bayesian estimation of a dsge model for norway

Bayesian Estimation of a DSGE Model

for Norway

Monetary Policy and the Exchange Rate in a Small Open Economy

Christian Presterud

Thesis submitted for the degree of

Master of Economic Theory and Econometrics

30 credits

Department of Economics

Faculty of Social Sciences

University of Oslo

Spring 2020

Bayesian Estimation of a DSGE Model for Norway

Monetary Policy and the Exchange Rate in a Small Open Economy

Christian Presterud

© 2020 Christian Presterud

Bayesian Estimation of a DSGE Model for Norway

http://www.duo.uio.no

Printed: Reprosentralen, University of Oslo

Abstract

This thesis investigates the welfare effects of using the real exchange rate as a deciding

factor in the conduction of monetary policy in a small open economy. A New Keynesian

dynamic stochastic general-equilibrium model for a small open economy is presented.

A baseline monetary policy that respond to the inflation rate and output is compared

to monetary polices that additionally respond to the real exchange rate. The dynamic

responses of the economy to various shocks are examined, and using a loss function the

welfare loss of the economy is estimated in order to compare the policies. Prior information

in the literature together with Norwegian quarterly data from 1995Q1 to 2019Q4 are

combined using Bayesian methods to estimate the parameters of the model.

The results shows that including the real exchange rate as a deciding factor in the

reaction function of the central bank reduces the welfare loss introduced by various shocks.

Exchange rate intervention reduces observed volatility in output, inflation and the interest

rate. A small open economy should use exchange rate intervention instead of letting its

currency float freely.

i

Preface

This thesis marks the end of my five years as a student at the Department of Economics,

University of Oslo. I am grateful for all the experiences and knowledge that have followed

through.

I would like to thank my supervisor professor Ragnar Nymoen for invaluable sugges-

tions and comments, as well as always leaving the door to the office open to other engaging

discussion of various topics. It has been an absolute honor.

For two years I worked as a research assistant at the Depart of Economics. I would

like to thank my former colleagues at the department, as well as all my fellow students

for an unforgettable five years.

Most of the coding has been done i Matlab/Dynare. The code can be made available

upon request.

All remaining errors are my own.

Oslo, May 2020

Christian Presterud

[email protected]

ii

Contents

Abstract i

Preface ii

List of Figures vii

List of Tables ix

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The theoretical model 6

2.1 Household problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Terms of trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 The real exchange rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Uncovered interest rate parity . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Foreign consumption, inflation and the interest rate . . . . . . . . . . . . . 13

2.6 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6.1 Production technology and marginal cost . . . . . . . . . . . . . . . 14

2.6.2 Calvo-type price setting of domestic firms . . . . . . . . . . . . . . 15

2.7 Incomplete pass-through and imported goods . . . . . . . . . . . . . . . . . 17

2.8 Import and export sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.9 Monetary policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.9.1 Monetary policy in Norway . . . . . . . . . . . . . . . . . . . . . . 19

2.9.2 A baseline monetary policy . . . . . . . . . . . . . . . . . . . . . . 19

2.9.3 A monetary policy with exchange rate . . . . . . . . . . . . . . . . 20

2.10 Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 The log linearized model 22

3.1 Basic principles of log linearization . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Important identities . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Log linearizing household conditions . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Labor-leisure choice . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

iii

3.2.2 Consumption Euler equation . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Log linearizing terms of trade, domestic inflation and CPI inflation . . . . 25

3.3.1 Terms of trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.2 Consumer price index . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Log linearizing the real exchange rate . . . . . . . . . . . . . . . . . . . . . 26

3.5 Log linearizing the law of one price gap . . . . . . . . . . . . . . . . . . . . 27

3.6 Log linearizing uncovered interest rate parity . . . . . . . . . . . . . . . . . 28

3.7 Log linearizing foreign consumption, inflation and the interest rate . . . . . 29

3.8 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.8.1 Log linearizing production technology . . . . . . . . . . . . . . . . . 30

3.8.2 Log linearizing marginal cost . . . . . . . . . . . . . . . . . . . . . . 30

3.8.3 Log linearizing optimal price setting . . . . . . . . . . . . . . . . . . 31

3.9 Log linearizing imports and exports . . . . . . . . . . . . . . . . . . . . . . 31

3.10 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.10.1 Demand and output . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.10.2 Inflation dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.11 Log linearizing monetary policy . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Calibration 36

4.1 Steady state and observed variables . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 Growth variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.2 Linear detrending . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Non-growth variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.1 Priors for the households . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.2 Priors for the firms . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.3 Priors for the foreign economy . . . . . . . . . . . . . . . . . . . . . 40

4.3.4 Shock processe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.5 Priors for the monetary policy . . . . . . . . . . . . . . . . . . . . . 42

5 Estimation 44

5.1 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.1.1 An introduction to Bayesian estimation . . . . . . . . . . . . . . . . 45

iv

5.1.2 Bayesian estimation and DSGE models . . . . . . . . . . . . . . . . 47

5.2 Posterior distributions of the parameters . . . . . . . . . . . . . . . . . . . 51

5.3 Simulation and impulse response functions . . . . . . . . . . . . . . . . . . 52

6 Results 56

7 Concluding remarks 59

8 Extension 60

A Appendix A 61

A.1 Constant elasticity of substitution (CES) . . . . . . . . . . . . . . . . . . . 62

A.2 Optimal consumption of households . . . . . . . . . . . . . . . . . . . . . . 63

A.3 Solving the household problem . . . . . . . . . . . . . . . . . . . . . . . . . 65

A.4 Obtaining the uncovered interest rate parity condition . . . . . . . . . . . . 68

A.5 Finding the optimal price setting . . . . . . . . . . . . . . . . . . . . . . . 69

B Appendix B 70

B.1 Log linearizing household conditions . . . . . . . . . . . . . . . . . . . . . 71

B.2 Log linearizing terms of trade, domestic inflation and CPI inflation . . . . 71

B.2.1 Terms of trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

B.2.2 CPI inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

B.3 Log linearizing the uncovered interest parity condition . . . . . . . . . . . . 72

B.4 Log linearizing optimal price . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B.5 Log linearizing the inflation dynamics . . . . . . . . . . . . . . . . . . . . . 76

C Appendix C 78

C.1 Priors and posteriors monetary policy 1 . . . . . . . . . . . . . . . . . . . . 79

C.2 Priors and posteriors, monetary policy 2 . . . . . . . . . . . . . . . . . . . 81

C.3 Priors and posteriors monetary policy 3 . . . . . . . . . . . . . . . . . . . . 83

D Appendix D 85

D.1 Impulse responses monetary policy 1 . . . . . . . . . . . . . . . . . . . . . 86

D.2 Impulse responses monetary policy 2 . . . . . . . . . . . . . . . . . . . . . 91

D.3 Impulse reponses monetary policy 3 . . . . . . . . . . . . . . . . . . . . . . 96

v

E Appendix E 101

E.1 Variances monetary policy 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 102



F Appendix F 105

F.1 Posterior results monetary policy 1 . . . . . . . . . . . . . . . . . . . . . . 106



G Appendix G 112

G.1 Data description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

G.2 Plots of the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

vi

List of Figures

5.2.1 Priors and posteriors for a selection of structural parameters, Monetary

Policy 1, (MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3.1 Impulse response functions to a shock in the real exchange rate, monetary

policy 1, (MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


policy 2, (MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


policy 3, (MP3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

C.1.1 Priors and posteriors, monetary policy 1, (MP1). . . . . . . . . . . . . . . 79












D.1.1 Impulse response functions to a shock in the terms of trade, monetary

policy 1, (MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

D.1.2 Impulse response functions to a shock in the real exchange rate, monetary

policy 1, (MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

D.1.3 Impulse response functions to a shock in the law of one price gap, mone-

tary policy 1, (MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

D.1.4 Impulse response functions to a shock in the import inflation, monetary

policy 1, (MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

D.1.5 Impulse response functions to a shock in the domestic inflation, monetary

policy 1, (MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

vii

D.1.6 Impulse response functions to a shock in the domestic interest rate, mon-

etary policy 1, (MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

D.1.7 Impulse response functions to a productivity shock, monetary policy 1,

(MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

D.1.8 Impulse response functions to a shock in the foreign output, monetary

policy 1, (MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

D.1.9 Impulse response functions to a shock in the foreign interest rate, mone-

tary policy 1, (MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

D.1.10Impulse response functions to a shock in the foreign inflation, monetary

policy 1, (MP1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90


policy 2, (MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91


policy 2, (MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91


tary policy 2, (MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

D.2.4Impulse response functions to a shock in the import inflation, monetary

policy 2, (MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


policy 2, (MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


etary policy 2, (MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


(MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


policy 2, (MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


tary policy 2, (MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

D.2.10Impulse response functions to a shock in the foreign interest rate, mone-

tary policy 2, (MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


policy 3, (MP3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

viii


policy 3, (MP3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96


tary policy 3, (MP3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

D.3.4 Impulse response functions to a shock in the import inflation, monetary

policy 3, (MP3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


policy 3, (MP3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98


etary policy 3, (MP3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98


(MP3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99


policy 3, (MP3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99


tary policy 3, (MP3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

D.3.10Impulse response functions to a shock in the foreign inflation, monetary

policy 3, (MP3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

G.2.1Plots of the data used in the model . . . . . . . . . . . . . . . . . . . . . . 115

List of Tables

4.3.1 Priors for the households . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.2 Priors for the firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.3 Priors for the foreign economy . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.4 Shocks to endogenous variables . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.5 Shocks to exogenous variables . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.6 Priors for the autocorrelation parameters . . . . . . . . . . . . . . . . . . . 42

4.3.7 Priors for the shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.8 Priors for monetary policy rule 1 (MP1) . . . . . . . . . . . . . . . . . . . 43



ix

6.0.1 Welfare loss for the different policies when there are shocks to domestic

variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.0.2 Welfare loss for the different policies when there are shocks to foreign variables 57

E.1.1The variance of the average response function, monetary policy 1 (MP1) . 102



F.1.1Posterior results of the structural parameters, monetary policy 1 . . . . . . 106

F.1.2Posterior results of the shocks, monetary policy 1 . . . . . . . . . . . . . . 107





G.1.1Data description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

x

1 Introduction

Over the last decades, understanding monetary economics has been a among the most

central areas within macroeconomics. The effort of many researchers within the field has

been to develop a framework to understand the relationship between monetary policy,

inflation, and the business cycle in order to implement an efficient monetary policy strat-

egy. When conducting monetary policy, a key concern for the central banks are usually

to keep low and stable inflation.1 How to achieve this is not straight forward, and is likely

to differ across countries. Small open economies (SOE) import a larger amount of goods

relative to GDP than larger economies. As a result, a large number of papers has tried

to clarify how to successfully implement monetary policy in small open economies, see

e.g. Batini et al. (2003), Ball (1999a) and Leitemo and Soderstrom (2001). Of particular

interest is the exchange rate, and how movements in the exchange rate affects central

banks in its policymaking process, see e.g. Taylor (2001).

A key variable in a small open economy is the exchange rate. Through this channel

the economy is exposed to fluctuations in international markets. External shocks in

international markets generates fluctuations which can alter the real exchange rate and

thus affect the overall price level in the economy through the prices on imported goods

used in both consumption and production. As a result, the expected path of domestic

inflation might change and the central bank has to adjust accordingly.

In this thesis, a Dynamic Stochastic General Equilibrium (DSGE) model for a small

open economy is specified. Bayesian techniques are used to estimate all equations and

shocks simultaneously. The model is estimated using Norwegian historical time series

data, and the goal is to analyse how the monetary policy reacts when the model is exposed

to external shocks. Impulse response functions are used in order to give an insight in the

dynamics of the model and how structural variables behave when exposed to different

shocks.

Four questions are addressed. 1) When deciding the monetary policy strategy for a

SOE, what are the welfare effects of using the real exchange rate as an argument in the

decision functions? 2) Should central banks in SOE with floating currencies use exchange

rate intervention? 3) When should central banks use exchange rate interventions, i.e, in

response to which shocks? In order to investigate these questions, three different monetary

1The central bank of Norway, Norges Bank, conducts monetary policy to keep inflation around 2%.

1

policies are compared. A baseline monetary policy that allows the currency to float freely

and responds to output and inflation. A second monetary policy that additionally respond

to the exchange rate level. Finally, a third monetary policy that additionally respond to

the rate of change in the exchange rate. 4) How do these different monetary policies

compare in terms of welfare loss?

The rest of the thesis is organised as follows. Section 1.1 provides general background

information and previous studies. In section 2 the small open economy model is formu-

lated, and section 3 presents the log linearized version of the model. In section 4 the model

is calibrated and the data is presented. Section 5 presents basics on Bayesian estimation

and the estimation process of the model. Results of the estimation are presented in sec-

tion 6. Section 7 concludes, and section 8 presents possible extensions of the theoretical

framework presented here.

2

1.1 Background

During the 80s the Real Business Cycle (RBC) theory following the papers of Kydland

and Prescott (1982) and Prescott (1986) became the main framework for analysing eco-

nomic fluctuations and the core of modern macroeconomics. The RBC theory established

the use of DGSE models as foundation for macroeconomic analysis. Earlier used behavior

equations describing the aggregate variables was replaced by formal optimality condi-

tions from intertemporal optimization problems faced by consumers and firms. Rational

expectations replaced earlier more ad hoc assumptions about expectations.

Although the RBC framework had a major impact among researchers, it had very

limited impact on central banks due to its lack of reference to monetary factors. Solutions

to this problem where attempted in the literature, see e.g. Cooley and Hansen (1989).

They introduced a monetary sector in the conventional RBC framework while keeping

the assumptions on nominal wages and flexible prices. However, this approach did not

yield a framework that was relevant for policy analysis. Often referred to as the classical

monetary model, the framework predicts near neutrality of monetary policy with respect

to real variables, which is not in line with the common beliefs about monetary policy’s

power (at least short run power) to affect output and developments in the employment

sector. Such beliefs are also supported by empirical work, see e.g Christiano et al. (1999),

or a more narrative evidence from Friedman and Schwartz (1963).

The shortcomings of the classical monetary model are the main motivation behind

the introduction of short run Keynesian assumptions, such as monopolistic competition,

nominal rigidities following Calvo (1983) staggered price-setting theory and Taylor (1980)

staggered wage contracts, and short run non-neutrality of monetary policy. This frame-

work is often referred to as the New Keynesian model or new-neoclassical synthesis, see

e.g Clarida et al. (1999) and Gali and Gertler (2007).

It is worth mentioning that these three assumptions were already in the making in

the 1970s, and developed alongside the RBC theory in the 1980s. However, the earlier

framework was put under pressure by the famous Lucas critique, following the papers

of Lucas (1976) and Lucas and Sargent (1979). Who argued that private agents should

adjust their supposed behavior in response to policy announcements when they behave

according to a dynamic optimization approach and use all available information rationally.

The earlier framework were often static and used reduced form equilibrium conditions not

3

stemming from dynamic optimization problems faced by firms and households, and thus

subjected to the critique. However, as mentioned above, DSGE models are based on

dynamic optimizing firms and households and are therefor less susceptible to the critique.

The modern New Keynesian framework has become the fundamental in monetary

policy analysis. It has earned great popularity and grown with tremendous speed in

both research and central banks. In recent years, there has been a growing interest for

open economy macroeconomic models building on the New Keynesian framework. Among

some institutions that have developed such an open economy framework are Norges Bank,

Sveriges Riksbank, US federal reserve, IMF and the European Central Bank, see e.g. Tovar

(2009).

As mentioned above, much of the literature on small open economies focuses on

whether the central banks respond to the exchange rate or not. The literature is however

not conclusive.

Lubik and Schorfheide (2007) estimates a small-scale structural general equilibrium

model of a small open economy using Bayesian methods. Their main focus is on the

conduction of monetary policy in Australia, Canada, New Zealand and UK. Generic

Taylor-type rules are considered, where monetary policy reacts to output, inflation and

exchange rate movements. Results shows that the monetary authorities in Canada and

UK include nominal exchange rate in their policy rule, while Australia and New Zealand

do not.

Collaborative results can be found in Bjoernland (2009) and Bjoernland and Halvorsen

(2014). Structural vector autoregressive (SVAR) models that allow for simultaneous re-

sponse between monetary policy and exchange rate changes are considered. They apply

their analysis to six open economies with floating currencies; Australia, Canada, New

Zealand, Norway, Sweden and UK. When doing so, they find an instantaneous reaction

in the exchange rate following a monetary policy shock in all six countries. However,

monetary policy respond to exchange rate shocks in four of the six countries; Canada,

New Zealand, Norway and Sweden. This suggest that exchange rate movements are not

equally important for monetary policy setting in all countries, but accounting for potential

interaction is still important when identifying monetary policy shocks in open economies

structural VARs.

Garcia and Gonzalez (2010) presents a DSGE model for small open economies and es-

4

timate how monetary policy work with an inflation targeting central bank using Bayesian

methods. They find that risk premium shocks (shocks from international financial mar-

kets) explain most of the variance in the exchange rate. In the paper they also show

that in order to reduce the observed volatility of the inflation rate and the output gap,

exchange rate intervention is necessary.

Contradictory results can be found in e.g. Bergin et al. (2007), Ball (1999b) and

Svensson (2000). Bergin et al. (2007) provides a welfare analysis which produce contradi-

cotory results depending on the model used. Ball (1999b), Svensson (2000) and Batini

et al. (2003) find that having monetary authorities respond to the real exchange rate

marginally improves performance of the central banks.

5

2 The theoretical model

The economy is described by a New Keynesian dynamic stochastic general-equilibrium

model (NKMDSGE) for a small open economy (SOE). As a foundation for the small open

economy model, earlier litterateur on the subject is used.2 A small open economy is an

economy participating on the international market, but is very small compared to the rest

of the economies. As a results, its policies has no effect on world prices, interest rates or

incomes.

The home country, i.e, the small open economy, consists of a continuum of households,

a continuum of firms, and an inflation-targeting central bank. Government spending is

exogenous and financed purely through lump-sum taxes in each period, such that fiscal

policy is passive. Goods markets feature monopolistic competition, labour market features

perfect competition and the capital stock is assumed to be fixed. In the economy, there

also exists an import/export sector. The real exchange rate is modelled through the

uncovered interest rate parity condition.

The firms uses labour services supplied by individual households together with Cobb-

Douglas technology to produce a differentiated intermediate good, which gives firms mar-

ket power. Firms set prices according to the Calvo (1983) staggered price-setting theory.

Given demand and wages, each firm choose factor inputs to minimize its cost and maxi-

mize its profits. Firms are assumed to be engaged in international price discrimination and

goods are priced in the currency of the country they are sold. This allows for incomplete

pass-through of exchange rate developments into prices in the short run.

