h. Hence, when conducting fore-

casts with a DSGE model there are four sources of uncertainty affecting the predictive

distribution in (3.26). First, there is parameter uncertainty reflected in the posterior

distribution p(θ|Y1:T ). Second, since the vector of state variables contains variables

that are unobservable to the econometrician, there is uncertainty about the current

state of the economy, captured by p(sT |θ,Y1:T ). Third, there is uncertainty about the

future states of the economy and the realization of shocks as reflected by p(ST+1:T+h).

Fourth, since the observables are being measured with error there is uncertainty

about the true values of the observed time series, p(YT+1:T+h |ST+1:T+h). Bayesian

inference provides predictive distributions that take into account all these sources

of uncertainty. Since a closed-form expression of (3.26) is not available this must

instead be approximated by drawing y(i )T+h (based on a parameter draw θi ), where

i = 1, ..., Nsi m .

Often, the primary object of interest is the mean of the posterior predictive dis-

tribution. In fact, if the loss function associated with forecast errors is quadratic the

mean will minimize the expected loss. Using (3.23) and (3.24), the h-step ahead point

forecast based on the mean is given as

E[yt+h |θ,Y1:T

]=Ψ0(θ)+Ψ1(θ)[Φ1(θ)

]hE[st |θ,Y1:T

], (3.27)

where E[st |θ,Y1:T

]is found using the Kalman filter and numerical integration with

respect to the parameter vector θ. Alternatively, a plug-in estimate of θ can be used

(e.g. the posterior mode or mean), which speeds up the computation but ignores the

true Bayesian distribution, not taking into account parameter uncertainty. Amisano

and Geweke (2013) find that using the full Bayesian predictive distribution substan-

tially improves the model predictability relative to the plug-in estimate. However, this

is will be dependent of the specific application, for example how much density of the

3.7. DSGE MODEL-BASED FORECASTING 107

posterior distribution is near the plug-in estimate. Overall, any plug-in estimate will

underrate the dispersion of p(YT+1:T+h |Y1:T ) and therefore, in the section below, the

full Bayesian forecasts will be used.

3.7.1 Forecasting Performance

The forecasting performance of the DSGE model and the alternative models is as-

sessed by a rolling estimation and forecast procedure in the following way: First, the

model parameters are estimated using data up until time T . Second, the estimated

model is used to compute point forecasts and forecast densities H quarters ahead,

yT+1,yT+2, ...,yT+H . Then, the information set is updated as the estimation sample

is increased to Y1:T+1 and new forecasts are made. These steps are iterated for the

entire sample where the initial estimation sample is 1990Q2-2001Q4 and the hold-out

sample is 2002Q1-2008Q2 (the Great Recession episode starting in 2008Q3 will be

examined separately below). The forecast horizons h = 1,2, ...,8 will be considered.

Hence, there are 26 observations for the 1-step ahead and 19 observations for the

8-step ahead forecasts. The variables considered below are (real) GDP, wages, con-

sumption, investment, imports, exports, the output deflator and the consumer price

index.

The point forecast accuracy is based on the mean of the posterior predictive

distribution given in (3.27). As is common in the literature the root mean square errors

(RMSE) will be used as the assessment criteria. The forecasts of the DSGE model

will be compared with an AR(1) model and the random walk, i.e. constant growth

rates. Because of the simplicity of these alternative models few further assumptions

that will affect the relative model performances have to be made. Further, it is well

known that it is not an easy task for structural models to beat the AR(1) model in

out-of-sample predictions so this will be a good benchmark to measure the DSGE

model against.

Figure 3.4 and 3.5 show the RMSEs, based on yearly growth rates as a function of

the forecast horizon from one to eight quarters ahead. Generally, the DSGE model

does well in forecasting the different variables, better than the random walk and

comparable in performance to the AR(1) model. Thus, the results suggest that it

is possible to apply a DSGE model incorporating a fixed exchange rate regime for

forecasting and obtain reasonable forecasts.