Households consume a bundle of both foreign and domestic intermediate goods pro-

duced by the individual firms. In each period the households must decide how much to

consume of the final good, how much to invest in domestic bonds (in domestic currency),

how much to invest in foreign bonds (in foreign currency). Taking wages as given, it

chooses how much labor services to supply.

2Earlier literature are e.g. Gali and Monacelli (2005), Monacelli (2005).

6

2.1 Household problem

Consider an economy consisting of many identically, infinitely-lived households, with mea-

sure normalized to one. A representative household3 has an instantaneous and time sep-

arable utility function

Ut =C1−σt

1− σ−N

1+ψt

1 + ψ. (2.1.1)

Where the private consumption level at time t is denoted Ct, Nt represents labor at time

t, and Et is the expectation operator. σ > 0 is the curvature of the utility function, or

the inverse elasticity of intertemporal substitution (EIS). It measures the responsiveness

of consumption to changes in the interest rate. ψ > 0 is the curvature of labor disutility,

or the inverse Frisch-elasticity. It governs how elastic labor supply is to changes in the

wages. The representative household maximize lifetime utility and discounts the future

proportionally by a factor β

Et

{

∞∑

t=0

βtUt(Ct, Nt)}

β ∈ (0, 1). (2.1.2)

Private consumption is a composite consumption index consisting of both home and for-

eign goods

Ct ≡[

(1− ǫB)1

ǫH CǫH−1

ǫH

Ht + ǫ1

ǫH

B CǫH−1

ǫH

Ft

]

ǫHǫH−1

. (2.1.3)

Where ǫB ∈ [0, 1] measures the degree of openness in the economy. The closer ǫB is to

zero, the more closed the economy is. If ǫB = 0 the model collapses to a closed economy.

ǫH > 0 measures the substitutiability between foreign and domestic goods from the point

of a domestic consumer.

Each country produces a continuum of differentiated goods over the unit interval [0, 1].

CHt is the index for consumption of domestic goods, given by the constant elasticity of

substitution (CES) function4

CHt ≡(

∫ 1

0

C

ǫp−1

ǫp

Hit di)

ǫp

ǫp−1

. (2.1.4)

Where i ∈ [0, 1] denotes the good variety and ǫp > 1 is the elasticity of substitution

between goods produced within any given country j, including the home country. The

3It is possible to aggregate households to a representative household due to Gorman-form of indirectutility.

4See Appendix A.1 for a brief discussion on the CES function.

7

index for imported goods is given by the CES function

CFt ≡(

∫ 1

0

CǫF−1

ǫF

jit dj)

. (2.1.5)

Where ǫF is the elasticity of substitution between goods produced in different foreign

countries.5 Cjt is the index of the quantity of good i imported from country j and

consumed by the domestic household at time t, given by the CES function

Cjt ≡(

∫ 1

0

C

ǫp−1

ǫp

jit di)

ǫp

ǫp−1

. (2.1.6)

It is convenient to note that utility is a nested function of Cjit, where Ct is increasing in

Cjit, and utility represented by (2.1.2) is increasing in Ct. Utility is thus increasing in Cjit.

(2.1.3) is a CES-aggregate, and goods becomes imperfect substitutes, such that firms have

market power, i.e, monopolistic competition. The nested equation system (2.1.2) - (2.1.6)

characterizes preferences of the representative household. The representative household

maximize lifetime utility subject to a sequence of flow budget constraints

∫ 1

0

PHitCHitdi+

∫ 1

0

∫ 1

0

PjitCjitdidj + Et∆t,t+1Bt+1 + Et∆∗t,t+1Et+1B

∗t+1 ≤

Bt + EtB∗t +WtNt +Πt + Tt.

For t = 1, 2, . . . ,∞. Where the domestic price on good i is denoted PHit. The price on

good i imported from country j is Pjit. Households invest in both domestic and foreign

bonds, Bt and B∗t respectively. Bt+1 is the nominal payoff in period t+ 1 of the portfolio

held at the end of period t. ∆t,t+1 is the stochastic discount factor on nominal payoffs. Wt

is the nominal wage, such that WtNt denotes the nominal labor-income of the household,

Πt are dividends from ownership in firms. Finally, Tt denotes any netto transfers from

the government. Note that this can be negative (Lump-sum taxation).

The price indexes and optimal demands for every specific consumption unit at each

stage in the nested system are aggregated. The aggregate price index for home goods is

5Note that it is assumed that ǫp = ǫF , i.e, the elasticity of substitution between varieties of goods isthe same in the foreign and home economies. This is however irrelevant, since the domestic consumptionof foreign goods has negligible effect on foreign economy. See e.g Gali and Monacelli (2005) for a similarassumption.

8

given by the CES function

PHt =(

∫ 1

0

P1−ǫpHit di

)1

1−ǫp. (2.1.8)

Where ǫp is defined as above, the elasticity of substitution between goods produced within

any given country. The optimal demand for home good i is given by

CHit =(PHit

PHt

)ǫp

CHt. (2.1.9)

In a similar manner, the aggregate price index for imported goods from country j is given

by the CES function

Pjt =(

∫ 1

0

P1−ǫpjit di

)1

1−ǫp. (2.1.10)

The optimal consumption of good i imported from country j is given by

Cjit =(Pjit

Pjt

)ǫp

Cjt (2.1.11)

The aggregate price index for all imported goods is given by the CES function

PFt =(

∫ 1

0

P 1−ǫFjit di

)1

1−ǫF . (2.1.12)

Where ǫF is defined as above, the elasticity of substitution between goods produced in

different foreign countries. The optimal basket of import consumption from country j is

given by

Cjt =( Pjt

PFt

)ǫFCFt. (2.1.13)

Finally, the aggregate consumption price index (CPI), in the home country is given by

Pt ≡[

(1− ǫB)1

ǫH PǫH−1

ǫH

Ht + ǫ1

ǫH

B PǫH−1

ǫH

Ft

]

ǫHǫH−1

. (2.1.14)

Where ǫH is defined as above, the substitutiability between domestic and foreign goods.

Notice that under equal price indexes for both domestic and foreign goods, ǫB corresponds

to the share of consumption allocated to imported goods. In this sense, ǫB becomes a

9

natural openness index. Optimal consumption of home goods are given by6

CHt = (1− ǫB)(PHt

Pt

)−ǫHCt. (2.1.15)

Optimal consumption of imported goods are given by

CFt = ǫB(PFt

Pt

)−ǫHCt. (2.1.16)

The total consumption expenditures by a domestic household can be derived using the

market equilibrium for all the aggregates. Such that the budget constraint can be rewrit-

ten as

PtCt + Et∆t,t+1Bt+1 + Et∆∗t,t+1Et+1B

∗t+1 ≤ Bt + EtB

∗t +WtNt +Πt + Tt. (BC)

Where∫ 1

0PHitCHitdi +

∫ 1

0PjitCjitdidj = PHtCHt + PFtCFt = PtCt. Total consumption

expenditures by the domestic household is a sum of the product of the price indexes times

the quantity indexes for both foreign and domestic goods.

The household must decide on the allocation of total consumption, labor and how

much to invest in foreign and domestic bonds. It maximize lifetime utility given by

(2.2.1), and the utility function takes the specific form given by (2.1.1) subject to their

budget constraint (BC). The household optimization problem can be written as

maxCt,Nt,Bt+1,B

∗t+1

Et

{

∞∑

t=0

βtUt(Ct, Nt)}

s.t (HP)


∗t+1 ≤ Bt + EtB

∗t +WtNt +Πt + Tt.

First-order conditions (FOC) of the household problem can be combined to yield the

6Derivation of (2.1.15) and (2.1.16) can be found in Appendix A.2

10

following intertemporal optimality conditions7

C−σt = β∆−1

t,t+1Et

{

C−σt+1

Pt

Pt+1

}

(2.1.17)

C−σt = β∆−1∗

t,t+1Et

{

C−σt+1

Pt

Pt+1

Et+1

Et

}

(2.1.18)

Wt

Pt= Cσ

t Nψt . (2.1.19)

The risk-free gross nominal return on a risk-less one-period discount bond is given by

Rt = (Et∆−1t,t+1). Notice that Et∆t,t+1 is the price of the bond. Notice also thatRt = 1+R

′

t,

where R′

t is the nominal interest rate in percent. (2.1.17) is the standard Euler equation

for intertemporal consumption. (2.1.19) is the intratemporal labor-leisure choice. It

states that the marginal utility of consumption is equal to the marginal value of labour

at any point in time. Together, these equations determine the forward-looking, rational

household allocation decision.

2.2 Terms of trade

The domestic economy is small compared to the rest of the world, and the foreign econ-

omy is considered exogenous. It does not affect the foreign economy through monetary

policy decisions, imports nor exports. In the following section, connections to the foreign

economy through exchange rate and other variables are derived.

Let the bilateral terms of trade (TOT) between the domestic and foreign country j be

defined as the price of country j goods in terms of the home good, i.e,

Sjt =Pjt

PHt. (2.2.1)

The effective terms of trade are given by8

St =PFt

PHt=

(

∫ 1

0

S1−ǫFjt dj

)1

1−ǫF . (2.2.2)

Terms of trade is the ratio between the price of imports and exports, in other words, the

7See derivation of household problem in appendix A.3.8The formulation of the TOT is common in the literature. See e.g Haider and Khan (2008), Gali and

Monacelli (2005) and Liu (2006).

11

domestic currency relative price of imports. It becomes a natural measure of competitive-

ness of the home economy. Increased terms of trade is equivalent to an increase in the

competitiveness of the domestic economy. This is due to increased foreign prices, and/or

a fall in domestic prices.

2.3 The real exchange rate

Let Et be the nominal exchange rate, measured in units of foreign currency to domestic

currency. An increase in Et is associated with an appreciation of the domestic currency.

Let the real exchange rate be defined as

Qt = EtPt

P ∗t

. (2.3.1)

Where P ∗t is the foreign price level measured in units of foreign currency. An Increase in

the real exchange rate, Qt, is associated with an appreciation of the domestic currency.

Under incomplete pass-through, the law of one price (LOP) does not hold. Let the

law of on price gap (LOP-gap) be defined as9

Φt =P ∗t

EtPFt. (2.3.2)

Where Φt is a measure of the deviation from the law of one price. The law of one price gap

can be thought of as a wedge between the foreign price of goods and the domestic price

on these imported goods. In other words, an inverse mark-up on foreign goods. If Φt = 1,

the LOP holds.10 Then the import price index PFt is the foreign price index divided by

the nominal exchange rate. I.e, PFt =P ∗t

Et. These equations are further examined when

they are log linearized in section 3.

2.4 Uncovered interest rate parity

Households are allowed to invest in both domestic and foreign bonds. The optimality

conditions of the household can be combined to yield an expression for the uncovered

9e.g Liu (2006) uses the same formulation of the LOP-gap10Note that when Φt = 1, the purchasing power parity (PPP) holds. This implies imperfect pass-

through from exchange rate movements to domestic currency prices of imports. Since importers seek toadjust prices such that revenue from consumers are extracted optimally.

12

interest parity (UIP),11

∆∗t,t+1

∆t,t+1

= Et

{Et+1

Et

}

. (2.4.1)

This condition summarizes the optimal holdings of domestic and foreign bonds. In equi-

librium, it should not be possible to increase portfolio returns by choosing a different

compositions of bonds. In other words, the expected nominal return from risk-free bonds,

measured in domestic currency, must be equal to the expected return from foreign goods,

in domestic currency terms.

2.5 Foreign consumption, inflation and the interest rate

When modelling a small open economy, it is common to assume that the foreign sector

is exogenous, the domestic economy has negligible effect on the world economy.12 The

domestic economy takes foreign consumption, foreign inflation and the foreign interest rate

as given. These variables are subject to shocks, such that we can analyse the response of

the domestic economy, but they revert back to steady state at a pace determined by an

autocorrelation parameter ρ.13 For simplicity it is assumed that there exist a steady state

for the variables, which is further discussed in section 3.

Let foreign total private consumption be defined as

C∗t

C∗=

(C∗t−1

C∗t

)ρC∗

eǫtC∗

. (2.5.1)

Where C∗t and C are foreign total private consumption and its steady state respectively.

ρC∗ ∈ (0, 1) is the autocorrelation parameter, and ǫC∗

t is white noise with mean zero and

variance σ2ǫC

∗

t

.14

Let foreign inflation be defined in the same way,

Π∗t

Π∗=

(Π∗t−1

Π∗t

)ρΠ∗

eǫtΠ∗

. (2.5.2)

Where Π∗t =

P ∗t

P ∗t−1

is the foreign inflation rate, and Π∗ is the steady state. ρΠ∗ ∈ (0, 1) is

11See derivations in Appendix A.4.12See e.g Clarida et al. (2001), Gali and Monacelli (2005) and Liu (2006).13Notice that when the equations for the foreign economy are log linearized, they will become AR(1)

processes.14The shock process is further explained in section 4.3.4, and white noise is formally discussed in

section 4.1.2.

13

the autocorrelation parameter, and ǫΠ∗

t is white noise with variance σ2Π∗ .

Finally, let the foreign interest rate be defined similarly as

R∗t

R∗=

(R∗t−1

R∗t

)ρR∗

eǫtR∗

. (2.5.3)

Where R∗t = 1+R∗′

t is the foreign interest, R∗′

t is the foreign nominal interest in percent,

and R∗ is the steady state value. ρR∗ ∈ (0, 1) is the autocorrelation parameter, and ǫR∗

t is

white noise with variance σ2Π∗ .

2.6 Firms

So far, the demand side of the economy has been covered. In this section, the supply

side of the economy is covered. In the economy there exist a continuum of identical

monopolistically competitive domestic firms, indexed by i ∈ [0, 1]. Domestic firms take

the business environment as given, including the state of the world in foreign economies

and use homogeneous labor in production. The capital stock is treated as fixed, and

investment is zero in short run.15 Prices are assumed to be sticky, following staggered

type price-setting theory according to Calvo (1983). Only a randomly selected amount of

firms set new prices each period.

2.6.1 Production technology and marginal cost

A typical domestic firm i produces a differentiated good using only labor as input with

technology represented by the production function

Yit = AtNit. (2.6.1.1)

Where Yit is the output produced by firm i in period t, At is the technology level common

to all domestic firms,16 and Nit is the labor force used by firm i in the production.

Final goods are a CES aggregate of intermediate goods, i.e,

Yt ≡(

∫ 1

0

Y

ǫp−1

ǫp

it di)

ǫp

ǫp−1

. (2.6.1.2)

15McCallum and Nelson (1999) argued that capital do not play a central role in monetary policy andbusiness cycle analysis.

16At can also be thought of as the specific labor productivity, which will become clear when themarginal cost of the firm is obtained.

14

Due to the staggered price setting, there is a probability that firms will not be able to

maximize profits every period. However, the firms will find it optimal to minimize the

costs regardless of the price of the good. The firm problem can be split into two parts.

Firms minimize cost subject to output, and then given cost minimization, they maximize

profits. Firms are price takers in input markets. Taking wages, Wt as given, the cost-

minimization problem of firm i can be represented as

min{Nit}

WtNit s.t Yit ≤ AtNit, (2.6.1.3)

and the F.O.C of this problem is

Wt = ΨitAt. (2.6.1.4)

Note that Ψit is the nominal marginal cost of firm i in period t. The nominal marginal

cost by using labor is the wage. Marginal gain in income to the firm is the price times the

marginal increase in the production due to a marginal increase in the labor. Let MPNnt

be the marginal productivity of labor, given by

MPNnt =

∂Yit

∂Nit

= At ∀t. (2.6.1.5)

Firms hire labor facing the same nominal wage. It follows from (2.6.1.5) that the marginal

cost of increasing output are equal across all firms, regardless of any heterogeneity in the

output stemming from differences in prices.17 Let MCrHt =

Ψit

PHt= Ψt

PHt∀t, be the real

marginal cost of a domestic firm in period t. Insert the nominal marginal cost from the

cost-minimization problem to obtain the real marginal cost18

MCrHt =

(Ψt

PH

)

=Wt

PHMPNnt

=Wt

PHtAt. (2.6.1.6)

2.6.2 Calvo-type price setting of domestic firms

Domestic firms set prices freely, however, a priori they do not know when they can

reset prices. The firms faces a probability to be stuck with a price that is not optimal,

the "wrong" price. Let θH denote the probability of being stuck with the same price

17This is also due to the fact we have constant returns to scale in the production function. Marginalcosts are independent of any production level, and common across all firms.

18Notice that since capital is omitted, the marginal cost is also the total cost of the firm.

15

as they had last period, i.e, the "wrong" price for a domestic firm, and 1 − θH be the

probability that they can reset prices. The probability of re-setting the price in any given

period is assumed to be independent of the time elapsed since last re-setting. As a result,

θH becomes an index of the price stickiness in the economy. The firms have to take into

account that setting the optimal price today determines the expected profits in the future,

due the probability of being stuck with today’s price k periods ahead is θkH .

Following Gali and Monacelli (2005), all firms resetting prices in any given period will

choose the same price, i subscripts can be dropped. Let PHt denote the price level an

optimizing firm sets. Under this particular pricing structure, the dynamics of the domestic

price index are described by

PHt ≡[

θHP1−ǫpHt−1 + (1− θH)P

1−ǫpHt

]1

1−ǫp. (2.6.2.1)

Where ǫp is defined as above, the substitutiability between varieties of goods produced

within any given country. A firm that re-optimizes in period t chooses a price PHt such

that current market value of the profits generated while the price remains effective are

maximized. The firms solves the following maximization problem,

maxPHt

∞∑

k=0

(θkH)Et{

∆t,t+kYt+k[PHt − PHt+kMCrHt+k]

}

(2.6.2.2)

s.t Yit+k =( PHt

PHt+k

)−ǫpYt+k.

Where the real marginal cost is measured in terms of home output, and is exogenous to

firm i. The firms optimal price setting behavior is given by19

∞∑

k=0

(θkH)Et{

∆t,t+kYt+k[PHt −MΨHt+k]}

= 0. (2.6.2.3)

Notice that under fully flexible prices, i.e, θH → 0, this expression reduces to a friction-less

mark-up rule PHt = MΨHt. Where M = ǫpǫp−1

is the markup if prices were fully flexible.

Divide through by PHt−1, and let ΠHt+k =PHt+k

PHt−1. Rewrite (2.6.2.3) as

∞∑

k=0

(θkH)Et

{

∆t,t+kYt+k

[ PHt

PHt−1

−MΠHt+kMCrHt+k

]

}

= 0, (2.6.2.4)

19See derivations in Appendix A.5.

16

and use the fact that the stochastic discount factor can be derived from the household

optimization problem as ∆t,t+k = βkUc,t+k

Uc,t. Use this to rewrite (2.6.2.4) as

∞∑

k=0

(βkθkH)Et

{

C−σt+k

Pt+kYt+k

[ PHt

PHt−1

−MΠHt+kMCrHt+k

]

}

= 0. (2.6.2.5)

The pricing decision in the model is forward-looking. The firms that adjust prices

in any period recognizes that the prices they set will remain for a number of periods (in

expectation). They set prices as mark-up over a weighted average of current and expected

future marginal cost.20 The future get less weight, and periods with high demand get

higher weight. This equation is further examined when the model is log linearized in

section 3.

2.7 Incomplete pass-through and imported goods

The assumption that the LOP holds is made for the wholesale for imports.21 However,

endogenous short run fluctuations in the purchasing power parity (PPP) arise due to

monopolistic competitive importers. Monopolistic importers purchase foreign goods to

world market prices, such that the LOP holds "at the dock". Foreign goods are sold

to domestic consumers as a markup over marginal cost, the LOP fails to hold at retail

level for domestic imports due to this. Since prices on import are charged to domestic

households as a markup over marginal cost, there exist a wedge between domestic and

import prices on foreign goods, measured in terms of domestic currency.

The domestic market is populated by local retailers who import differentiated goods.

The pricing behavior of the monopolistic domestic importers can be summarized following

a similar Calvo-pricing argument as earlier, and in similar fashion to (2.6.2.5). Thus

domestic importers set prices according to22

∞∑

k=0

(βkθkF )Et

{

C−σt+k

Pt+kYt+k

[ PFt

PFt−1

−MΠFt+kΦt+k

]

}

= 0. (2.7.1)

Where Φt+k is the LOP defined as above. The stickiness parameter θF ∈ [0, 1] is defined

as above, but now for importing retailers that do not re-optimize prices. This equation is

20This becomes clear when the pricing decision is log linearized in section 3.8.3.21Following Monacelli (2005) and Gali and Monacelli (2005).22Liu (2006) and Haider and Khan (2008) uses similar form of the price level on imported goods.

17

also further examined in section 3.8.3.

2.8 Import and export sector

The import relationship for the economy is defined by (2.1.16), repeated here for simplicity

CFt = ǫB(PFt

Pt

)−ǫHCt. (2.8.1)

The amount of imports depend on the elasticity of substitution between foreign and

domestic goods ǫH , the degree of openness ǫB, the ratio of the price level for imported

goods, the aggregate price level PFt

Ptand the total level of private consumption Ct.

The export relationship is obtained by first writing the import function for the foreign

economy, and realising that this is also the export function of the domestic economy,

similar to (2.1.16)

C∗Ht = ǫB

(

EtPHt

P ∗t

)−ǫHC∗t . (2.8.2)

Where C∗Ht is the import of the foreign economy, and thus the export of the domestic

economy. ǫ∗B is the degree of openness for the foreign economy, ǫ∗H is the elasticity of

substitution between foreign and domestic goods, now seen from the foreign economy.

Two simplifying assumptions are made. The degree of openness and the elasticity of

substitution between domestic and foreign goods for the foreign economy is the same as

for the SOE, i.e, ǫ∗B = ǫB and ǫ∗H = ǫH . These assumptions might be unreasonable. It is

possible to to think that the foreign economy (the rest of the world) have more substitutes

to choose from than the SOE, and should have a higher elasticity of substitution.23 Finally,

C∗t is the total level of private consumption in the foreign economy.

2.9 Monetary policy

In order to close the model, it is necessary to specify a monetary policy rule. In this

section three different monetary policy rules for the monetary authorities is defined.

23See e.g. Liu (2006) where the same assumption is used.

18

2.9.1 Monetary policy in Norway

As a start, the monetary policy in Norway is investigated. Norges Bank´ main objective

is to keep low and stable inflation.24 The goal is to keep inflation around its target at

2%.25 Inflation targeting shall be forward-looking and flexible so that it can contribute to

high and stable output and employment, and to counteract financial imbalances. Norges

Banks main monetary policy instrument is the policy rate. The policy rate is the interest

rate on banks’ deposits in Norges Bank up to a specific threshold. Since 2001, Norway

have formally had a floating currency. Before 2001 Norges Bank had like many other

countries exchange rate targeting. The goal of the monetary policy was to stabilise the

Norwegian Krone’ value against other currencies.

As a supporting tool in making monetary policy decisions, Norges Bank has developed

the Norwegian Economy Model (NEMO), which is a DSGE model. NEMO describes how

the interest can be set to bring the economy from its current state, back to a long-term

equilibrium path.26

Monetary policy in NEMO can be solved under a Taylor rule of the kind27

Rt = ωRRt−1 + (1− ωR)(ωP πt + ωY YNAT,t) + ZRN3M,t.

Where Rt is the money market rate, ωR is a parameter that measures interest rate per-

sistence and ωP and ωY are weights on inflation and output respectively. YNAT,t is the

output level. Finally, ZRN3M,t represents a monetary policy shock that follows an AR(1)

process.

2.9.2 A baseline monetary policy

The baseline monetary policy rule is constructed similarly to what Norges bank conduct.

Let the baseline monetary policy, Monetary policy 1 (MP1) be defined in a similar way

as the one described for NEMO28

Rt = RωR

t−1

{

RrΠT(Πt+1

ΠT

)ωΠ(

YtYt−1

¯GDP

)ωY}1−ωR

. (MP1)

24See e.g https://www.norges-bank.no/en/topics/Monetary-policy/Mandate-monetary-policy/25Until 2018 Norges Bank had an inflation target at 2.5%.26More information on NEMO can be found in Brubakk et al. (2006) and Kravik and Mimir (2019).27See Kravik and Mimir (2019).28See Kravik and Mimir (2019).

19

Where Rt is the domestic interest rate in period t. ωR ∈ [0, 1] governs the interest rate

smoothing, and ωΠ ≥ 0 and ωY ≥ 0 is the weight on inflation and output respectively.

Πt+1 is the rational expected inflation in period t + 1, defined as Πt = 1 + Πt

′

, where Π′

is the percent change in domestic inflation. Π is the Central Bank’ inflation target, and

¯GDP is steady state growth of GDP, which is set to 1 in this model. Finally, Rr is the

steady state real interest rate. Rr = 1 + R′

r, where R′

r is the steady state real interest in

percentage.

2.9.3 A monetary policy with exchange rate

The second monetary policy will additionally have the central bank react to the real

exchange rate. Let the second monetary policy, Monetary Policy 2 (MP2) be defines as

Rt = RωR

t−1

{

RrΠT(Πt+1

ΠT

)ωΠ(

YtYt−1

¯GDP

)ωY(Qt

Q

)−ωQ1}1−ωR

. (MP2)

Where Qt and Q is the real exchange rate from (2.3.1) and the steady state of the real

exchange rate respectively. ωQ1≥ 0 is the weight on the real exchange rate. When

following this monetary policy, the central bank seeks to stabilise the level of the real

exchange rate in addition to output and inflation. Note that there is a minus in front of

ωQ1due to a rise in the real exchange rate is an appreciation of the domestic currency

and the interest should be lowered to make it less tractable.

A final monetary policy will additionally have the central bank respond to changes

in the real exchange rate. Let the third monetary policy, Monetary Policy 3 (MP3) be

defined as

Rt = RωR

t−1

{

RrΠT(Πt+1

ΠT

)ωΠ(

YtYt−1

¯GDP

)ωY(Qt

Q

)−ωQ1( Qt

Qt−1

)ωQ2}1−ωR

. (MP3)

Where ωQ2≥ 0 is the weight on the rate of change in the real exchange rate. Notice that

if both ωQ1and ωQ2

are set to zero, the baseline policy is obtained.

2.10 Welfare

To compare the three monetary policies a loss function is used. The loss functions mea-

sures the welfare loss of the economy. The lower the value of the loss function, the greater

20

the welfare of the economy. Let the loss function, LF, be defined as

LF = σ2π + γyσ

2y + γrσ

2r . (LF)

Where σ2Π, σ2

y and σ2r denotes the variance of the deviation of inflation, output and the

interest rate from its steady state respectively. γy denotes weight put on the output, and

γr denotes weight put on the interest rate. Following Garcia and Gonzalez (2010), γy =12

and γr =15, such that the LF becomes

LF = σ2π +

1

2σ2y +

1

5σ2r .

This completes the presentation of the theoretical model. In the next section the model

is log linearized.

21

3 The log linearized model

The theoretical model presented in section 2 is a complex system of non-linear equations

that needs to be solved simultaneously. In general, such systems has computationally

intractable solutions, or cannot be solved analytically. A solution to this problem is

the method of log linearization. Log linearization converts a non-linear equation into a

linear equation in terms of log-deviation from each of the variables steady state value.

Log-deviations from steady states have convenient economic interpretations. For small

deviations from steady state, they are approximately equal to percentage deviations from

steady state.

This section covers basics on the method of log linearization and presents important

identities that is used when linearizing the model.29 The log linear form of the model

which is used in the estimation process below is derived.

3.1 Basic principles of log linearization

The method of log linearization is taking log-deviation around a steady state value. Con-

sider a variable Xt, let X be the steady state value of Xt. Define the log-deviation of Xt

from its steady state value as

xt ≡ lnXt − lnX. (3.1.1)

Where lower-case letter denotes log-deviation from steady state of the variable. Rewrite

equation (3.1.1) as

xt ≡ ln(Xt

X

)

= ln(

1 +Xt − X

X

)

.

Notice that ln(

1+Xt−XX

)

can be approximated by a first-order Taylor expansion at steady

state Xt = X

ln(

1 +Xt − X

X

)

≈ ln1 +1

X(Xt − X) =

Xt − X

X,

29See e.g. Ziet (2006) for an introduction on this method in DSGE models.

22

such that

xt ≈Xt − X

X=Xt

X− 1. (3.1.2)

Equation (3.1.2) states that the log-deviation of the variable of interest xt from its steady

state value, is approximately equal to the percentage difference between Xt and its steady

state value X. Note that this approximation method holds for small deviations from

steady state only, implying that log-linearization is a local approximation method.

However, this locality is not an issue. In DSGE models one takes logs, and then

linearize log of the variables around a steady state path. Along the steady state path,

all real variables are growing at the same rate. On average, the stochastic economy will

fluctuate around the steady state path, such that the approximation is more likely to be

accurate. The linearization results in a set of linear equations that are in log-deviations

from steady state. Variables are expressed in terms of percentage deviation from the

steady state path, and the coefficients in the the system can be thought of as elasticities.

3.1.1 Important identities

Other important identities that is used when linearizing the model is presented in this

section. Note again that lowercase letter denotes log-deviation from steady state.

xt ≡ lnXt − ln(X) ≈Xt − X

X. (3.1.2.1)

Every variable can be written as

Xt = XXt

X= Xext , (3.1.2.2)

after taking first order Taylor approximation this becomes:

Xt = Xext ≈ X(1 + xt). (3.1.2.3)

Another important approximation that is used is

XtYt ≈ XY (1 + xt)(1 + yt) ≈ XY (1 + xt + yt). (3.1.2.4)

23

Notice that the product xtyt is set to zero. Which is a good approximation since small

deviations from steady state is considered, and the product of small numbers is negligible.

A final important identity that is used is

ln(1 + xt) ≈ xt and ln(1 + xt + yt) ≈ xt + yt. (3.1.2.5)

The rest of this section presents the log linearized model. Basic steps are shown in

appendix 3 in some more involved cases, but methods and identities motioned in the

above section are mainly used.

3.2 Log linearizing household conditions

3.2.1 Labor-leisure choice

Log linearizing the labor-leisure choice of the households (2.1.19) yields

wt − pt = σct + ψnt. (3.2.1.1)

Where identity (3.1.2.1) is used, i.e that wt = ln(Wt) − ln(W ), pt = ln(Pt) − ln(P ) and

so on. Notice that wt − pt is the real wage.

3.2.2 Consumption Euler equation

Log linearizing the consumption Euler equation (2.1.17) yields30

ct = Et{ct+1} −1

σ(rt − Et{πt+1} − ρ). (3.2.2.1)

Where ρ = β−1 − 1 and πt+1 = pt+1 − pt.

30See derivations in Appendix B.1.

24

3.3 Log linearizing terms of trade, domestic inflation and CPI

inflation

3.3.1 Terms of trade

Recall that the effective terms of trade are given by (2.2.2). Log linearizing the effective

terms of trade around a symmetric steady-state, Sjt = Sj = 1 ∀ j yields 31

st = pFt − pHt ≈

∫ 1

0

sjtdj. (3.3.1.1)

Taking first difference of this equation, i.e, subtracting the equation in period t− 1 yields

st − st−1 = ∆st = pFt − pFt−1 −(

pHt − pHt−1

)

(3.3.1.2)

= πFt − πHt.

Where the fact that πt = pt − pt−1 is used. In order to investigate the dynamics of the

model to a shock in the TOT, a shock variable is added. Such a shock can be thought of

as a measurement error. Let zst be the shock variable defined as

zst = ρszst−1 + ǫst .

That is, an AR(1) process.32 Where ρs ∈ (0, 1) is the autocorrelation parameter, and ǫst

is white noise with zero mean and variance σ2s . The final expression becomes

∆st = πFt − πHt + zst . (3.3.1.3)

3.3.2 Consumer price index

Log linearizing the CPI (2.1.14) yields33

pt = (1− ǫB)pHt + ǫBpFt. (3.3.2.1)

31See Appendix B.2.132Autoregressive process with one lag.33See derivations in Appendix B.2.2.

25

Taking first difference in order to obtain the log linearized expression for aggregate overall

inflation. The aggregate overall inflation evolve according to

πt = (1− ǫB)πHt + ǫBπFt, (3.3.2.2)

Notice that (3.3.2.2) can be rewritten as

πt = πHt + ǫB(

πFt − πHt)

.

Combine (3.3.2.2) and (3.3.1.2) to obtain the link between the domestic inflation and the

CPI inflation according to

πt = πHt + ǫB∆st. (3.3.2.3)

The gap between the CPI inflation and the domestic inflation is proportional to the change

(percent) in the terms of trade. The proportionality coefficient is given by the openness

of the economy ǫB. Notice that (3.3.2.1) and (3.3.1.1) can be combined to yield

pt = pHt + ǫBst, (3.3.2.4)

which will be an important identity when the LOP is log linearized below.

3.4 Log linearizing the real exchange rate

Log linearizing the real exchange rate (2.3.1) yields

qt = et + pt − p∗t . (3.4.1)

Where qt = ln(Qt) and so on. In order to investigate the dynamics of the model to a

shock in the real exchange rate, a shock variable is added. Let zqt be the shock variable

defined similarly as above

zqt = ρqz

qt−1 + ǫ

qt .

26

Where ρq ∈ (0, 1) is the autocorrelation parameter, and ǫqt is white noise with zero mean

and variance σ2q . The final expression for the real exchange rate becomes

qt = et + pt − p∗t + zqt . (3.4.2)

3.5 Log linearizing the law of one price gap

Log linearizing the LOP-gap (2.3.1) yields

θt = p∗t − et − pFt. (3.5.1)

Notice that (3.4.1) can be used to to eliminate p∗t and et, such that

θt = −qt + pt − pFt. (3.5.2)

Use (3.3.2.4) to eliminate pt to obtain

θt = −qt + pHt + ǫBst − pFt.

Finally, using (3.3.1.1) to rewrite pFt − pHt, the final expression for the log linearized

LOP − gap is obtained

θt = −qt − (1− ǫB)st. (3.5.3)

The LOP-gap depends inversely on the real exchange rate and the degree of competitive-

ness for the domestic economy.

Notice that the real exchange rate can be written as

qt = −(1− ǫB)st − θt + zqt . (3.5.4)

There are two sources of deviation from aggregate purchasing power parity. The first de-

viations is captured by the term (1−ǫB)st. This stems from the degree of competitiveness

of the domestic economy, as long as ǫB < 1. The second deviations is due to deviations

from the law of one price, captured by the term θt. Notice that if the LOP holds, Θt = 1

27

and θt = 0, the real exchange rate (3.5.4) collapses to

qt = −(1− ǫB)st + zqt .

The LOP-gap contributes to exchange rate volatility. In order to investigate the dynamics

of the model to a shock in the LOP-gap, a shock variable is added. Such a shock can be

thought of as a measurement error. Let zθt be the shock variable defined as

zθt = ρθzθt−1 + ǫθt .

Where ρθ ∈ (0, 1) is the autocorrelation parameter, and ǫθt is white noise with zero mean

and variance σ2θ . The final expression for the LOP-gap becomes

θt = −λt − (1− ǫB)st + zθt . (3.5.5)

3.6 Log linearizing uncovered interest rate parity

Log linearizing the uncovered interest parity condition (2.4.1) yields

rt = r∗t + Et∆et+1. (3.6.1)

The nominal interest rate at home is equal to the world nominal interest rate plus an

expected depreciation rate of the home currency. Using (3.4.1) to eliminate et and et+1

such that

qt+1 − qt = r∗t − π∗t+1 − (r − πt+1) + zq. (3.6.2)

Notice also that combining (3.3.1.1) and (3.5.1) st = pFt − pHt = et + p∗t − pHt yields

Etst+1 − st = Etet+1 − et + Etp∗t+1 − p∗t − EtpHt+1 + pHt

st = Et∆et+1 − Etπ∗t+1 + EtπHt+1 + Etst+1.

28

Given stability of the model, and that terms of trade are uniquely identified in the perfect

foresight steady state combined with unit relative prices, solve forward to obtain34

st = Et{

∞∑

n=0

[(r∗t+n − π∗t+n)− (rt+n − πHt+n)]

}

. (3.6.3)

This results tells that the terms of trade are an expected sum of the real interest rate

difference between the domestic economy and the world economy.

3.7 Log linearizing foreign consumption, inflation and the interest

rate

The log linearized equations will become AR(1) processes. Such that log linearizing (2.5.1)

yields the following AR(1) process,35

c∗t = ρc∗c∗t−1 + ǫc

∗

t .

Log linearizing (2.5.2) yields

π∗t = ρπ∗π∗

t−1 + ǫπ∗

t .

Finally, log linearizing equation (2.5.3) yields

r∗t = ρr∗r∗t−1 + ǫr

∗

t .

Where the parameters ρj and ǫjt for j = c∗, π∗, r∗ are the autocorrelation parameters and

white noise processes defined similarly as above respectively.

34See derivations in Appendix B.3.35Notice that by aggregating over all countries, it is possible to derive a world market clearing condition,

y∗t ≡∫

1

0yitdi =

∫

1

0citdi ≡ c∗t . See e.g Gali and Monacelli (2005).

29

3.8 Firms

3.8.1 Log linearizing production technology

Log linearizing the production function (2.6.1.1) around a symmetric equilibrium across

all i firms yields

yt = at + nt. (3.8.1.1)

Where identity (3.1.2.1) is used, i.e, yt = ln(Yt) − ln(Y ) and so on. The technology

in log-terms, at = ln(At) follows an AR(1) process, at = ρaat−1 + ǫat . Where ρa is the

autocorrelation parameter, and ǫat is white noise with zero mean and variance σ2ǫa

.