For GDP, the DSGE and AR(1) model have almost the same predictive ability for

short horizons, although for forecasts four to six quarters ahead, the DSGE model is

more accurate. Both models are better than the random walk and this is especially

pronounced at longer forecast horizons. For aggregate consumption, the RMSE of the

three models follow a similar pattern. Hence, it seems that the random walk is hard

to beat for consumption at short horizons. However, at longer horizons the other


models (and especially the DSGE) clearly outperform constant growth rates predic-

tions. Similarly, the random walk is outperformed for all horizons for investment

growth. In this case, the DSGE model seems more accurate than the AR(1) model for

shorter horizons while the opposite is the case for forecasts longer than a year. For

consumer price inflation it is again hard to beat the random walk, although the DSGE

model seems to be slightly more accurate across forecasting horizons in general. It is

particularly encouraging to observe the forecasting performance of the open econ-

omy DSGE model for the growth rates of imports and exports. For these variables,

the model performs well compared to the alternatives, especially pronounced at

shorter horizons. Finally, for real wage growth and the domestic price deflator, the

DSGE model does a poor job at forecasting beyond two quarters when compared to

the other two models. However, it should be noted that the RMSEs are somewhat

contaminated by a few periods with very poor forecasts, in which the medium-run

variations in these series are too large.

3.7.2 Forecasting During the Great Recession

After having analyzed the properties and forecasting performance of the fixed ex-

change rate DSGE model, specific attention will be given to the Great Recession

period in Denmark, which I define as starting in 2008Q3. For the sake of brevity, the

subsequent analysis will focus on the model’s ability to predict GDP growth.

Figure 3.6 shows the realized yearly growth rates and forecasts for the DSGE model

(upper row) and the AR(1) model (lower row). As a general pattern we see that both

models tend to be too optimistic throughout the Great Recession by over-predicting

the growth in real GDP. Initially, both models strongly underestimate the severity

of the downturn but instead predict a steady return to the balanced growth path

when forecasting from 2008Q3 and 2008Q4. This is consistent with results found

from a similar exercise performed for the Smets-Wouters model in Del Negro and

Schorfheide (2013) and as such not surprising. In 2009Q1, the GDP forecasts of the

DSGE model start to differ from the AR(1) model. Here, it can be seen from the figure

that the DSGE model is more pessimistic. At this point, real GDP in Denmark had

already contracted for three consecutive quarters, underlining the severity of the

crisis. The DSGE model correctly predicts a continued decline in GDP growth in the

second quarter of 2009, although actual growth decreased even more. From 2009Q2,

the DSGE model captures the gradual recovery of the Danish economy with fairly

accurate growth predictions while the AR(1) model predicts a stronger recovery than

what occurred.

A major benefit of using DSGE models is that they provide a structural interpre-

tation of business cycles. Figure 3.7 shows the shock decomposition for yt . The line

depicts the deviations from the steady state of the filtered variable and the bars the

3.8. CONCLUSION 109

contribution of the smoothed structural shocks. Hence, we can interpret this as the

ex post interpretation of the crisis when seen through the lens of the estimated DSGE

model. Since Denmark is largely affected by the Euro zone economy it is interesting to

examine whether the prediction errors during the crisis were caused by domestic or

foreign sources. According to the figure, output fell below its steady state in the third

quarter of 2008. This is consistent with the fact that real GDP growth in Denmark was

negative in this quarter, compared to the previous quarter and the year before. It is

noteworthy that the foreign variables contributed positively to Danish GDP in all the

pre-crisis quarters, reflecting the boom in the Euro area which supported exports.

From the onset of the crisis however, the foreign variables make a strong negative

contribution to output. Thus, to a wide extent the crisis was initially imported from

the rest of the world. The large residuals in the foreign VAR might reflect that model-

ing of the foreign economy does not sufficiently account for nonlinearities. Domestic

demand makes a significant negative contribution throughout the downturn. In fact,

this group of shocks continue to affect output negatively, even as the Euro area recov-

ered temporarily in 2010. This underlines that the Danish economy recovered more

slowly than the Euro zone, in part because domestic consumption and investment

were suppressed. However, after the initial downturn, the main contributors to the

negative output gap in Denmark are negative markup shocks.