3.8.2 Log linearizing marginal cost

Log linearizing the real marginal cost of the domestic firms (2.6.1.6) yields

mcrt = wt − pHt −mpnnt = wt − pHt − at. (3.8.2.1)

Add and subtract pt from (3.8.2.1) to obtain

mcrt = (wt − pt) + (pt − pHt)− at. (3.8.2.2)

Insert equation (3.2.1.1) for pt − wt, and insert (3.3.2.4) for pt such that

mcrt = σct + ψnt + ǫBst − at.

Use the fact that from (3.8.1.1), nt = yt − at, and rearrange to get

mcrt = σct + ψyt + ǫBst − (1 + ψ)at. (3.8.2.3)

The marginal cost is an increasing function in consumption ct, output yt, and TOT st,

but is inversely related to labour productivity at.

30

3.8.3 Log linearizing optimal price setting

Log linearizing the optimal pricing behavior of a domestic firm (2.6.2.5) yields36

pHt = pHt−1 +∞∑

k=0

(βkθkH)Et

{

πHt+k + (1− βθH)mcrHt+k

}

. (3.8.3.1)

Where pHt denotes the newly set price by a domestic firms. Firms set prices in a forward-

looking way and choose a price that reflect their desired markup over a weighted average of

their current and expected marginal cost. Notice that it is possible to obtain an alternative

expression for the optimal price37

pHt = µ+ (1− βθ)∞∑

k=0

(βkθkH)Et

{

ψrHt+k

}

.

Where µ ≡ ǫp1−ǫp

is the log of the markup.

Similarly, the log linearized expression for the optimal pricing behaviour of the domes-

tic importer (2.7.1) is

pFt = pFt−1 +∞∑

k=0

(βkθkF )Et

{

πFt+k + (1− βθF )θt+k

}

. (3.8.3.2)

Where θF ∈ [0, 1] as before, the fraction of importers that do not re-optimize its price. A

domestic retailer resetting its price is concerned with future inflation as well as the LOP-

gap. The LOP-gap, θt, is the margin over and above the wholesale import price. If the

LOP does not hold, meaning θt 6= 1, there exist a wedge between the world and domestic

import prices given by the LOP-gap. This becomes the incomplete import pass-through

mechanism in the short run. Changes in the world import prices have an affect on the

domestic economy.

3.9 Log linearizing imports and exports

Log linearizing the imports (2.8.1) yields

cFt = −ǫH(pFt − pt) + ct.

36More details can be found in Appendix B in Gali and Monacelli (2005).37See derivations in Appendix B.4.

31

Use (3.3.2.4) and (3.3.1.1) to rewrite this as

cFt = −ǫH(pFt − pHt − ǫBst) + ct

= −ǫH((1− ǫB)st) + ct. (3.9.1)

Note that by using (3.5.4), (3.9.1) can be also be rewritten as

cFt = ǫH(θt + qt) + ct. (3.9.1b)

Log linearizing exports (2.8.2) yields

c∗Ht = −ǫH(et + pHt − p∗t ) + c∗t .

Use (3.5.1) and (3.3.1.1) to rewrite this as

c∗Ht = −ǫH(pHt − pFt − θt) + c∗t

= ǫH(st + θt) + c∗t . (3.9.2)

From (3.9.2), it follows that an increased competitiveness of the domestic economy on the

world market (i.e, an increased terms of trade) will have foreign households substitute out

foreign goods into consumption of domestic goods. The elasticity of substitution between

foreign and domestic goods ǫH determines the magnitude of the substitution.

Recall that the domestic demand for domestic goods is given by (2.1.15), and can be

written in log linearized form as

cHt = −ǫH(pHt − pt) + ct.

Use (3.3.2.4) to rewrite this as

cHt = ǫHǫBst + ct, (3.9.3)

Equation (3.9.3), will have domestic households substitute out foreign consumption into

domestic consumption. The substitution depends on the elasticity of substitution between

foreign and domestic goods ǫH ; and the openness index ǫB.

32

3.10 Equilibrium

3.10.1 Demand and output

Market clearing for goods in the domestic economy requires domestic output to be equal

to the sum of domestic consumption and foreign consumption of home produced goods,

implying

yt = (1− ǫB)cHt + ǫBc∗Ht. (3.10.1.1)

Insert (3.9.3) and (3.9.2) to obtain

yt = (1− ǫB)ct + ǫBc∗t + (2− ǫB)ǫHǫBst + ǫBǫHθt. (3.10.1.2)

3.10.2 Inflation dynamics

In section 3.8.3 the optimal pricing behaviour of the economy were derived. These results

can be used to derive the inflation dynamics of the economy. Recall that under this

particular price-setting behavior, the dynamics of the domestic price evolves according to

(2.6.2.1). Log linearizing around a zero-inflation steady state yields38

πHt = (1− θH)(pHt − pHt−1). (3.10.2.1)

Combining this result with the log linearized expression for the optimal pricing behavior

of a domestic firm (3.8.3.1), domestic inflation evolve according to39

πHt = (1− θH)EtπHt+1 + λHmcrt . (3.10.2.2)

Where λH = (1−βθH)(1−θH)θH

, which is a standard New Keynesian Philips Curve (NKPC),

and mcrt is the real marginal cost. A shock variable zπHt is added to the expression in

order to investigate the dynamics of the model to a shock in the domestic inflation. The

final expression for the domestic inflation becomes

πHt = (1− θH)EtπHt+1 + λHmcrt + zπHt . (3.10.2.2a)

38See derivations in Appendix B.5.39See derivations in Appendix B.5.

33

Where the shock variable is zπHt is defined similarly as above

zπHt = ρπHzπHt−1 + ǫπHt .

Where zπHt is an AR(1) process. ρπH ∈ (0, 1) is the autocorrelation parameter, and ǫπHt is

white noise with zero mean and variance σ2πH

.

Similarly, using (3.8.3.2) in the determination of πFt, under this particular pricing

behavior import inflation evolve according to

πFt = (1− θF )EtπFt+1 + λF θt. (3.10.2.3a)

Where λF = (1−βθF )(1−θF )θF

, and θt is the LOP. In addition, a shock variable zπFt is added

to the to expression in order to investigate the dynamics of the model to a shock in the

import inflation. The final expression for the import inflation becomes

πFt = (1− θF )EtπFt+1 + λF θt + zπFt .. (3.10.2.3)

The shock variable zπFt is defined in similar manner as

zπFt = ρπF zπFt−1 + ǫπFt .

Where zπFt is an AR(1) process .ρπF ∈ (0, 1) is the autocorrelation parameter, and ǫπFt is

white noise with zero mean and variance σ2πF

.

The inflation dynamics of the SOE are given by (3.3.2.2), (3.10.2.2) and (3.10.2.3). In

such sticky price models, the inflation dynamics arises from firms’ preference for smoothing

their pricing decision. If prices were fully flexible, these nominal rigidities would not occur.

A social planner in this economy would seek to minimize deviations of marginal cost and

LOP-gap from steady state.

34

3.11 Log linearizing monetary policy

Log linearizing monetary policy 1 (MP1) yields40

rt = ωRrt−1 + (1− ωR){

ωΠ(πt+1) + ωY (∆yt)}

.

As above, rt = ln(Rt) − ln(R) and so on. Notice that ∆yt = yt − yt−1, and not to be

confused with the stochastic discount factor. Log linearizing Monetary Policy 2 (MP2)

yields

rt = ωRrt−1 + (1− ωR){

ωΠ(πt+1) + ωY (∆yt)− ωQ1qt

}

.

Finally, log linearizing Monetary Policy 3 (MP3) yields41

rt = ωRrt−1 + (1− ωR){

ωΠ(πt+1) + ωY (∆yt)− ωQ1qt + ωλ2∆qt

}

.

Where ∆qt = qt − qt−1. A shock variable is added to all three Monetary Policies in order

to investigate the dynamics of the model to a monetary policy shock. Let the monetary

policy shock zrt be defined in similar manner as above

zrt = ρrrt−1 + ǫrt .

This completes the log linearization of the model. In the next section the model is

calibrated.

40This is similar to the interest rate rule described by Kravik and Mimir (2019) for NEMO.41This type of interest rate rule is also used by Garcia and Gonzalez (2010).

35

4 Calibration

Section 3 presented the log linearized version of the model that is used in the estimation

in section 5 below. The log linearized version of the model presents the dynamics of the

model in terms of deviations from steady state. In this section, the model is calibrated,

i.e, prior distributions of the parameters and shocks are presented.

4.1 Steady state and observed variables

The variables in the model are defined in terms of deviations from steady state. As before,

variables with a bar denotes the steady state value, and will be calibrated using historical

time series data of the observables.

4.1.1 Growth variables

Norwegian quarterly time series data from 1995Q1 - 2019Q4 are used for the domestic

variables. Description of the data used can be found in Appendix G.1. Plots of the data

used can be found in Appendix G.2. An approximation that there is a constant steady

state for the whole period is used. Observables that experience growth are Yt, Ct and C∗t .

In order to approximate a constant steady state for these variables, the growth must be

filtered out. To filter out the trend in the observables that experience growth, the method

of linear detrending is used.42

4.1.2 Linear detrending

A time series {Yt; t = 1, 2 . . . , T} is said to be a determenestic trend (DT) (e.g linear

trend) if the data generating process (DGP) is43

Yt = β0 + γt+ ǫt, γ 6= 0.

Where {ǫt; t = 1, 2 . . . , T} denotes a white noise time series with variance σ2ǫ . Formally,

a process is white noise if E(ǫt) = 0, V ar(ǫt) = E(ǫ2t ) = σ2ǫ , and Cov(ǫt, ǫt−i) = δj =

42See e.g. Liu (2006) and Karunaratne and Pathberiya (2014), where the same method is used.43More technicalities can be found in Nymoen (2019) chapter 9.

36

0, ∀ j 6= 0. Under a deterministic trend the expectation of Yt is a function of the time

E(Yt) = β0 + γt,

such that Yt is non-stationary. However, the variance of Yt is constant, and independent

of the time

V ar(Yt) = σ2ǫ .

The non-stationarity only affects the expectation of the series. When the non-stationarity

only resides in the expectation of the process, the model can be brought to stationary

realm by purging the deterministic part of from the series using regression. The residuals

from a regression of the variable of interest on a constant and a time trend are de-

meaned(have mean zero) and are linearly de-trended. This is done for all the variables

that experience growth under the period. It is important to keep in mind that linear

de-trending arbitrarily fix a linear trend, which might not reflect the true trend in data

and not give the stationary result that is sought.

4.2 Non-growth variables

In the model there are variables that do not experience any particular trend in the given

time period. For further reference, these are called non-growth variables. The domestic

interest rate Rt, the foreign interest rate R∗t , the CPI Πt, foreign inflation Π∗

t , import

inflation ΠF and the real exchange Qt are such observables.

A time series on the CPI for Norway is used for the CPI inflation, and Πt =Pt

Pt−1.

The import inflation is defined as ΠFt =PFt

PFt−1. A time series on the import price index

for Norway is used. Recall that the model is log linearized, such that all variables are

in percent deviations from steady state and the steady state is in fact 0. The data

corresponding to πt in the model is

Πobst = ln(Πdata

t )− ln(Π) = πt.

Where Πobst is the final data transformation of the inflation used to link the empirical

data and the model variables. Πdatat is empirically observed data on inflation, πt as before

37

denotes deviation from steady state, Π is the steady state value. The tricky part for the

data computation is to compute the percent deviations from the steady state (in this case

Π), since the steady state is usually unknown. A common way to deal with this issue is

to assume, as done in this thesis, that the means (i.e, averages) of the time series data

corresponds to the steady state values of the variables of the model, i.e, Π = mean(Πobst ).44

Implying45

Πobst = ln(Πdata

t )−mean(ln(Π)) = πt.

This method is used for all the non-growth observables.

A time series on the three month money market rate for Norway (NIBOR) is used for

the domestic interest rate Rt.46

For simplicity, it is assumed that the foreign economy consist of only the Euro Area

(EA) and the United states (US). Around 80% of the Norwegian exports are shipped

to European countries, while a smaller 5% is shipped to the US. However, I find this

assumption a bit concerning. It would be more accurate to divide the foreign economy

into top exporting countries, like Sweden, Denmark, Germany, UK etc. The foreign

interest rate is defined as

R∗t = 0.7REA

t + 0.3RUSt .

Where REAt is the Euro Area interest rate and RUS

t is the United states interest rate. As

done for Norway, a time series for the three months money market rate in both the EA

and the US are used as interest rates (EURLIBOR and USLIBOR respectively).

The foreign inflation is defined in the same way as the foreign interest rate as

Π∗t = 0.7ΠEA

t + 0.3ΠUSt .

Where ΠEAt is the inflation in the Euro Area and ΠUS

t is the inflation in the United States.

A time series for the CPI for both the EA and the US is used for the foreign inflation.

44See e.g Pfeifer (2013).45Note that another issue arising here is Jensen´s Inequality. Log-differences is only up to first order

equivalent to using percent deviations. If Π = mean(Πobst ) is used to compute πt = lnΠ−lnΠ the resulting

series would not be mean 0. Therefor Jensen´s Inequality is ignored and the log and the mean-operatoris interchanged to use πt = lnΠobs

t −mean(lnΠobst ), which is mean 0.

46See e.g Kravik and Mimir (2019) where the same interest rate is used.

38

4.3 Priors

Loosely speaking, priors formally incorporates prior knowledge about the problem at hand

in the model.47 Such prior information can reflect strongly held believes about the validity

of economic theories. In practice, one often chooses priors based on existing literature on

the problem, such as microeconometric studies or other macroeconomic studies. When

choosing priors, existing literature on Bayesian estimation for the Norwegian economy

is used to obtain information on the structural characteristics of Norway,48 as well as

other existing literature on the subject.49 The choice of priors also reflect restrictions on

the parameters, concerning for example interval restrictions. The parameters that are

restricted on the unit-interval are assumed to be random variables described by the Beta

distribution. For parameters that are in R+ are assumed to be random variables described

by the Gamma distribution, while the Inverse Gamma distribution is used for modeling

of the shocks.50

4.3.1 Priors for the households

For the households, priors for the inverse EIS σ, the inverse Frisch-elasticity ψ, and the

elasticity of substitution between domestic and foreign goods ǫH must be chosen. Priors

are chosen in line with earlier literature and shown in Table 4.3.1. The rate of time

preference (the discount factor) is fixed to 0.99.51

Parameter Distribution Mean S.E Domain

σ Gamma 2.00 0.10 R+

ψ Gamma 1.00 0.30 R+

ǫH Gamma 1.00 0.30 R+

Table 4.3.1: Priors for the households

47Priors and Bayesian methods are discussd more formally in section 5.48E.g Kravik and Mimir (2019) and Brubakk et al. (2006).49E.g Liu (2006), Haider and Khan (2008) and Garcia and Gonzalez (2010).50This selection of parameter distributions are also used in Liu (2006) and Haider and Khan (2008),

as well as in Brubakk et al. (2006).51This value on the discount factor is also used in Kravik and Mimir (2019). Liu (2006), as well as

Karunaratne and Pathberiya (2014) fixes the discount factor.

39

4.3.2 Priors for the firms

For the firms, priors on the price setting fractions θH and θF must be chosen. These priors

are chosen in line with earlier literature52, and are shown in Table 4.3.2.


θH Beta 0.50 0.25 [0, 1]

θF Beta 0.50 0.25 [0, 1]

Table 4.3.2: Priors for the firms

4.3.3 Priors for the foreign economy

For the foreign economy, prior for the elasticity of substitution between domestic and

foreign goods seen from the foreign economy ǫ∗H must be chosen. The degree of openness

ǫB is fixed at 0.4.53 The elasticity of substitution between domestic and foreign goods

seen from the foreign economy is assumed to be the same as for the domestic economy,

see section 3. Priors for the foreign economy are shown in table 4.3.3.


ǫ∗H Gamma 1.00 0.30 R+

Table 4.3.3: Priors for the foreign economy

4.3.4 Shock processe

In order to investigate the dynamics of the model, shock-variables were added to the

variables in section 3. By applying shocks, it is possible to investigate the dynamic

responses of the model and compare the different monetary policies. A shock in the model

can only be temporary, since it is of a stationary stochastic kind. Therefor, a permanent

shock cannot be accommodated, the model must convert back to its constant steady

state. Shocks follows an AR(1) process, hit the economy today (at t = 0) and propagates

throughout the economy until it has recovered back to steady state (in expectation the

shock is zero). The shocks are added to the model equations as "latent shock variables",

see for example equation (3.3.1.3), where the "latent shock variable" zst is added to the

52See e.g Garcia and Gonzalez (2010) and Karunaratne and Pathberiya (2014).53Liu (2006) uses the same fixed value for the degree of openness.Karunaratne and Pathberiya (2014)

fixes the degree of openness at 0.35.

40

TOT. The equation is repeated below for simplicity of the explanation

∆st = πFt − πHt + zst .

Where zst follows an AR(1) process according to

zst = ρszst−1 + ǫst .

Where ρs ∈ (0, 1) is the autocorrelation parameter. It can be thought of as the persistence

of the shock. If ρs is close to 1, then the shock is persistent and resides in the economy

for a long time. On the other hand, if ρs is close to 0, the shock is not persistent. Note

that ρs must be smaller than 1 in absolute value to avoid an explosive shock (|ρs| > 1),

or a random walk (|ρs| = 1). ǫst is white noise with zero mean and variance σ2s .

To avoid stochastic singularity, there must be at least as many shocks as observed

variables.54 The priors for the shocks are separated between endogenous and exogenous

variables, and shown in table 4.3.4 and 4.3.5 respectively.

Variable Shock

∆st zst = ρszst−1 + ǫst

λt zλt = ρλzλt−1 + ǫλt

θt zθt = ρθzθt−1 + ǫθt

πFt zπFt = ρπF z

πFt−1 + ǫ

πFt

πHt zπHt = ρπHz

πHt−1 + ǫ

πHt

rt zrt = ρrzrt−1 + ǫrt

Table 4.3.4: Shocks to endogenous variables

Variable Shock

at at = ρaat−1 + ǫat

c∗t c∗t = ρc∗c∗t−1 + ǫc

∗

t

r∗t r∗t = ρr∗r∗t−1 + ǫr

∗

t

π∗t π∗t = ρπ∗π∗t−1 + ǫπ∗

π

Table 4.3.5: Shocks to exogenous variables

54More technicalities on Bayesian estimation in DSGE models can be found in e.g Hamilton (1994).