3.8 Conclusion

This paper estimates an open-economy DSGE model in which the nominal exchange

rate is fixed and examines its forecasting properties. While the predictive abilities of

DSGE models have previously been examined in various papers, these have focused

on either closed-economy models or models where the exchange rate is flexible.

Fixing the exchange rate has implications for the relative importance of shocks and

their propagation through the model. Specifically, foreign shocks assume a greater

role since the nominal exchange rate can not serve as a buffer and since monetary

policy can no longer be used as a macroeconomic stabilization tool.

Using Danish data from 1990Q2-2008Q2 a sequential out-of-sample forecast

evaluation of the DSGE model is constructed for (real) GDP, wages, consumption,

investment, imports, exports, the output deflator and the consumer prices index,

considering forecasting horizons of one to eight quarters. The DSGE model generally

delivers point estimates that are comparable to an AR(1) model and better than the

random walk when measured by the root mean square errors.

Finally, the Great Recession episode from 2008Q3 is examined specifically. In

line with previous research the DSGE model largely underestimates the severity of

the downturn initially. However, from 2009Q1 the DSGE model correctly predicts

a continued decline in GDP growth and subsequently relatively modest increase


in growth. This can be compared to the AR(1) model which consistently predicts a

stronger recovery than what occurred. A historical shock decomposition of output

deviations shows that the initial sharp decline in Danish GDP is primarily caused by

foreign shocks. The continued downturn however is mainly due to shocks to domestic

demand and markups. Generally, the large contribution to fluctuations in several

variables by markup shocks in the model might suggest some sort of misspecification

and is an obvious topic for future research.

Acknowledgments

The author gratefully acknowledges support from Aarhus University, Department

of Economics and Business Economics and from CREATES - Center for Research

in Econometric Analysis of Time Series (DNRF78), funded by the Danish National

Research Foundation. Comments from Frank Schorfheide are also gratefully acknowl-

edged. Further, I am grateful to Jesper Pedersen and Søren Hove Ravn for sharing the

dataset used in their paper.

3.8. CONCLUSION 111

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CHRISTIANO, L. J., M. EICHENBAUM, AND C. L. EVANS (2005): “Nominal Rigidities and

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DEL NEGRO, M. AND F. SCHORFHEIDE (2004): “Priors from General Equilibrium

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——— (2013): “DSGE Model-Based Forecasting,” Chapter in Handbook of Economic

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Open Economy? Evidence from an Estimated DSGE Model of the Danish Economy,”

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proach,” American Economic Review, 97.

3.8. CONCLUSION 113

Appendix A: Model overview

1. πdt = β

1+κdβπd

t+1 + κd1+κdβ

πdt−1 +

(1−ξd )(1−βξd )ξd (1+κdβ) (mcd

t + λdt ),

2. r kt = wt + Rt−1 + Ht − kt + µz,t ,

3. mcdt =αr k

t + (1−α)(wt + Rt−1)− εt ,

4. πm,ct = β

1+κm,cβπm,c

t+1 +κm,c

1+κm,cβπm,c

t−1 +(1−ξm,c )(1−βξm,c )ξm,c (1+κm,cβ) (mcm,c

t + λm,ct ),

5. mcm,ct =−mcx

t − γx,∗t − γmc,d

t ,

6. πm,it = β

1+κm,iβπm,i

t+1 +κm,i

1+κm,iβπm,i

t−1 +(1−ξm,i )(1−βξm,i )ξm,i (1+κm,iβ) (mcm,i

t + λm,it ),

7. mcm,it =−mcx

t − γx,∗t − γmi ,d

t ,

8. πxt = β

1+κxβπx

t+1 + κx1+κxβ

πxt−1 +

(1−ξx )(1−βξx )ξx (1+κxβ) (mcx

t + λxt ),

9. wt =− 1η1

[η0wt−1 +η2wt+1 +η3πt +η4πt+1 +η5π

ct−1 +η6π

ct +η7ψz,t

+η8Ht +η9τyt

]+ ζht ,

10. kt+1 = 1−δµz

(kt − µz,t )+ (1− 1−δµz

)(it + Υt ),

11. ct = 1µ2

z+b2β(bβµz ct+1 +bµz ct−1 −bµz (µz,t −βµz,t+1)