41

Bayesian methods are applied to estimate the atocorrelation paramaters, ρi for i =

s, λ . . . π∗. As done in previous literature, the autocorrelation parameters are assumed to

all follow the same distribution.55 Table 4.3.6 contains the priors for the autocorrelation

parameters.


ρi Beta 0.50 0.10 [0, 1]

Table 4.3.6: Priors for the autocorrelation parameters

In the simulation of the model, random shocks are drawn from the prior distribution

and applied to the economy at t = 0. Following the shock, response functions are esti-

mated. The response functions are often referred to as impulse response functions (IRF),

since a shock is applied at t = 0 as an impulse only, not at later times. This is presented

in more detail in section 5. Table 4.3.7 contains the priors for the shocks.


σst InvGamma 2.00 ∞ R+

σqt InvGamma 2.00 ∞ R

+

σθt InvGamma 2.00 ∞ R+

σπFt InvGamma 2.00 ∞ R

+

σπHt InvGamma 2.00 ∞ R

+

σrt InvGamma 2.00 ∞ R+

σat InvGamma 2.00 ∞ R+

σc∗

t InvGamma 2.00 ∞ R+

σr∗


σπ∗


Table 4.3.7: Priors for the shocks

4.3.5 Priors for the monetary policy

For the three policy rules, there are a total of five parameters. Priors for the weight on

the interest rate ωR, weight on the inflation ωΠ, weight on the level of the exchange rate

ωQ1, and weight on the change in the exchange rate ωQ2

must be chosen. The priors are

55See E.g Haider and Khan (2008) and Garcia and Gonzalez (2010).

42

chosen in line with earlier literature.56 Table 4.3.8, 4.3.9 and 4.3.10 contains the priors

that are used for the weights in monetary policy 1, 2 and 3 respectively.


ωR Beta 0.70 0.20 R+

ωΠ Gamma 1.50 0.25 R+

ωY Gamma 0.50 0.10 R+

Table 4.3.8: Priors for monetary policy rule 1 (MP1)


ωR Beta 0.70 0.20 R+

ωΠ Gamma 1.50 0.25 R+

ωY Gamma 0.50 0.10 R+

ωQ1Gamma 0.25 0.10 R

+



ωR Beta 0.70 0.20 R+

ωΠ Gamma 1.50 0.25 R+

ωY Gamma 0.50 0.10 R+


+


+


56See e.g Garcia and Gonzalez (2010) where the same priors for the inflation and the exchange rate areused. Liu (2006) and Karunaratne and Pathberiya (2014) uses the same priors for the weight on interestand output.

43

5 Estimation

The log linearized model from section 3 is simulated for a number of periods, and the

dynamic responses of the model to various shocks is examined. Posterior distributions of

the parameters is estimated, as well as the impulse response functions (IRF). The model

is simulated with the three different monetary policy rules that were introduced earlier,

and the difference in responses of the model under the different rules are examined.

A brief description of the estimation process is as follows.57 The log linearized model

is shocked at period t = 0. Shocks and parameters are drawn from the prior distributions.

Given the model and parameters, historical time series data for Norway described in

section 4 are used to find the likelihood function that fits the density of the data. To

obtain the posterior distributions of the parameters, the priors together with the maximum

likelihood approach are combined using Bayesian methods. After obtaining the posterior

distributions, the model is simulated. The simulation is iterated a number of times,

drawing shocks and parameters from the posterior distributions, and for each draw an

impulse response is generated. Repeating this process a number of times, an average

response of the model is estimated. The response is used to calculate the welfare loss

using the (LF) between the different policy rules. Results of the estimation are further

discussed in section 6.

5.1 Bayesian Estimation

The New Keynesian DSGE literature provides different ways to determine the model pa-

rameters. They range from pure calibration as done by Kydland and Prescott (1982),

Generalized Method of Moments of general equilibrium relationship estimation as done

by Christiano and Eichenbaum (1992), Full Information Maximum Likelihood as done by

Leeper and Sims (1994). Bayesian estimation has grown with great popularity in macroe-

conomics, and more recent literature on New Keynesian DSGE models uses Bayesian

methods to estimate the models.58 Due to improvements in computational power and

technical and practical reasons, it has become a more frequently used tool. Central banks

around the world uses DSGE models that are estimated by Bayesian estimation as a

57More details on the estimation process can be found in the Dynare reference manual Adjemian et al.(2011).

58See e.g. Smets and Wouters (2003) and Lubik and Schorfheide (2005).

44

helping tool when conducting monetary policy.59

In "classical" or "frequentist" statistics one usually treats parameters as fixed quan-

tities. Bayesian methods differ in this area, and treats parameters as random variables.

As mentioned above, Bayesian estimation allows formally the use of prior information

on the distribution of the parameters stemming from microeconometric studies or other

macroeconomic studies in the estimation procedure. Due to the stylized nature and the

resulting misspecification of DSGE models the likelihood function often peaks in regions

of the parameter space which are contradictory with common observations, leading to

dilemma of absurd parameter estimates.60 The likelihood function is re-weighted by the

priors. The priors can bring forth information that is not contained in the estimation

sample such that the likelihood does not peak in the areas that are at odds with common

observations.

This creates a link between previous pure calibration based literature and the more

recent literature, where a formal econometric framework is applied when estimating the

parameters. Due to this capability to relate inference statements to the actual observa-

tions that are collected, Bayesian methods are sometimes preferred by researchers over

traditional methods where inference statements are related to what will happen when for

example similar experiments are repeated many times.

5.1.1 An introduction to Bayesian estimation

In this section, basic ideas on the Bayesian method is provided.61 At a very basic level,

Bayesian estimation can be seen as a combination of calibration and maximum likelihood

estimation. Calibration of the model is done through the specification of priors, or loosely

speaking prior knowledge about the model. The Maximum Likelihood approach enters

in the estimation process based on confronting the model with data. The priors are used

on order to give more importance (weight) to certain areas of the parameter subspace.

In more technical manner, the priors and the likelihood function are linked according to

Bayes rule.

59E.g NEMO used in Norges Bank is estimated by Bayesian methods, see Kravik and Mimir (2019)and Brubakk et al. (2006).

60See An and Schorfheide (2007).61For more information on Bayesian Methods the reader is referred to the literature, e.g. Hamilton

(1994) and Lutkepohl (2005).

45

Let the priors be defined by the density function

p(θM|M). (5.1.1.1)

Where M is a specific model, θM is a vector containing the parameters of the model M,

and p(.) is a probability density function (pdf). Given the model and its parameters, the

density of the model is described by the likelihood function

L(θM|yT ,M). (5.1.1.2)

Where yT is available data, a vector of observation until time T . Recursively the likeli-

hood function can be written as

p(yT |θM,M) = p(y0|θM,M)T∏

t=1

p(yt|yt−1,θM,M). (5.1.1.3)

Taking a step back, we have a prior density p(θ) on one hand, and on the other hand

we have the likelihood function p(yT |θ). We are interested in obtaining the posterior

distribution, p(θ|yT ). Using Bayes’ Theorem and knowing the data, it is possible obtain

the density of the parameters. Generally

p(θ|yT ) =p(θ;yT )

p(yT ). (5.1.1.4)

In similar manner

p(yT |θ) =p(θ;yT )

p(θ)⇔ p(θ|yT ) = p(yT |θ)p(θ). (5.1.1.5)

After deriving these important identities, it is possible to combine the prior density and

the likelihood function discussed above to obtain the posterior density as

p(θM|yT ,M) =p(yT |θM,M)p(θM|M)

p(yT |M). (5.1.1.6)

Where p(yT |M) is the marginal density of the data conditional on the model, i.e,

p(yT |M) =

∫

ΘM

p(θM;yT |M)dθM. (5.1.1.7)

46

The posterior kernal, i.e, the un-normalized posterior density, corresponds to the numer-

ator of the posterior density

p(θM|yT ,M) ∝ p(yT |θM,M)p(θM|M) ≡ K(θM|yT ,M). (5.1.1.8)

Where ∝ denotes proportional to. Using this fundamental equation, it is possible to

rebuild all posterior moments of interest. In order to obtain the likelihood, the trick

is to use the help of Kalman Filter and then simulate the posterior kernel by a Monte

Carlo method such as the Metropolis-Hastings. Matlab, which is a numerical computing

environment,62 is used to do the calculation together with a toolbox called Dynare.63

Dynare is a toolbox for estimating and solving models such a DSGE models. Below,

further details on the estimation process in Dynare as well as the topics on Kalman filter

and Metropolis-Hastings are discussed.

5.1.2 Bayesian estimation and DSGE models

One can think of DSGE models as a collection of first order and equilibrium conditions.

It can be written in general form as

Et{f(yt+1, yt, yt−1, ut)} = 0

E(ut) = 0

E(utu′

t) = Σu.

Where y is the vector of endogenous variables taking any dimension, and u is the vector

of exogenous stochastic shocks taking any dimension. Define further the policy function

as

yt = g(yt−1, ut).

A solution to this system is a set of equations linking variables in the current period to

the past state of the system and current shocks, that satisfy the original system. The

policy function defined above is such a solution. The solution to the DSGE model can be

62See www.mathworks.com for more information on Matlab.63Dynare is pre-processor which solves non-linear model with forward looking variables. See

www.dynare.org.

47

written as a system of the form

y∗t =My(θ) +My +N(θ)xt + ǫt

y = gy(θ)yt−1 + gu(θ)ut

E(ǫtǫ′

t) = V (θ)

E(utu′

t) = Q(θ).

Where yt denotes variables in deviations from their steady state value. y denotes the

steady state values of the variables. θ denotes the vector of structural parameters needed

to be estimated. Notice that the second equation is the policy function mentioned above.

In the system, only y∗t is observable, and is related to the true variables with an error

ǫt. N(θ)xt captures any potential trend in the system. It allows for the most general

case of trend, where the trend depends of the structural parameters. Notice also that the

first and second equation together make up a system of measurement and state equations

respectively.

The next step is to estimate the likelihood of the DSGE solution system above. Note

that the equations are non-linear in structural parameters, but they are linear in the

endogenous and exogenous variables. In order to evaluate the likelihood function, a linear

prediction error algorithm such as the Kalman Filter can be used.

The Kalman Filter recursion can be represented as64

For t = 0, 1, . . . , T given initial values y1 and P1 the recursion is as follows

vt = y∗t − y∗ −Myt −Nxt

Ft =MPtM′

+ V

Kt = gyPtg′

tF−1t

yt+1 = gyyt +Ktvt

Pt+1 = gyPt(gy −KtM)′

+ guQg′

u.

Given the Kalman filter recursion, it is possible to derive the log-likelihood function given

64See Hamilton (1994) for more technicalities on the Kalman Filter.

48

by

lnL(θ|y∗

T) = −

Tk

2ln(2π)−

1

2

T∑

t=1

|Ft| −1

2v

′

tF−1t vt. (5.1.1.9)

Where θ is vector containing the parameters to be estimated, i.e, θ, V (θ) and Q(θ). y∗T

denotes the set of observable endogenous variables yt found in the measurement equation.

The log-likelihood derived above, (5.1.1.9), is one step closer to obtaining the posterior

distribution of the parameters. Let the log posterior kernel be defined as

lnK(θ|y∗T ) = lnL(θ|y∗

T ) + lnp(θ). (5.1.1.10)

Notice that both terms on the right hand side is known. The first term are known after

carrying out the Kalman filter recursion, and the last terms, i.e the priors, are under the

modellers control, and thus known.

The next step is to obtain an important parameter, the mode of the posterior distri-

bution. To find the mode, one must maximize the above log posterior kernel given by

equation, (5.1.1.10). However, this is not straight forward since the log-likelihood function

is not Gaussian w.r.t to θ, but functions of θ from the state space equation. Meaning

that we are not able to obtain an explicit form of the posterior distribution. In order to

obtain the posterior distribution numerical methods are used. A common sampling-like

method used in the literature is the Metropolis-Hastings algorithm, which is regarded as

particularly efficient. The Metropolis-Hastings algorithm is a "rejection sampling algo-

rithm" used to generate a sequence of samples65 from a distribution that is unknown at

the outset. By doing so the algorithm simulates the posterior distribution.

Note that so far, only the posterior mode has been obtained. Usually, one are more

interested in the mean and the variance of the estimators of θ. In order to obtain the

mean and the variance, the algorithm builds on the fact that under general conditions

the distribution of the structural parameters are asymptotically normal. The algorithm

constructs a Gaussian approximation around the posterior mode and uses a scaled version

of the asymptotic covariance matrix as the covariance matrix for the proposal distribu-

tion. This allows for an efficient exploration of the posterior distribution, at least in the

65This is also known as a Markov Chain process.

49

neighborhood of the mode.66 The Metropolis-Hastings algorithm can be summarized in

the following 4 steps

1. θ0 is chosen as a starting point, typically the posterior mode. Then loop over the

steps 2− 3− 4.

2. θ∗ is drawn as a proposal from a jumping distribution

J(θ∗|θt−1) = N (θt−1, cΣm).

Where Σm is the inverse Hessian computed at the mode, c denotes the scale factor

3. Compute the acceptance ratio according to

r =p(θ∗|yT )

p(θt−1|yT )=

K(θ∗|yT)

K(θt−1|yT ).

4. In the final step the proposal θ∗ is either accepted or rejected according to the

following rule

θt =

θ∗ with probability min(r,1)

θt−1 otherwise

,

if necessary, update from the jumping distribution.

In step 1, a candidate parameter θ∗ is chosen from the Normal distribution with

mean θt−1. In step 2, the value of the posterior kernel for that candidate parameter

is calculated and compared to the value of the Posterior Kernel from the mean of the

drawing distribution. Step 3 decides whether or not to keep the candidate parameter.

The candidate parameter is kept if the acceptance ratio is greater than one. If not, go

back to the candidate from last period.67 Then the mean of the drawing distribution is

updated and the value of the parameter that is retained is noted. After repeating these

steps sufficient times, the final step is to build a histogram of the retained values. The

histogram will after sufficient iterations of the above process be the posterior distribution.

66See An and Schorfheide (2007) page 19/20.67Note that in fact the candidate is only kept with a probability less than one.

50

Such a complicated acceptance rule is necessary in order to visit the whole domain of

the posterior distribution. In the case of a candidate giving a lower value of the posterior

kernel we don’t want to throw it out to early. This is in case using the lower value

candidate for the mean of the drawing distribution allows us to leave a local maximum

and instead approach the global maximum. In simpler words one could say the idea is

to allow the search to turn away from small steps up and let it take a few steps down in

order to take bigger steps up in the future. The variance of the jumping distribution, and

in particular the scale factor c, plays a central role in the search process. Should the scale

factor be too small, the acceptance rate, i.e, the fraction of candidate parameters that are

accepted in, will be too high and the Markov Chain of candidates will mix slowly. Then

the distribution will take to long to converge too the posterior distribution, since the chain

is likely to be stuck around a local maximum and not visit the tails of the distribution.

Should on the other hand the scale factor be too high, the the acceptance rate will be

very low. Candidates are then more likely to land in regions of low probability density,

and the chain will spend to much time in the tails of the posterior distribution.

This completes the introduction to Bayesian estimation in DSGE models. Using these

methods, it is possible to obtain the posterior distribution of the parameters. The next

section calculates the posterior distributions for the parameters and the shocks in the

SOE.

5.2 Posterior distributions of the parameters

Taking the log linearized model from section 3 together with the prior and empirical data

from section 4 using the Bayesian methods discussed above in 5.1, the posterior distribu-

tions of all the structural parameters and shocks are obtained. The Metropolis-Hastings

algorithm is used with 5 parallel chains, each with a length of 500.000 replications to

generate the Markov chain. The reason behind using 5 parallel chains is due the improve-

ments in the computation between group variance of the parameter means, which is a

key criteria to evaluate the efficiency of the Metropolis-Hastings to evaluate the posterior

distribution.68 Liu (2006) uses 1.500.000 replications. However, I find no significant differ-

ence in the results by increasing the amount of replications to this extent.69 As discussed

68More details can be found in Griffoli (2010).69Except from the computation time.

51

in section 5.1.1, the scale factor c of the jumping distribution is crucial when estimating

the posteriors. The literature has settled on a value between 0.2 and 0.4 to obtain an

ideal acceptance rate of around 25%.

In the model, there are three different monetary policies, (MP1), (MP2) and (MP3)

respectively, thus the estimation needs to be done three times, one for each policy. The

numerical results from the estimation can be found in Appendix F. Below, a figure con-

taining graphical result for a few priors and posteriors are shown. Graphical results of all

priors and posteriors under all three different policies can be found in Appendix C.

2 4 6 8 10

0

50

100

150

SE_eps_pistar

1.5 2 2.5

0

20

40

sigma

1 1.5 2 2.5

0

10

20

psi

0.5 1 1.5 2

0

20

40

60

epsilon_H

0.2 0.4 0.6 0.8

0

20

40

thetah

0.2 0.4 0.6 0.8

0

20

40

60

thetaf

0.2 0.4 0.6 0.8

0

20

40

60

omegaR

0.4 0.6 0.8

0

50

100

omegaY

1 1.5 2

0

20

40

omegaPI

Figure 5.2.1: Priors and posteriors for a selection of structural parameters, MonetaryPolicy 1, (MP1).

5.3 Simulation and impulse response functions

The estimation process were discussed in section 5, but repeated briefly here for simplicity.

The model is shocked applying all the ten shocks defined in table 4.3.7 at period t = 0.

Shocks and parameters are randomly drawn from the posterior distributions, and for each

draw an impulse response is generated. By repeating this a number of times, an average

52

response of the model is estimated. The model is simulated for T = 80 periods.

The simulation is done for all three monetary policies. In the model there are ten

shocks, such that there is a total of thirty impulse responses. We are most interested in

looking at the impulse response to the domestic variables Yt,Πt, Rt, Qt,ΠFt, St and Ct.

Below, three figures of the impulse responses to a shock in the real exchange rate are

shown. Appendix D contains figures of the impulse response functions to all shocks.

y

20 40 60 80

0.02

0.04

0.06

0.08

0.1

0.12

c

20 40 60 80

-10

-8

-6

-4

-2

10-3 r

20 40 60 80

2

4

6

8

10

12

1410

-3

pi

20 40 60 80

0

0.02

0.04

pif

20 40 60 80

0

0.02

0.04

0.06

q

20 40 60 80

-0.4

-0.2

0

s

20 40 60 80

0.01

0.02

0.03

Figure 5.3.1: Impulse response functions to a shock in the real exchange rate, monetarypolicy 1, (MP1).

53

y

20 40 60 80

0.05

0.1

0.15

c

20 40 60 80

-0.15

-0.1

-0.05

r

20 40 60 80

0.02

0.04

0.06

0.08

0.1

0.12

pi

20 40 60 80

0

0.01

0.02

0.03

pif

20 40 60 80

0

0.05

0.1

q

20 40 60 80

-0.8

-0.6

-0.4

-0.2

0

s

20 40 60 80

0.02

0.04

0.06

0.08

0.1

0.12


y

20 40 60 80

0

0.05

0.1

c

20 40 60 80

-0.04

-0.03

-0.02

-0.01

r

20 40 60 80

5

10

15

10-3

pi

20 40 60 80

-5

0

5

10

15

10-3 pif

20 40 60 80

-0.01

0

0.01

0.02

0.03

q

20 40 60 80

-0.3

-0.2

-0.1

0

s

20 40 60 80

0.01

0.02

0.03


A depreciation of the real exchange rate leads to increased import prices and reduced

export prices, measured in domestic currency. Such that the domestic economy might

54

experience an increase in competitiveness. When the exchange rate depreciates, it leads

to an increased overall inflation due to the price on imported goods rise. This is what is

called cost-push inflationary pressure. Also, the increased exports and fall in imports lead

to increased output. In the figure consumption suffers due to the depreciation. Much of

private consumption in Norway is consumption of foreign goods such that increased import

prices leads to decreased total private consumption. To counter the effects, the central

bank increases the interest rate, bringing the inflation back down, as well as bringing the

exchange rate back up and thus import prices back down.

The dynamics of the model to a shock in the real exchange rate is similar for all the

three policies, which is to be expected. However, what is interesting is which of the three

policies that results is the least welfare loss for the economy, i.e, the policy that results

in least deviations of the variables from steady state. In order to investigate which of the

three policies that results in least welfare loss, the variance of deviation from steady state

for the variables must be obtained.

The model have been estimated under various shocks and the average responses from

all the shocks of the variables of interest for 80 periods have been obtained. In the next

section, these variances are computed and results of the estimation are presented and

discussed.

55

6 Results

In section 5 the model was estimated using Bayesian methods. Ten shocks were applied to

the model and the posterior distributions were estimated. After obtaining the posterior

distributions, shocks and parameters were randomly drawn in order to obtain the impulse

response functions of the model.

Section 2.10 introduced a loss function (LF) that is used to compare the different

monetary polices. It is repeated her for simplicity

LF = σ2π +

1

2σ2y +

1

5σ2r .

Where σ2Π, σ2

y and σ2r denotes the variance of the deviations of inflation, output and interest

rate from its steady state respectively. The average response of the model estimated in the

previous section provides the deviations from steady state over 80 periods. By examining

all the figures of the impulse responses in Appendix D, one can see that in most cases all

variables have converged back to steady state after being hit by a shock after 80 periods

have passed. It is worth noting that most variables reach steady state way before all 80

periods have passed.

In order to compare the different policies, the variance of the deviations from steady

state of the variables must be obtained. This is obtained by calculating the variance of

the average responses to the different shocks. The variance of all the responses can be

found in Appendix E. Note that sometimes the variance is zero, this is because the SOE

has no effect on the foreign economy (i.e, the rest of the world).

To calculate the welfare loss, the variance of the output, the inflation and the interest

rate is used. The results are presented in table 6.0.1 and table 6.0.2.

56

Welfare loss

Shock MP1 MP2 MP3

σst 5,428E-07 1,124E-06 3,496E-06

σqt 5,420E-08 7,325E-07 7,419E-08

σθt 9,585E-08 5,135E-07 4,095E-07

σπFt 1,552E-07 3,097E-07 1,264E-06

σπHt 2,681E-05 1,005E-06 3,783E-06

σrt 1,410E-04 2,808E-05 3,246E-06

σat 2,081E-05 5,805E-06 1,613E-04

Average: 2,706E-05 5,367E-06 2,480E-05

Table 6.0.1: Welfare loss for the different policies when there are shocks to domesticvariables

Welfare loss

Shock MP1 MP2 MP3

σc∗

t 2,221E-06 1,871E-06 8,196E-07

σr∗

t 1,677E-04 1,126E-05 8,271E-07

σπ∗

t 2,128E-07 1,956E-08 7,380E-08

Average: 5,670E-05 4,382E-06 5,735E-07

Table 6.0.2: Welfare loss for the different policies when there are shocks to foreign variables

The results are reported for domestic and foreign shocks separately. When the economy

experiences shocks to domestic variables, the results differ quite a bit between which of

the three policies that results in the lowest welfare loss. On average, (MP2) results in the

lowest welfare loss indicating that the the level of the real exchange rate is important in

the SOE when reacting to shocks in domestic variables.

When the economy experiences shocks to foreign variables, (MP2) and (MP3) yields

on average lower welfare loss. Note that (MP2) and (MP3) are always better regardless

of which foreign shock the economy experiences. Further, on average (MP3) results in

the smallest welfare loss, indicating that both the level of the real exchange rate and

changes in the real exchange are important in the SOE when reacting to shocks in foreign

variables.

57

A results that deserves a comment is when the economy is hit by a shock to the real

exchange rate. From table 6.0.1 (MP1) results in the lowest welfare loss. This is due to

the significant increase in overall inflation when (MP2) and (MP3) are used compared

to when (MP1) is used. Indicating that exchange rate intervention increase welfare loss

when the economy is faced with a depreciation shock due to the effects that intervention

has on inflation in the short run.

58

7 Concluding remarks

In this thesis Bayesian methods were used to combine prior information with historical

data of the Norwegian economy to present a theoretical small open economy model with

staggered price setting for Norway. The main focus in this thesis has been to compare

three different monetary policy rules for the central bank, and investigate if using the real

exchange rate as a deciding factor in the policy rule help reduce welfare loss. The baseline

monetary policy (MP1), had the central bank adjust interest according to past interest,

inflation and output. While (MP2) and (MP3) introduced the level of the real exchange

rate and the change in the real exchange rate as deciding factors respectively.

When the domestic economy experiences shocks to domestic variables, the results

differ between which of the polices that results in lowest welfare loss. On average, (MP2)

results in the lower loss, indicating that using the real exchange rate as a deciding factor

help reduce welfare loss.

The welfare loss is always smaller when (MP2) and (MP3) is used to react to shocks

in foreign variables. On average, (MP3) results in smallest welfare loss, indicating that

using the level of the real exchange and changes in the real exchange rate reduces welfare

loss in response to shocks in foreign variables.

In total, the results suggest that a small open economy such as Norway should use

the real exchange rate as a deciding factor in the policy rule, and use exchange rate

interventions instead of letting its currency float freely. Exchange rate intervention help

reduce the observed volatility in output, inflation and the interest rate in responds to

various shocks, and especially shocks to foreign variables. The welfare loss introduced by

external shocks are reduced significantly.

59

8 Extension

The theoretical framework presented here is restricted to quit simple specification of the

model, with a linear production function and a simple role for the monetary policy. Fur-

ther extensions of the framework could be the incorporation of staggered wage contracts

as formulated by Taylor (1980), capital accumulation with variable capital utilization as

in Christiano et al. (2005). See also Brubakk et al. (2006). It would also be possible to

include habit formation to further investigate the consumption pattern in the economy as

done by e.g Liu (2006), Haider and Khan (2008) and Karunaratne and Pathberiya (2014).

Another possible extension is to introduce an explicit government sector undertaking a

role for fiscal policy and interactions with monetary policy, see e.g von Thadden and Leith

(2006).

Introduction of a housing sector together with a model of the financial market and

the banking sector is another possible extension. Then it is possible to investigate if the

central bank should monitor asset price inflation along with consumer price inflation. It is

possible to analyse the effects of exchange rate intervention on inflation and asset prices.

60

A Appendix A

Appendix A contains mathematical derivations of the theoretical model presented in sec-

tion 2.

61

A.1 Constant elasticity of substitution (CES)

An important economic property that is frequently used when defining the model is con-

stant elasticity of substitution, herafter CES. This property has some desirable and con-

venient properties for some economic applications. CES refers to a particular aggregation

function which combines two or more inputs into an aggregated output. In general form

y =[

∫ n

0

aǫ1i nǫ2i di

]1

ǫ2. (A.1.1)

Where y is the output, ni is input-factor (e.g labor), ai is a parameter that measures the

share of input i, 0 ≤ ai < 1. ǫ1 and ǫ2 define the shape of the function, ǫ1 ∈ R and

−∞ < ǫ1 < 1. ǫ2 measures the substitutiability between inputs. As ǫ2 → 0 the aggregate

function approaches the functional form of the familiar Cobb-Douglas. If one chooses

ǫ1 = 1s

and ǫ2 = s−1s

the general form of the CES production function is obtained. s is

the elasticity of substitution between inputs in the production, s > 0,. The particular

aggregation function exhibits constant elasticity of substitution, and thus referred to as

the CES function. The elasticity of substitution is the percentage in the input ratio

divided by the marginal rate of technical substitution, output being fixed.

62

A.2 Optimal consumption of households

The optimal expression for consumption of domestic, CHt, and foreign goods, CFt. The

maximization problem of the household is maximizing utility subject to their budget

constraint. The Lagrangian of the problem can be written as:

L = Et

{

∞∑

t=0

βtUt(Ct, Nt)}

(A.2.1)

− λt(PtCt + Et∆t,t+1Bt+1 + Et∆∗t,t+1Et+1B

∗t+1 − Bt − EtB

∗t −WtNt − Πt − Tt)

Use (2.1.3) to rewrite the budget constraint by eliminating Ct as:

L = Et

{

∞∑

t=0

βtUt(Ct, Nt)}

(A.2.2)

− λt

(

Pt

[

(1− ǫB)1

ǫH CǫH−1

ǫH

Ht + ǫ1

ǫH

B CǫH−1

ǫH

Ft

]

ǫHǫH−1

+ Et∆t,t+1Bt+1 + Et∆∗t,t+1Et+1B

∗t+1 −Bt − EtB

∗t −WtNt − Πt − Tt

)

Notice also that we can use the result PtCt = PHtCHt + PFtCFt to rewrite the budget

constraint as:

L = Et

{

∞∑

t=0

βtUt(Ct, Nt)}

(A.2.3)

− λt(PHtCHt + PFtCFt + Et∆t,t+1Bt+1 + Et∆∗t,t+1Et+1B



Taking partial derivative w.r.t domestic consumption, CHt, of the second Lagrangian, i.e

(A.2.2), to obtain:

∂L

∂CHt= λtPt

[

(1− ǫB)1

ǫH CǫH−1

ǫH

Ht + ǫ1

ǫH

B CǫH−1

ǫH

Ft

]

ǫHǫH−1

−1

(1− ǫB)1

ǫH

ǫH−1

ǫH CǫH−1

ǫH−1

Ht (A.2.4)

+ βt∂U(Ct, Nt)

∂CHt= 0

Use (2.1.3) to rewrite the optimality condition as:

∂L

∂CHt= −λtPt(1− ǫB)

1

ǫH CǫH−1

ǫH−1

Ht CtC−

ǫH−1

ǫH

t + βt∂U(Ct, Nt)

∂CHt= 0 (A.2.5)

63

Now take partial derivative w.r.t domestic consumption, CHt of the third Lagrangian, i.e

(A.2.3), to obtain:

∂L

∂CHt= −λtPHt + βt

∂U(Ct, Nt)

∂CHt= 0 (A.2.6)

Combine these results and solving for CHt to get:

CHt = (1− ǫB)(PHt

Pt

)−ǫHCt (A.2.7)

which completes the derivation of (2.1.15). Taking Partial derivative w.r.t foreign con-

sumption, CFt, of (A.2.2) and (A.2.3), and following the same process as done for domestic

consumption a similar expression:

CFt = ǫB(PFt

Pt

)−ǫHCt (A.2.8)

which completes the derivation of (2.1.16), and is left as an exercise to the reader to show.

64

A.3 Solving the household problem

Recall that the households solve the maximization problem given by equation (HP):

maxCt,Nt,Bt+1,B

∗t+1

Et

{

∞∑

t=0

βtUt(Ct, Nt)}

s.t


∗t+1 ≤ Bt + EtB

∗t +WtNt +Πt + Tt

The Lagrangian of this problem can be written as:

L = Et

∞∑

t=0

βt(C1−σ

t

1− σ−N

1+ψt

1 + ψ

)

− λt(PtCt + Et∆t,t+1Bt+1 + Et∆∗t,t+1Et+1B



Taking partial derivative w.r.t Ct, Nt, Bt+1, B∗t+1 yields the following first order conditions:

∂L

∂Ct→ βtCσ

t = λtPt (A.3.1)

∂L

∂Nt

→ βtNψt = λtWt (A.3.2)

∂L

∂B∗t+1

→ λtEt∆∗t,t+1Et+1 = λt+1Et (A.3.3)

∂L

∂Bt+1

→ λtEt∆t,t+1 = λt+1 (A.3.4)

Combining (A.3.1) and (A.3.4)

First note that (A.3.1) can be written as:

βtC−σt

Pt= λt (A.3.1a)

This also implies that:

βt+1C−σt+1

Pt+1

= λt+1 (A.3.1b)

Insert (A.3.1a) and (A.3.1b) into (A.3.4), and we can get:

βtC−σt

Pt= Et∆

−1t,t+1β

t+1C−σt+1

Pt+1

65

Rearranging yields:

C−σt = β∆−1

t,t+1Et

{

C−σt+1

Pt

Pt+1

}

(A.3.5)

Which completes the derivation of (2.1.17).

Combining (A.3.1) and (A.3.3) in the same manner as before:

βtC−σt

PtEt+1 = Et∆

−1∗t,t+1β

t+1C−σt+1

Pt+1

Et

Rearranging this yields:

C−σt = β∆−1

t,t+1Et

{

C−σt+1

Pt

Pt+1

Et+1

Et

}

(A.3.6)


Finally combine (A.3.1) and (A.3.2)

First notice that (A.3.2) can be written as:

βtNψt

Wt

= λt (A.3.2a)

Set (A.3.2a) and (A.3.1a) equal to eliminate λt, and we get:

βtNψt

Wt

= βtC−σt

Pt

Rearranging this yields:

Wt

Pt= Cσ

t Nψt (A.3.7)


Notice also that we can get the stochastic discount factor from (A.3.5). Rearranging

this yields:

βEt

{(Ct+1

Ct

)−σ( Pt

Pt+1

)}

= Et

{

∆t,t+1

}

(A.3.8)

66

It is also useful to note that (A.3.8) can be written as:

C−σt = βRtEt

{

C−σt+1

Pt

Pt+1

}

(A.3.8b)

67

A.4 Obtaining the uncovered interest rate parity condition

Notice first that both (2.1.17) and (2.1.18) can be written respectively as:

1 = β∆−1t,t+1Et

{(Ct+1

Ct

)−σ Pt

Pt+1

}

(A.4.1)

1 = β∆−1∗t,t+1Et

{(Ct+1

Ct

)−σ Pt

Pt+1

Et+1

Et

}

(A.4.2)

Set (A.4.1) equal to (A.4.2):

β∆−1∗t,t+1Et

{(Ct+1

Ct

)−σ Pt

Pt+1

}

= β∆−1∗t,t+1Et

{(Ct+1

Ct

)−σ Pt

Pt+1

Et+1

Et

}

Rearranging yields:

∆∗t,t+1

∆t,t+1

= Et

{Et+1

Et

}

(A.4.3)


68

A.5 Finding the optimal price setting

Recall that a domestic firm solves the following optimization problem:

maxPHt

∞∑

k=0

(θkH)Et{


}

s.t Yit+k =( PHt

PHt+k

)−ǫpYt+k

The Lagrangian of this problem can be written as:

L =∞∑

k=0

(θkH)Et{


}

(A.5.1)

− λt

(

Yit+k −( PHt

PHt+k

)−ǫpYt+k

)

Take partial derivative w.r.t PHt:

∂L

∂PHt→

∞∑

k=0

(θkH)Et{

∆t,t+kYt+k[PHt −MΨHt+k]}

= 0 (A.5.2)

where ΨHt+k =MCrHt+kPHt+k is the nominal marginal cost measured in domestic prices,

and M = ǫpǫp−1

is real marginal cost if prices were fully flexible, i.e a friction-less mark-up.

This completes the derivation of (2.6.2.3).

69

B Appendix B

Appendix B contains mathematical derivations of the log linearized model in section 3.

70

B.1 Log linearizing household conditions

Log linearizing the Households conditions:

C−σ(ect)−σ = βEt∆−1t,t+1Et

{

C−σ(ect+1)−σ( Pept

Pept+1

)}

(B.1.1)

−σct ≈ lnβ + ln∆t,t+1 + Et

{

− σct+1 + pt − pt+1

}

ct ≈ Et{ct+1} −1

σ(rt − Et{πt+1} − ρ}

Which completes the derivation of (3.2.2.1)

B.2 Log linearizing terms of trade, domestic inflation and CPI

inflation

B.2.1 Terms of trade

First order approximation around symmetric steady state Sjt = Sj = 1 ∀t:

St =PFt

PHt=

(

∫ 1

0

S1−ǫFjt dj

)1

1−ǫF (B.2.1.1)

A first order approximation of this:

St ≈(

∫ 1

0

S1−ǫFj dj

)1

1−ǫF +1

1− ǫF

(

∫ 1

0

(1− ǫF )S−ǫFj (Sjt − Sj)dj

)

st = pFt − pHt ≈

∫ 1

0

st − 1

1dj ≈

∫ 1

0

sjtdj (B.2.1.2)

Which completes the derivation of (3.3.1.1).

71

B.2.2 CPI inflation

Log linearaizing the CPI inflation (2.1.14) around symmetric steady state

PH = PF = P :

Pt ≈[

(1− εB)P1−ǫH + ǫBP

1−ǫH]

1

1−ǫH

+1

1− ǫH

[

(1− ǫB)P1−ǫH + ǫBP

1−ǫH ]1

1−ǫH−1[

(1− ǫB)(1− ǫH)P−ǫH (PHt − P )

+ ǫB(1− ǫH)P−ǫH (PFt − P )

]

= P + [(1− ǫB)(PHt − P ) + ǫB(PFt − P )] (B.2.2.1)

Then we can obtain:

Pt − P

P≈ (1− ǫB)

PHt − P

P+ ǫB

PFt − P

P

pt = (1− ǫB)pHt + ǫBpFt (B.2.2.2)

Which completes the derivation of (3.3.2.1).

B.3 Log linearizing the uncovered interest parity condition

Recall that we have terms of trade given by:

st = Et∆et+1 − Etπ∗t+1 + EtπHt+1 + Etst+1

Combine this with rt = r∗t + Et∆et+1 to get:

st = (r∗t − Etπ∗t+1)− (rt − EtπHt+1) + Etst+1 (B.3.1)

72

It follows by the assumptions of the model that limT → ∞ EtST = 0. Then we can

solve iterative forward to get:

st = (r∗t − Etπ∗t+1)− (rt − EtπHt+1) + (r∗t+1 − Etπ

∗t+2)− (rt+1 − EtπHt+2)

+ (r∗t+2 − Etπ∗t+3)− (rt+2 − EtπHt+3) + (r∗t+3 − Etπ

∗t+4)− (rt+3 − EtπHt+4)

+ . . .