+(µz −bβ)(µz −b)ψz,t + τc

(1+τc ) (µz −bβ)(µz −b)τct + (µz −bβ)(µz −b)γc,d

t

−(µz −b)(µz ζct −bβζc

t+1)),

12. it = µ2z F ′′

(µ2z F ′′)(1+β)

(it−1 +βit+1 − µz,t +βµz,t+1 + Pk,t − γi ,dt + Υt ),

13. ψz,t + µz,t+1 − ψz,t+1 − β(1−δ)µz

Pk,t+1 + Pk,t − µz−β(1−δ)µz

r kt+1 + τk

(1−τk )µz−β(1−δ)

µzτk

t+1,

14. ∆St+1 = Rt − R∗t +φa at + ˆφt ,

15. (1−ωc )(γc,d )ηc cy (ct +ηc γ

c,dt )+ (1−ωi )(γi ,d )ηi i

y (it +ηi γi ,dt )

+ gy g t + y∗

y (y∗t −η f γ

x,∗t + ˆz∗

t ) =λd (εt +α(kt − µz,t )+ (1−α)Ht ),

16. at = y∗(−mcxt −η f γ

x,∗t + y∗

t + ˆz∗t )+ (cm + i m)γ f

t

−cm(−ηc (1−ωc )(γc,d )ηc−1γmc,dt + ct )

−i m(−ηi (1−ωi )(γi ,d )ηi−1γmi ,dt + it )+ R

πµzat−1,

17. Rt = ρR Rt−1 + (1−ρR )(κππct−1 +κy yt−1 +κs∆St ),

18. γc,dt =ωc (γmc,c )(1−ηc )γmc,d ,

19. γi ,dt =ωi (γmi ,c )(1−ηi )γmi ,d ,

20. γmc,dt = γmc,d

t−1 πm,ct − πd

t ,

21. γmi ,dt = γmi ,d

t−1 πm,it − πd

t ,

22. γx,∗t = γx,∗

t−1 + πxt − π∗

t ,

23. mcxt = mcx

t−1 + πt − πxt −∆St ,

24. γft = mcx

t + γx,∗t ,

25.−37 xt+1 = ρx xt +σxεxt+1,

38.−40. Xt = A1Xt−1 +A2Xt−2 +A3Xt−3 +A4Xt−4.


The expectation operator has been omitted for notational simplicity. The relative

prices are defined as

γc,dt = P c

t

Pt,

γi ,dt = P i

t

Pt,

γmc,dt = P m,c

t

Pt,

γmi ,dt = P m,i

t

Pt,

γx,∗t = P x

t

P∗t

,

γft = Pt

St P∗t

.

Appendix B: Model consistent data measurement

For the measurement equations, some of the model variables are rewritten to be

consistent with the data. For example, consider consumption which is CES aggregated

in the model but Ct ≡C dt +C m

t in the data. This implies that (using the consumption

demands in (3.8))

Ct = (1−ωc )

(Pt

P ct

)−ηc

Ct +ωc

(P m,c

t

P ct

)−ηc

Ct .

Log-linearizing the stationary relationship yields

ct = cd

cd + cm(ηc γ

c,dt + ct )+ cm

cd + cm(−ηc (1−ωc )(γc,d )ηc−1γmc,d

t + ct ).

A similar measurement equation can be constructed for investment (It = I dt + I m

t ).

Imports are given as Mt ≡C mt + I m

t . Using (3.8) and (3.9) this relates to the model in

the following way

Mt =ωc

(P m,c

t

P ct

)−ηc

Ct +ωi

P m,it

P it

−ηi

It .

Log-linearizing the stationary relationship yields

mt = cm

cd + cm(−ηc (1−ωc )(γc,d )ηc−1γmc,d

t + ct )

+ im

id + im(−ηi (1−ωi )(γi ,d )ηi−1γmi ,d

t + it ).

3.8. CONCLUSION 115

For the consumer price index, the data definition is given as P ct ≡ Pt C d

t +P m,ct C m

t

C dt +C m

t.

This implies the following relationship with the stationarized and log-linearized

model

ˆπct =

cd

cd +λmc cmπt + λmc cm

cd +λmc cmπm,c

t

+((

cd

cd +λm,c cm− cd

cd + cm)ηcωc (γc,mc )ηc−1

− (λm,c cm

cd +λm,c cm− cm

cd + cm)ηc (1−ωc )(γc,d )ηc−1

)(γmc,d

t − γmc,dt ).

Appendix C: Measurement equations

yt =

∆lnY d at at

∆l nC d at at

∆l nI d at at

∆lnX d at at

∆lnM d at at

∆ln(

WtPt

)d at a

πy,d at at

πc,d at at

Rd at at

Y ∗,d at at

π∗,d at at

R∗,d at at

=

µz −1

µz −1

µz −1

µz −1

µz −1

µz −1

π−1

π−1

0

0

0

0

+

µz,t + yt − yt−1

µz,t + ˆct − ˆct−1

µz,t + ˆit − ˆit−1

µz,t + xt − xt−1

µz,t + ˆmt − ˆmt−1

µz,t + wt − wt−1

πtˆπc

t

Rt

y∗t

π∗t

R∗t

+ut .


Table 3.1. Calibrated parameters

Parameter Description Valueβ Discount factor 0.99δ Capital depreciation rate 0.025α Capital share in production 0.30ηc Substitution elasticity of consumption 2.50µz Mean growth rate 1.01π Mean inflation rate 1.005AL Constant in disutility of labor 7.5σL Elasticity of labor supply 1.00λw Wage markup 1.05ωc Imported consumption share 0.40ωi Imported investment share 0.40

0 5 10 15 20

xt+

h

×10-4

02468

GDP

0 5 10 15 20

×10-4

02468

Real wages

0 5 10 15 20

xt+

h

×10-4

02468

Consumption

0 5 10 15 20

×10-4

0

5

10

15Investment

0 5 10 15 20

xt+

h

×10-4

-2024

Imports

0 5 10 15 20

×10-4

0

2

4Exports

Impulse horizon, h0 5 10 15 20

xt+

h

×10-4

-6-4-20

GDP deflator


×10-4

-4

-2

0

CPI

Figure 3.1. Impulse responses: Productivity shock εtThe impulse respones are shown in log-deviations from the steady state, xt = l og xt − log xss ,

and show the response of an increase in εt of one standard deviation, based on the posterior

mode.

3.8. CONCLUSION 117

Table 3.2. Prior and posterior distributions of estimated parameters

The table shows the prior distributions and posterior estimates based on the initial sample

1990:Q2-2001:Q4.

Prior distribution Posterior distribution

Parameter Density Param 1 Param 2 Mode Mean 5 pct. 95 pct.