= Et{

∞∑

n=0

[(r∗t+n − π∗t+n)− (rt+n − πHt+n)]

}

(B.3.2)

Which is the same as (3.6.3).

73

B.4 Log linearizing optimal price

Have PHt∑∞

k=0(βθH)kEt

{

C1−ǫpt+k P

ǫp−1Ht+k

}

= M∑∞

k=0(βθH)kEt

{

C1−ǫpt+k P

ǫpHt+kMCr

t+k

}

. Steady

state conditions Ct+k = Ct = C,PHt+k = PHt = P and MCrt+k =MCr.

Taylor expansion on LHS first:

LHS ≈P∞∑

k=0

(βθH)kEt[C

1−σP ǫp−1] (B.4.1)

+∞∑

k=0

(βθH)kEt[C

1−σP ǫ−1][PHt − P ]

+ P

∞∑

k=0

(βθH)kEt[C

1−σ(ǫp − 1)P ǫp−2][PHt+k − P ]

+ P

∞∑

k=0

(βθH)kEt[C

−σ(1− σ)P ǫp−1][Ct+k − C]

= C1−σP ǫp

∞∑

k=0

(βθH)kEt

{

1 +PHt − P

P+ (ǫp − 1)

PHt+k − P

P+ (1− σ)

Ct+k − C

C

}

= C1−σP ǫp

∞∑

k=0

(βθH)kEt {1 + (pHt − p) + (ǫp − 1)(pHt+k − p) + (1− σ)(ct+k − c)}

Taylor expansion on the RHS:

RHS ≈M∞∑

k=0

(βθH)kEt[C

1−σP ǫpMCr] (B.4.2)

+M∞∑

k=0

(βθH)kEt[C

1−σǫpPǫp−1MCr][PHt+k − P ]

+M∞∑

k=0

(βθH)kEt[C

−σ(1− σ)P ǫpMCr][Ct+k − C]

+M∞∑

k=0

(βθH)kEt[C

1−ǫpP ǫp ][MCrt+k −MCr]

= C1−σP ǫp

∞∑

k=0

(βθH)kEt

{

1 + ǫpPHt+k − P

P(1− σ)

Ct+k − C

C+MCr

Ht+k −MCr

MCr

}

= C1−σP ǫp

∞∑

k=0

(βθH)kEt

{

1 + ǫp(pHt+k − p) + (1− σ)(ct+k − c) + (mcrt+k −mcr)}

74

Setting both sides equal to get:

C1−σP ǫp

∞∑

k=0

(βθH)kEt {1 + (pHt − p) + (ǫp − 1)(pHt+k − p) + (1− σ)(ct+k − c)} (B.4.3)

= C1−σP ǫp

∞∑

k=0

(βθH)kEt

{

1 + ǫp(pHt+k − p) + (1− σ)(ct+k − c) + (mcrt+k −mcr)}

Divide by C1−σP ǫp and collect terms:

Et

{

∞∑

k=0

(βθH)k[pHt − pHt+k

}

= Et

{

∞∑

k=0

(βθH)k[mcrt+k −mcr]

}

Et

{

∞∑

k=0

(βθH)kpHt

}

= Et

{

∞∑

k=0

(βθH)k[mcrt+k + pt+k −mcr]

}

(B.4.4)

Use the fact that µ = −mcr and ψHt+k = mcrt+k + pHt+k. Insert, solve for pHt to

obtain:∞∑

k=0

(βθH)kpHt = Et

{

∞∑

k=0

(βθH)k[ψHt+k + µ]

}

(B.4.5)

Use that∑∞

k=0(βθ)k = 1

1−βθ. Notice that µ does not depend on k and can be taken outside

the sum. Also Et{µ} = µ. Finally solve for pHt to obtain:

pHt

1− βθH=

µ

1− βθH+ Et

{

∞∑

k=0

(βθH)k[ψHt+k]

}

(B.4.6)

Multiply by (1− βθH) on both sides to obtain the finale expression:

pHt = µ+ (1− βθH)Et

{

∞∑

k=0

(βθH)k[ψHt+k]

}

(B.4.7)

75

B.5 Log linearizing the inflation dynamics

Recall that the domestic price evolves according to (2.6.2.1), which can be log linearized

around a symmetric steady state in similar manner as when the CPI were log linearized

in Appendix B.2.2. Such that:

pHt = (1− θH)pHt + θHpHt−1 (B.5.1)

the domestic inflation is given by:

πHt = pHt − pHt−1 (B.5.2)

add and subtract pHt−1 from (B.5.1) to obtain domestic inflation (3.10.2.1).

pHt = (1− θH)pHt + θHpHt−1 + pHt−1 − pHt−1

pHt − pHt−1 = (1− θH)pHt + (θH − 1)pHt−1

πHt = (1− θH)(pHt − pHt−1) (B.5.3)

In order to derive the NKPC for a domestic firm given by (3.10.2.2), first rewrite the

optimal price setting (3.8.3.1).

pHt = pHt−1 +∞∑

k=0

(βkθkH)Et

{

πHt+k + (1− βθH)mcrt+k

}

(B.5.4)

pHt = pHt−1 + πHt + (1− βθH)mcrt + (βθ)

∞∑

k=0

(βkθkH)Et

{

πHt+k+1 + (1− βθH)mcrt+k+1

}

Notice that this step splits the summation into two parts. One starting at time t and the

other from t+ 1 to ∞. For the next step, rewrite using the original expression.

pHt = pHt−1 + πHt + (1− βθH)mcrt + (βθ)(pHt+1 − pHt) (B.5.5)

where pHt+1 − pHt =∑∞

k=0(βkθkH)Et

{

πHt+k+1 + (1− βθH)mcrt+k+1

}

. Finally rearrange to

obtain the familiar NKPC.

pHt − pHt−1 = (βθH)EtπHt+1 + πHt + 1− βθHmcrt (B.5.6)

76

Finally combine (B.5.3) and (B.5.6) to obtain the evolution of domestic inflation (3.10.2.2).

πHt = (1− θH)EtπHt+1 + λHmcrt (B.5.7)

Which completes the derivations of the inflation dynamics of the model, (3.10.2.2). Similar

derivations holds for πFt.

77

C Appendix C

Appendix C contains all graphical results for all priors and posteriors for all structural

parameters and shocks under the three different monetary policies. In the graphs the grey

line represents the prior, the black line represents the posterior, and the dotted green line

represents the mode of the posterior distribution.

78

C.1 Priors and posteriors monetary policy 1

2 4 6 8 10

0

10

20

SE_eps_s

2 4 6 8 10

0

50

100

150

SE_eps_q

2 4 6 8 10

0

10

20

SE_eps_theta

2 4 6 8 10

0

5

10

SE_eps_pih

2 4 6 8 10

0

10

20

SE_eps_pif

2 4 6 8 10

0

20

40

60SE_eps_r

2 4 6 8 10

0

2

4

6

8

SE_eps_a

2 4 6 8 10

0

5

10

15

SE_eps_ystar

2 4 6 8 10

0

10

20

SE_eps_rstar

Figure C.1.1: Priors and posteriors, monetary policy 1, (MP1).

2 4 6 8 10

0

50

100

150

SE_eps_pistar

1.5 2 2.5

0

20

40

sigma

1 1.5 2 2.5

0

10

20

psi

0.5 1 1.5 2

0

20

40

60

epsilon_H

0.2 0.4 0.6 0.8

0

20

40

thetah

0.2 0.4 0.6 0.8

0

20

40

60

thetaf

0.2 0.4 0.6 0.8

0

20

40

60

omegaR

0.4 0.6 0.8

0

50

100

omegaY

1 1.5 2

0

20

40

omegaPI


79

0.2 0.4 0.6 0.8

0

20

40

rho_s

0.2 0.4 0.6 0.8

0

20

40

rho_q

0.2 0.4 0.6 0.8

0

20

40

60

rho_theta

0.2 0.4 0.6 0.8

0

20

40

rho_pif

0.2 0.4 0.6 0.8

0

20

40

60

rho_pih

0.2 0.4 0.6 0.8

0

20

40

60

rho_r

0.2 0.4 0.6 0.8

0

100

200

rho_a

0.2 0.4 0.6 0.8

0

50

100

rho_ystar

0.2 0.4 0.6 0.8

0

20

40

rho_rstar


0.2 0.4 0.6 0.8

0

20

40

60

80

rho_pistar


80

C.2 Priors and posteriors, monetary policy 2

0.2 0.4 0.6 0.8

0

20

40

rho_rstar

0.2 0.4 0.6 0.8

0

50

100

rho_pistar


0.2 0.4 0.6

0

50

100

omegaQ1

0.2 0.4 0.6 0.8

0

50

100

150

rho_s

0.2 0.4 0.6 0.8

0

20

40

60

rho_q

0.2 0.4 0.6 0.8

0

20

40

rho_theta

0.2 0.4 0.6 0.8

0

20

40

60

rho_pif

0.2 0.4 0.6 0.8

0

20

40

60

rho_pih

0.2 0.4 0.6 0.8

0

50

100

150

rho_r

0.2 0.4 0.6 0.8

0

20

40

rho_a

0.2 0.4 0.6 0.8

0

20

40

rho_ystar


81

2 4 6 8 10

0

10

20

SE_eps_pistar

1.5 2 2.5

0

20

40

sigma

1 1.5 2 2.5

0

20

40

psi

0.5 1 1.5 2

0

20

40

epsilon_H

0.2 0.4 0.6 0.8

0

10

20

30

thetah

0.2 0.4 0.6 0.8

0

20

40

60

thetaf

0.2 0.4 0.6 0.8

0

20

40

60

80

omegaR

0.4 0.6 0.8

0

50

100

omegaY

1 1.5 2

0

20

40

omegaPI


2 4 6 8 10

0

5

10

15

SE_eps_s

2 4 6 8 10

0

20

40

SE_eps_q

2 4 6 8 10

0

5

10

15

SE_eps_theta

2 4 6 8 10

0

20

40

SE_eps_pih

2 4 6 8 10

0

5

10

15

SE_eps_pif

2 4 6 8 10

0

100

200

SE_eps_r

2 4 6 8 10

0

2

4

SE_eps_a

2 4 6 8 10

0

10

20

SE_eps_ystar

2 4 6 8 10

0

10

20

SE_eps_rstar


82

C.3 Priors and posteriors monetary policy 3

0.2 0.4 0.6 0.8 1

0

20

40

rho_ystar

0.2 0.4 0.6 0.8

0

50

100

rho_rstar

0.2 0.4 0.6 0.8

0

50

100

150

rho_pistar


0.2 0.4 0.6

0

100

200

omegaQ1

0.2 0.4 0.6

0

100

200

omegaQ2

0.2 0.4 0.6 0.8

0

20

40

rho_s

0.2 0.4 0.6 0.8

0

20

40

60

rho_q

0.2 0.4 0.6 0.8

0

20

40

60

rho_theta

0.2 0.4 0.6 0.8

0

20

40

60

80

rho_pif

0.2 0.4 0.6 0.8

0

20

40

60

rho_pih

0.2 0.4 0.6 0.8

0

20

40

60

80

rho_r

0.2 0.4 0.6 0.8

0

50

100

150

rho_a


83

2 4 6 8 10

0

20

40

60

SE_eps_pistar

1.5 2 2.5

0

50

100

sigma

1 1.5 2 2.5

0

10

20

30

psi

0.5 1 1.5 2

0

20

40

epsilon_H

0.2 0.4 0.6 0.8

0

20

40

60

thetah

0.2 0.4 0.6 0.8

0

20

40

60

thetaf

0.2 0.4 0.6 0.8

0

50

100

omegaR

0.4 0.6 0.8

0

100

200

omegaY

1 1.5 2

0

20

40

omegaPI


2 4 6 8 10

0

5

10

15

SE_eps_s

2 4 6 8 10

0

20

40

SE_eps_q

2 4 6 8 10

0

10

20

SE_eps_theta

2 4 6 8 10

0

10

20

SE_eps_pih

2 4 6 8 10

0

50

100

SE_eps_pif

2 4 6 8 10

0

50

100

150

SE_eps_r

2 4 6 8 10

0

10

20

30

SE_eps_a

2 4 6 8 10

0

10

20

SE_eps_ystar

2 4 6 8 10

0

100

200

SE_eps_rstar


84

D Appendix D

Appendix D contains impulse response figures for all ten shocks under the three different

monetary policies.

85

D.1 Impulse responses monetary policy 1

y

20 40 60 80

0.01

0.02

0.03

0.04

0.05

c

20 40 60 80

-6

-4

-2

10-3 r

20 40 60 80

2

4

6

10-3

pi

20 40 60 80

2

4

6

8

10

10-3 pif

20 40 60 80

-0.08

-0.06

-0.04

-0.02

q

20 40 60 80

5

10

15

10-3

s

20 40 60 80

0.05

0.1

0.15

0.2

Figure D.1.1: Impulse response functions to a shock in the terms of trade, monetarypolicy 1, (MP1).

y

20 40 60 80

0.02

0.04

0.06

0.08

0.1

0.12

c

20 40 60 80

-10

-8

-6

-4

-2

10-3 r

20 40 60 80

2

4

6

8

10

12

1410

-3

pi

20 40 60 80

0

0.02

0.04

pif

20 40 60 80

0

0.02

0.04

0.06

q

20 40 60 80

-0.4

-0.2

0

s

20 40 60 80

0.01

0.02

0.03

Figure D.1.2: Impulse response functions to a shock in the real exchange rate, monetarypolicy 1, (MP1).

86

y

20 40 60 80

0.05

0.1

0.15

0.2

0.25

c

20 40 60 80

-0.02

-0.015

-0.01

-0.005

r

20 40 60 80

0.01

0.02

0.03

pi

20 40 60 80

0

0.05

0.1

pif

20 40 60 80

0

0.05

0.1

0.15

q

20 40 60 80

0.01

0.02

0.03

0.04

s

20 40 60 80

0.02

0.04

0.06

0.08

Figure D.1.3: Impulse response functions to a shock in the law of one price gap, monetarypolicy 1, (MP1).

y

20 40 60 80

0.01

0.02

0.03

0.04

c

20 40 60 80

-0.03

-0.02

-0.01

r

20 40 60 80

0.005

0.01

0.015

0.02

pi

20 40 60 80

0

0.05

0.1

pif

20 40 60 80

0

0.1

0.2

q

20 40 60 80

0.02

0.04

0.06

s

20 40 60 80

0.05

0.1

0.15

0.2

0.25

Figure D.1.4: Impulse response functions to a shock in the import inflation, monetarypolicy 1, (MP1).

87

y

20 40 60 80

-0.2

-0.15

-0.1

-0.05

c

20 40 60 80

-0.06

-0.04

-0.02

0

r

20 40 60 80

0

0.02

0.04

pi

20 40 60 80

0

0.2

0.4

pif

20 40 60 80

0.02

0.04

0.06

0.08

0.1

0.12

q

20 40 60 80

0

0.05

0.1

s

20 40 60 80

-0.5

-0.4

-0.3

-0.2

-0.1

Figure D.1.5: Impulse response functions to a shock in the domestic inflation, monetarypolicy 1, (MP1).

y

20 40 60 80

-3

-2

-1

0c

20 40 60 80

-2.5

-2

-1.5

-1

-0.5

r

20 40 60 80

0.1

0.2

0.3

pi

20 40 60 80

-2

-1

0pif

20 40 60 80

-2

-1.5

-1

-0.5

q

20 40 60 80

1

2

3

4

5

s

20 40 60 80

0.2

0.4

0.6

Figure D.1.6: Impulse response functions to a shock in the domestic interest rate, mon-etary policy 1, (MP1).

88

y

20 40 60 80

0.1

0.2

0.3

0.4

0.5

c

20 40 60 80

0.05

0.1

0.15

0.2

0.25

r

20 40 60 80

-0.08

-0.06

-0.04

-0.02

pi

20 40 60 80

-0.2

-0.15

-0.1

-0.05

pif

20 40 60 80

-0.04

-0.02

0

q

20 40 60 80

-0.5

-0.4

-0.3

-0.2

-0.1

s

20 40 60 80

0.2

0.4

0.6

Figure D.1.7: Impulse response functions to a productivity shock, monetary policy 1,(MP1).

y

20 40 60 80

0.02

0.04

0.06

0.08

0.1

0.12

0.14

c

20 40 60 80

-15

-10

-5

10-3 r

20 40 60 80

2

4

6

8

10

12

10-3

pi

20 40 60 80

0.01

0.02

0.03

pif

20 40 60 80

2

4

6

8

10-3 q

20 40 60 80

0.01

0.02

0.03

s

20 40 60 80

-0.05

-0.04

-0.03

-0.02

-0.01

ystar

20 40 60 80

0.1

0.2

0.3

0.4

Figure D.1.8: Impulse response functions to a shock in the foreign output, monetarypolicy 1, (MP1).

89

y

20 40 60 80

0.2

0.4

0.6

0.8

1

1.2

1.4

c

20 40 60 80

-1

-0.8

-0.6

-0.4

-0.2

r

20 40 60 80

0.1

0.2

0.3

0.4

rstar

20 40 60 80

0.2

0.4

0.6

0.8

pi

20 40 60 80

0.2

0.4

0.6

0.8

pif

20 40 60 80

0

0.5

1

1.5

q

20 40 60 80

-5

-4

-3

-2

-1

s

20 40 60 80

0.5

1

1.5

2

Figure D.1.9: Impulse response functions to a shock in the foreign interest rate, monetarypolicy 1, (MP1).

y

20 40 60 80

-0.25

-0.2

-0.15

-0.1

-0.05

c

20 40 60 80

0.005

0.01

0.015

0.02

r

20 40 60 80

-0.025

-0.02

-0.015

-0.01

-0.005

pi

20 40 60 80

-0.1

-0.05

0

pif

20 40 60 80

-0.15

-0.1

-0.05

0

q

20 40 60 80

0

0.2

0.4

0.6

0.8

s

20 40 60 80

-0.08

-0.06

-0.04

-0.02

pistar

20 40 60 80

0.2

0.4

0.6

0.8

Figure D.1.10: Impulse response functions to a shock in the foreign inflation, monetarypolicy 1, (MP1).