ξw Beta 0.750 0.050 0.7569 0.7493 0.6687 0.8323

ξd Beta 0.750 0.050 0.6980 0.6956 0.6107 0.7865

ξm,c Beta 0.750 0.050 0.7646 0.7589 0.6948 0.8235

ξm,i Beta 0.750 0.050 0.8218 0.8171 0.7624 0.8729

ξx Beta 0.750 0.050 0.8243 0.8167 0.7601 0.8726

κw Beta 0.250 0.050 0.2429 0.2513 0.1684 0.3312

κd Beta 0.250 0.050 0.1949 0.2020 0.1311 0.2717

κm,c Beta 0.250 0.050 0.2050 0.2119 0.1404 0.2860

κm,i Beta 0.250 0.050 0.2194 0.2256 0.1485 0.2991

κx Beta 0.250 0.050 0.2294 0.2339 0.1558 0.3083

λd Inverse gamma 1.200 0.050 1.1949 1.1967 1.1166 1.2794

λm,c Inverse gamma 1.200 0.050 1.0221 1.0365 1.0001 1.0699

λm,i Inverse gamma 1.200 0.050 1.1949 1.2006 1.1186 1.2829

F ′′ Normal 5.000 1.500 5.5635 5.7574 3.8295 7.6838

b Beta 0.5000 0.100 0.5664 0.5485 0.3917 0.7108

ηi Inverse gamma 1.500 2.000 1.3665 1.4679 0.8411 2.0586

η f Inverse gamma 1.500 2.000 1.2236 1.2421 0.9760 1.5052

ρλd Beta 0.250 0.100 0.1018 0.1330 0.0358 0.2274

ρλm,c Beta 0.250 0.100 0.1006 0.1257 0.0325 0.2111

ρλm,i Beta 0.250 0.100 0.1715 0.2068 0.0628 0.3457

ρλx Beta 0.250 0.100 0.2601 0.2819 0.1161 0.4428

ρµz Beta 0.500 0.100 0.4212 0.4268 0.2691 0.5790

ρε Beta 0.500 0.100 0.4969 0.4922 0.3305 0.6536

ρΥ Beta 0.500 0.100 0.3804 0.3903 0.2312 0.5440

ρ z∗ Beta 0.500 0.10 0.5101 0.5095 0.3512 0.6639

ρζc Beta 0.500 0.100 0.4059 0.4231 0.2669 0.5757

ρζh Beta 0.500 0.100 0.3594 0.3616 0.2269 0.4975

ρφ Beta 0.850 0.050 0.8917 0.8801 0.8161 0.9465


Table 3.3. Prior and posterior distributions of estimated parameters

The table shows the prior distributions and posterior estimates based on the initial sample

1990:Q2-2001:Q4.

Prior distribution Posterior distribution

Parameter Density Param 1 Param 2 Mode Mean 5 pct. 95 pct.

σλd Inverse Gamma 0.01 0.001 0.0094 0.0096 0.0083 0.0109

σλm,c Inverse Gamma 0.01 0.001 0.0126 0.0130 0.0111 0.0148

σλm,i Inverse Gamma 0.05 0.001 0.0506 0.0506 0.0490 0.0523

σλx Inverse Gamma 0.01 0.001 0.0144 0.0146 0.0123 0.0168

σµz Inverse Gamma 0.005 0.001 0.0047 0.0052 0.0035 0.0069

σε Inverse Gamma 0.01 2 0.0045 0.0069 0.0024 0.0118

σΥ Inverse Gamma 0.01 2 0.0261 0.0270 0.0207 0.0335

σz∗ Inverse Gamma 0.01 2 0.0026 0.0028 0.0019 0.0036

σζc Inverse Gamma 0.01 2 0.0091 0.0098 0.0066 0.0128

σζh Inverse Gamma 0.01 2 0.0037 0.0037 0.0027 0.0047

σφ Inverse Gamma 0.01 2 0.0017 0.0018 0.0014 0.0022

3.8. CONCLUSION 119

Tab

le3.

4.U

nco

nd

itio

nal

seco

nd

mo

men

ts

All

vari

able

sar

esh

own

inq

uar

terl

ygr

owth

rate

s.T

he

mo

men

tso

fth

eD

SGE

mo

del

com

pu

ted

bas

edo

nth

ep

ost

erio

rm

od

e.T

he

dat

ase

tuse

d

for

this

esti

mat

ion

isth

ein

itia

lsam

ple

,199

0:Q

2-2

001:

Q4.

Dat

aM

od

elV

aria

ble

Std

.dev

.A

uto

corr

elat

ion

Co

rr(∆

Yt)

Std

.dev

.A

uto

corr

elat

ion

Co

rr(∆

Yt)

∆ln

Yt

1.26

0.36

1.00

1.83

0.35

1.00

∆ln

( Wt

Pt

)0.

680.

06-

0.14

1.39

0.24

0.34

∆ln

Ct

1.21

0.10

0.33

1.41

0.42

0.73

∆ln

I t7.

280.

240.

626.

930.

350.

58∆

lnM

t2.

340.

120.

553.

080.

280.

23∆

lnX

t3.

060.

320.

542.

640.

240.

60π

d t0.

800.

63-

0.43

1.12

0.37

-0.

38π

c t0.

580.

070.

080.

850.

35-

0.12


Table

3.5.1-stepah

eadfo

recasterror

variance

deco

mp

ositio

n

Th

ed

ecom

po

sition

isco

mp

uted

based

on

the

po

sterior

mo

de

ofth

ep

arameters.M

easurem

enterro

rsh

aveb

eenexclu

ded

.

GD

PR

ealwages

Co

nsu

mp

tion

Investm

ent

Imp

orts

Exp

orts

GD

Pd

eflato

rC

PI

Fiscal

1.4780.004

0.0510.000

0.0010.000

0.0100.006

Do

mestic

marku

p11.351

49.18911.474

0.5662.783

0.16386.519

49.613

Imp

orted

con

sum

ptio

nm

arkup

9.6360.064

3.6210.399

2.7620.008

0.45940.822

Imp

orted

investm

entm

arkup

5.1730.016

9.5870.092

35.9210.008

0.2290.131

Exp

ortm

arkup

15.1470.113

1.6640.095

0.41275.231

0.5230.300

Pro

du

ctivity5.495

10.22912.171

0.3672.900

2.9111.303

0.747

Investm

entsp

ecific

30.8082.731

6.26297.572

53.8910.402

2.9201.675

Co

nsu

mp

tion

preferen

ce1.546

0.02429.768

0.0670.437

0.0000.019

0.011

Labo

rp

reference

3.25537.464

8.3100.352

0.0240.192

6.3713.653

Foreign

16.1100.164

17.0920.490

0.87121.084

1.6483.042

3.8. CONCLUSION 121

Tab

le3.

6.8-

step

ahea

dfo

reca

ster

ror

vari

ance

dec

om

po

siti

on

Th

ed

eco

mp

osi

tio

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s.M

easu

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ave

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ded

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PR

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ages

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nsu

mp

tio

nIn

vest

men

tIm

po

rts

Exp

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DP

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ato

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Fis

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1.65

20.

004

0.04

20.

000

0.00

10.

000

0.00

80.

005

Do

mes

tic

mar

kup

10.2

2341

.668

9.45

50.

693

3.69

50.

269

64.9

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.109

Imp

ort

edco

nsu

mp

tio

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p8.

249

0.51

52.

894

0.62

63.

799

0.02

70.

664

31.5

38

Imp

ort

edin

vest

men

tmar

kup

4.77

30.

019

8.24

80.

483

32.1

230.

035

0.30

40.

188

Exp

ort

mar

kup

14.0

280.

158

1.36

70.

136

0.53

072

.250

0.70

60.

436

Pro

du

ctiv

ity

7.44

013

.972

10.6

810.

874

4.81

04.

401

1.94

41.

200

Inve

stm

ents

pec

ific

30.6

0212

.478

17.0

2395

.308

53.2

762.

486

16.4

3410

.147

Co

nsu

mp

tio

np

refe

ren

ce1.

820

0.02

924

.368

0.11

40.

615

0.00

00.

023

0.01

4

Lab

or

pre

fere

nce

5.95

930

.891

10.0

991.

056

0.21

30.

797

8.99

55.

554

Fore

ign

15.2

540.

267

15.8

240.

712

0.93

519

.734

5.96

310

.809


0 5 10 15 20

xt+

h

×10-3

-6

-4

-2

0GDP

0 5 10 15 20

×10-3

-8-6-4-20

Real wages

0 5 10 15 20

xt+

h

×10-3

-4

-2

0Consumption

0 5 10 15 20

×10-3

-8-6-4-20

Investment

0 5 10 15 20

xt+

h

×10-3

-2024

Imports

0 5 10 15 20

×10-3

-2

-1

0Exports


xt+

h

×10-3

-202468

GDP deflator


×10-3

0

2

4

CPI

Figure 3.2. Impulse responses: Markup shock λdt

The impulse respones are shown in log-deviations from the steady state, xt = l og xt − log xss ,

and show the response of an increase in λt of one standard deviation, based on the posterior

mode.

3.8. CONCLUSION 123

0 5 10 15 20

xt+

h

×10-3

0

1

2

GDP

0 5 10 15 20

×10-4

-2024

Real wages

0 5 10 15 20

xt+

h

×10-3

0

1

2

Consumption

0 5 10 15 20

×10-3

0

2

4

Investment

0 5 10 15 20

xt+

h

×10-3

0

1

2

Imports

0 5 10 15 20

×10-3

0

2

4Exports


xt+

h

×10-4

-1012

GDP deflator


×10-4

-2

0

2

CPI

Figure 3.3. Impulse responses: Foreign output shock, Y ∗t

The impulse respones are shown in log-deviations from the steady state, xt = log xt − l og xss ,

and show the response of an increase in Y ∗t of one standard deviation, based on the posterior

mode.


1 2 3 4 5 6 7 8

RMSE(h)

1.2

1.4

1.6

1.8

2

2.2

2.4

GDP

DSGE AR(1) Random Walk

1 2 3 4 5 6 7 8

0.5

0.6

0.7

0.8

0.9

1

1.1

Wages

Forecast horizon, h1 2 3 4 5 6 7 8

RMSE(h)

1.61.82

2.22.42.62.83

3.2Consumption


7

8

9

10

11

12Investment

Figure 3.4. Out-of-sample predictive performanceThe RMSEs are shown for forecasting horizons h = 1, ...,8 and are based on yearly growth rates.

The recursive out-of-sample forecasts have been conducted for the period 2002Q1-2008Q2.

3.8. CONCLUSION 125

1 2 3 4 5 6 7 8

RMSE(h)

4

5

6

7

8

Imports

DSGE AR(1) Random Walk

1 2 3 4 5 6 7 82.53

3.54

4.55

5.56

6.5Exports


RMSE(h)

0.550.60.650.70.750.80.850.9

GDP deflator


0.350.40.450.50.550.60.650.70.75

CPI

Figure 3.5. Out-of-sample predictive performanceThe RMSEs are shown for forecasting horizons h = 1, ...,8 and are based on yearly growth rates.

The recursive out-of-sample forecasts have been conducted for the period 2002Q1-2008Q2.


2007Q1 Q2 Q3 Q4 2008Q1 Q2 Q3 Q4 2009Q1 Q2 Q3 Q4 2010Q1 Q2 Q3 Q4-8-6-4-20246

DSGE

Yearly GDP growthForecast

2007Q1 Q2 Q3 Q4 2008Q1 Q2 Q3 Q4 2009Q1 Q2 Q3 Q4 2010Q1 Q2 Q3 Q4-8

-6

-4

-2

0

2

AR(1)

Figure 3.6. Forecasting Danish GDP growth during the Great RecessionThe figure depicts historical values of real GDP growth in Denmark. Further, the sequential

forecasts for h = 1, ...,8 are shown for the DSGE model (upper row) and the AR(1) model (lower

row).

3.8. CONCLUSION 127

2007Q1 Q2 Q3 Q4 2008Q1 Q2 Q3 Q4 2009Q1 Q2 Q3 Q4 2010Q1 Q2 Q3 Q4

-0.15

-0.1

-0.05

0

0.05

Output gapFiscalMarkupsCapacityDomestic demandForeign

Figure 3.7. Historical shock decomposition of outputThe figure depicts the historical filtered values of deviations from the steady state of real output,

yt , as well as a decomposition of the structural shock contributions. The decomposition is

computed based on the posterior mode of the parameters and measurement errors have been

excluded.

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