90

D.2 Impulse responses monetary policy 2

y

20 40 60 80

0.01

0.02

0.03

0.04

c

20 40 60 80

-8

-6

-4

-2

10-3 r

20 40 60 80

2

4

6

8

10

12

14

10-3

pi

20 40 60 80

5

10

15

10-3 pif

20 40 60 80

-0.1

-0.08

-0.06

-0.04

-0.02

q

20 40 60 80

5

10

15

10-3

s

20 40 60 80

0.05

0.1

0.15

0.2


y

20 40 60 80

0.05

0.1

0.15

c

20 40 60 80

-0.15

-0.1

-0.05

r

20 40 60 80

0.02

0.04

0.06

0.08

0.1

0.12

pi

20 40 60 80

0

0.01

0.02

0.03

pif

20 40 60 80

0

0.05

0.1

q

20 40 60 80

-0.8

-0.6

-0.4

-0.2

0

s

20 40 60 80

0.02

0.04

0.06

0.08

0.1

0.12


91

y

20 40 60 80

0.05

0.1

0.15

0.2

0.25

c

20 40 60 80

-0.04

-0.03

-0.02

-0.01

r

20 40 60 80

0.02

0.04

0.06

0.08

pi

20 40 60 80

0.05

0.1

0.15

pif

20 40 60 80

0

0.1

0.2

q

20 40 60 80

0.02

0.04

0.06

0.08

s

20 40 60 80

0.02

0.04

0.06

0.08

0.1

0.12

0.14


y

20 40 60 80

0.01

0.02

0.03

0.04

0.05

c

20 40 60 80

-0.03

-0.02

-0.01

r

20 40 60 80

0.02

0.04

0.06

pi

20 40 60 80

0.05

0.1

0.15

pif

20 40 60 80

0

0.1

0.2

0.3

q

20 40 60 80

0.02

0.04

0.06

s

20 40 60 80

0.1

0.2

0.3


92

y

20 40 60 80

0.01

0.02

0.03

0.04

0.05

c

20 40 60 80

-0.03

-0.02

-0.01

r

20 40 60 80

0.02

0.04

0.06

pi

20 40 60 80

0.05

0.1

0.15

pif

20 40 60 80

0

0.1

0.2

0.3

q

20 40 60 80

0.02

0.04

0.06

s

20 40 60 80

0.1

0.2

0.3


y

20 40 60 80

-1

-0.5

0

c

20 40 60 80

-1.2

-1

-0.8

-0.6

-0.4

-0.2

r

20 40 60 80

0.1

0.2

0.3

pi

20 40 60 80

-0.8

-0.6

-0.4

-0.2

0pif

20 40 60 80

-0.6

-0.4

-0.2

q

20 40 60 80

0.5

1

1.5

2

s

20 40 60 80

0.05

0.1

0.15

0.2

0.25


93

y

20 40 60 80

0.02

0.04

0.06

0.08

0.1

c

20 40 60 80

0.01

0.02

0.03

0.04

0.05

r

20 40 60 80

-0.06

-0.04

-0.02

pi

20 40 60 80

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

pif

20 40 60 80

-0.06

-0.04

-0.02

q

20 40 60 80

-0.08

-0.06

-0.04

-0.02

s

20 40 60 80

0.05

0.1

0.15


y

20 40 60 80

0.02

0.04

0.06

0.08

0.1

c

20 40 60 80

-8

-6

-4

-2

10-3 r

20 40 60 80

0.005

0.01

0.015

0.02

pi

20 40 60 80

0.01

0.02

0.03

pif

20 40 60 80

5

10

15

10-3 q

20 40 60 80

5

10

15

10-3

s

20 40 60 80

-0.04

-0.03

-0.02

-0.01

ystar

20 40 60 80

0.1

0.2

0.3


94

y

20 40 60 80

0.02

0.04

0.06

0.08

0.1

0.12

c

20 40 60 80

-0.8

-0.6

-0.4

-0.2

r

20 40 60 80

0.05

0.1

0.15

0.2

rstar

20 40 60 80

0.2

0.4

0.6

0.8

pi

20 40 60 80

-0.1

-0.05

0

pif

20 40 60 80

0

0.1

0.2

q

20 40 60 80

-1.5

-1

-0.5

s

20 40 60 80

0.2

0.4

0.6


y

20 40 60 80

-0.025

-0.02

-0.015

-0.01

-0.005

c

20 40 60 80

0.02

0.04

0.06

r

20 40 60 80

-0.025

-0.02

-0.015

-0.01

-0.005

pi

20 40 60 80

-2

0

2

10-3 pif

20 40 60 80

-0.02

-0.01

0

q

20 40 60 80

0

0.1

0.2

s

20 40 60 80

-0.04

-0.03

-0.02

-0.01

pistar

20 40 60 80

0.05

0.1

0.15

0.2

Figure D.2.10: Impulse response functions to a shock in the foreign interest rate, mone-tary policy 2, (MP2).

95

D.3 Impulse reponses monetary policy 3

y

20 40 60 80

0.005

0.01

0.015

0.02

0.025

c

20 40 60 80

-6

-4

-2

10-3 r

20 40 60 80

2

4

6

8

10

10-3

pi

20 40 60 80

2

4

6

8

10

10-3 pif

20 40 60 80

-0.15

-0.1

-0.05

q

20 40 60 80

5

10

15

10-3

s

20 40 60 80

0.02

0.04

0.06

0.08

0.1


y

20 40 60 80

0

0.05

0.1

c

20 40 60 80

-0.04

-0.03

-0.02

-0.01

r

20 40 60 80

5

10

15

10-3

pi

20 40 60 80

-5

0

5

10

15

10-3 pif

20 40 60 80

-0.01

0

0.01

0.02

0.03

q

20 40 60 80

-0.3

-0.2

-0.1

0

s

20 40 60 80

0.01

0.02

0.03


96

y

20 40 60 80

0.1

0.2

0.3

0.4

c

20 40 60 80

-6

-4

-2

10-3 r

20 40 60 80

0.01

0.02

0.03

0.04

pi

20 40 60 80

0.02

0.04

0.06

0.08

0.1

pif

20 40 60 80

0

0.05

0.1

q

20 40 60 80

5

10

15

10-3

s

20 40 60 80

0.01

0.02

0.03


y

20 40 60 80

0.02

0.04

0.06

0.08

0.1

c

20 40 60 80

-0.04

-0.03

-0.02

-0.01

r

20 40 60 80

0.02

0.04

0.06

pi

20 40 60 80

0.05

0.1

0.15

0.2

pif

20 40 60 80

0

0.2

0.4

q

20 40 60 80

0.02

0.04

0.06

0.08

0.1

s

20 40 60 80

0.1

0.2

0.3


97

y

20 40 60 80

-0.2

-0.15

-0.1

-0.05

c

20 40 60 80

-0.04

-0.02

0

r

20 40 60 80

0

0.02

0.04

pi

20 40 60 80

0

0.1

0.2

pif

20 40 60 80

0.02

0.04

0.06

0.08

0.1

q

20 40 60 80

0

0.02

0.04

0.06

0.08

s

20 40 60 80

-0.3

-0.2

-0.1


y

20 40 60 80

-0.2

-0.15

-0.1

-0.05

c

20 40 60 80

-0.04

-0.02

0

r

20 40 60 80

0

0.02

0.04

pi

20 40 60 80

0

0.1

0.2

pif

20 40 60 80

0.02

0.04

0.06

0.08

0.1

q

20 40 60 80

0

0.02

0.04

0.06

0.08

s

20 40 60 80

-0.3

-0.2

-0.1


98

y

20 40 60 80

0.05

0.1

0.15

0.2

c

20 40 60 80

0.02

0.04

0.06

r

20 40 60 80

-0.08

-0.06

-0.04

-0.02

pi

20 40 60 80

-0.15

-0.1

-0.05

pif

20 40 60 80

-0.08

-0.06

-0.04

-0.02

q

20 40 60 80

-0.15

-0.1

-0.05

s

20 40 60 80

0.05

0.1

0.15

0.2

0.25


y

20 40 60 80

0

0.05

0.1

c

20 40 60 80

-2

-1

0

10-3 r

20 40 60 80

0

5

10

10-3

pi

20 40 60 80

0

0.01

0.02

pif

20 40 60 80

2

4

6

8

10-3 q

20 40 60 80

0

2

4

10-3

s

20 40 60 80

-0.025

-0.02

-0.015

-0.01

-0.005

ystar

20 40 60 80

0.1

0.2

0.3

0.4


99

y

20 40 60 80

0.05

0.1

0.15

c

20 40 60 80

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

r

20 40 60 80

0.01

0.02

0.03

rstar

20 40 60 80

0.05

0.1

0.15

0.2

pi

20 40 60 80

-0.01

0

0.01

0.02

pif

20 40 60 80

-0.02

0

0.02

0.04

0.06

q

20 40 60 80

-0.4

-0.2

0s

20 40 60 80

0.05

0.1

0.15


y

20 40 60 80

-0.04

-0.02

0

c

20 40 60 80

5

10

15

10-3 r

20 40 60 80

-6

-4

-2

10-3

pi

20 40 60 80

-6

-4

-2

0

2

10-3 pif

20 40 60 80

-10

-5

0

10-3 q

20 40 60 80

0

0.05

0.1

s

20 40 60 80

-12

-10

-8

-6

-4

-2

10-3 pistar

20 40 60 80

0

0.1

0.2

0.3

Figure D.3.10: Impulse response functions to a shock in the foreign inflation, monetarypolicy 3, (MP3).

100

E Appendix E

Appendix E contains tables of the variances of the average response function under the

three different monetary policies.

101

E.1 Variances monetary policy 1

Variable Shock Variance Variable Shock Variance

y σ2

s 1,02569E-06 y σ2

r 0,000123804

q σ2

s 1,12949E-07 q σ2

r 0,000325309

r σ2

s 1,78227E-08 r σ2

r 2,76222E-06

π σ2

s 2,63593E-08 π σ2

r 7,85194E-05

πF σ2

s 2,27956E-06 πF σ2

r 6,57348E-05

πH σ2

s 1,1206E-06 πH σ2

r 8,64066E-05

s σ2

s 1,78846E-05 s σ2

r 2,04764E-05

c σ2

s 2,82352E-08 c σ2

r 9,50001E-05

y∗ σ2

s 0 y∗ σ2

r 0

r∗ σ2

s 0 r∗ σ2

r 0

π∗ σ2

s 0 π∗ σ2

r 0

y σ2

q 8,63E-08 y σ2

a 3,7686E-05

q σ2

q 5,68E-07 q σ2

a 3,61224E-05

r σ2

q 4,19E-09 r σ2

a 1,00026E-06

π σ2

q 1,02E-08 π σ2

a 1,76452E-06

πF σ2

q 1,65E-08 πF σ2

a 7,71732E-07

πH σ2

q 7,84E-09 πH σ2

a 3,29445E-06

s σ2

q 1,43E-08 s σ2

a 8,00718E-05

c σ2

q 1,77E-09 c σ2

a 9,95237E-06

y∗ σ2

q 0 y∗ σ2

a 0

r∗ σ2

q 0 r∗ σ2

a 0

π∗ σ2

q 0 π∗ σ2

a 0

y σ2

θ 1,32886E-07 y σ2

y∗ 4,09073E-06

q σ2

θ 5,94835E-08 q σ2

y∗ 5,38309E-07

r σ2

θ 2,97967E-08 r σ2

y∗ 9,10729E-08

π σ2

θ 2,34507E-08 π σ2

y∗ 1,57744E-07

πF σ2

θ 3,5381E-08 πF σ2

y∗ 6,24112E-08

πH σ2

θ 2,03365E-08 πH σ2

y∗ 2,66113E-07

s σ2

θ 7,80577E-08 s σ2

y∗ 1,73024E-06

c σ2

θ 1,17022E-08 c σ2

y∗ 1,34872E-07

y∗ σ2

θ 0 y∗ σ2

y∗ 6,02003E-05

r∗ σ2

θ 0 r∗ σ2

y∗ 0

π∗ σ2

θ 0 π∗ σ2

y∗ 0

y σ2

πF5,08068E-08 y σ2

r∗0,000234938

q σ2

πF8,77084E-08 q σ2

r∗0,001985609

r σ2

πF1,7892E-08 r σ2

r∗5,67158E-05

π σ2

πF1,26207E-07 π σ2

r∗3,88416E-05

πF σ2

πF5,56947E-07 πF σ2

r∗9,10545E-05

πH σ2

πF3,82406E-08 πH σ2

r∗2,96855E-05

s σ2

πF1,58493E-06 s σ2

r∗0,001929689

c σ2

πF2,18239E-08 c σ2

r∗0,000392526

y∗ σ2

πF0 y∗ σ2

r∗0

r∗ σ2

πF0 r∗ σ2

r∗3,42532E-05

π∗ σ2

πF0 π∗ σ2

r∗0

y σ2

πH1,39185E-05 y σ2

π∗ 3,1367E-07

q σ2

πH3,57688E-06 q σ2

π∗ 3,24341E-06

r σ2

πH5,13086E-07 r σ2

π∗ 1,68311E-08

π σ2

πH1,97489E-05 π σ2

π∗ 5,26483E-08

πF σ2

πH3,89376E-06 πF σ2

π∗ 1,07209E-07

πH σ2

πH3,70128E-05 πH σ2

π∗ 3,44412E-08

s σ2

πH8,32148E-05 s σ2

π∗ 2,14433E-07

c σ2

πH8,98849E-07 c σ2

π∗ 2,83181E-08

y∗ σ2

πH0 y∗ σ2

π∗ 0

r∗ σ2

πH0 r∗ σ2

π∗ 0

π∗ σ2

πH0 π∗ σ2

π∗ 1,28966E-06

Table E.1.1: The variance of the average response function, monetary policy 1 (MP1)

102



y σ2

s 1,83857E-06 y σ2

r 2,60554E-05

q σ2

s 1,67503E-07 q σ2

r 9,16117E-05

r σ2

s 1,36223E-07 r σ2

r 3,42955E-06

π σ2

s 1,77567E-07 π σ2

r 1,43663E-05

πF σ2

s 6,68734E-06 πF σ2

r 1,25754E-05

πH σ2

s 4,14059E-06 πH σ2

r 1,54416E-05

s σ2

s 3,50743E-05 s σ2

r 2,1014E-06

c σ2

s 4,40861E-08 c σ2

r 2,20421E-05

y∗ σ2

s 0 y∗ σ2

r 0

r∗ σ2

s 0 r∗ σ2

r 0

π∗ σ2

s 0 π∗ σ2

r 0

y σ2

q 9,36664E-07 y σ2

a 5,96458E-06

q σ2

q 3,31162E-05 q σ2

a 4,87912E-06

r σ2

q 9,876E-07 r σ2

a 1,91214E-06

π σ2

q 6,66608E-08 π σ2

a 2,44048E-06

πF σ2

q 5,15969E-07 πF σ2

a 1,94659E-06

πH σ2

q 4,82004E-08 πH σ2

a 2,86114E-06

s σ2

q 3,08198E-06 s σ2

a 1,78118E-05

c σ2

q 2,3384E-06 c σ2

a 1,20059E-06

y∗ σ2

q 0 y∗ σ2

a 0

r∗ σ2

q 0 r∗ σ2

a 0

π∗ σ2

q 0 π∗ σ2

a 0

y σ2

θ 2,817E-07 y σ2

y∗ 3,09454E-06

q σ2

θ 4,8135E-07 q σ2

y∗ 1,64095E-07

r σ2

θ 3,92764E-07 r σ2

y∗ 1,50729E-07

π σ2

θ 2,94061E-07 π σ2

y∗ 2,93238E-07

πF σ2

θ 3,78755E-07 πF σ2

y∗ 1,57094E-07

πH σ2

θ 2,66657E-07 πH σ2

y∗ 4,19675E-07

s σ2

θ 1,00043E-06 s σ2

y∗ 1,56003E-06

c σ2

θ 1,81884E-07 c σ2

y∗ 4,06824E-08

y∗ σ2

θ 0 y∗ σ2

y∗ 4,19799E-05

r∗ σ2

θ 0 r∗ σ2

y∗ 0

π∗ σ2

θ 0 π∗ σ2

y∗ 0

y σ2

πF8,48042E-08 y σ2

r∗6,87253E-06

q σ2

πF1,4508E-07 q σ2

r∗9,93963E-05

r σ2

πF1,04164E-07 r σ2

r∗2,2807E-06

π σ2

πF2,46514E-07 π σ2

r∗7,36269E-06

πF σ2

πF7,98591E-07 πF σ2

r∗3,25053E-06

πH σ2

πF1,21868E-07 πH σ2

r∗1,33593E-05

s σ2

πF2,67401E-06 s σ2

r∗0,000198341

c σ2

πF4,5943E-08 c σ2

r∗7,40937E-05

y∗ σ2

πF0 y∗ σ2

r∗0

r∗ σ2

πF0 r∗ σ2

r∗1,60108E-05

π∗ σ2

πF0 π∗ σ2

r∗0

y σ2

πH4,4736E-07 y σ2

π∗ 1,88264E-08

q σ2

πH1,5244E-07 q σ2

π∗ 8,21489E-07

r σ2

πH1,37554E-07 r σ2

π∗ 4,22744E-08

π σ2

πH7,53913E-07 π σ2

π∗ 1,68843E-09

πF σ2

πH2,29731E-07 πF σ2

π∗ 1,79312E-08

πH σ2

πH1,29078E-06 πH σ2

π∗ 3,96994E-09

s σ2

πH3,17271E-06 s σ2

π∗ 2,41662E-07

c σ2

πH4,26272E-08 c σ2

π∗ 1,76558E-07

y∗ σ2

πH0 y∗ σ2

π∗ 0

r∗ σ2

πH0 r∗ σ2

π∗ 0

π∗ σ2

πH0 π∗ σ2

π∗ 3,63176E-07


103



y σ2

s 5,30327E-06 y σ2

r 3,57385E-06

q σ2

s 2,05648E-06 q σ2

r 7,79837E-06

r σ2

s 6,65419E-07 r σ2

r 1,22132E-07

π σ2

s 7,10889E-07 π σ2

r 1,4351E-06

πF σ2

s 0,000465534 πF σ2

r 1,15207E-06

πH σ2

s 0,000146298 πH σ2

r 1,61345E-06

s σ2

s 6,49077E-05 s σ2

r 4,15103E-07

c σ2

s 3,96928E-07 c σ2

r 1,18778E-06

y∗ σ2

s 0 y∗ σ2

r 0

r∗ σ2

s 0 r∗ σ2

r 0

π∗ σ2

s 0 π∗ σ2

r 0

y σ2

q 1,22194E-07 y σ2

a 0,000250332

q σ2

q 9,98574E-07 q σ2

a 0,000126413

r σ2

q 3,52807E-08 r σ2

a 2,81743E-05

π σ2

q 6,0339E-09 π σ2

a 3,05181E-05

πF σ2

q 1,5711E-08 πF σ2

a 2,81841E-05

πH σ2

q 1,20143E-08 πH σ2

a 3,25054E-05

s σ2

q 1,88235E-07 s σ2

a 0,000339922

c σ2

q 1,54925E-07 c σ2

a 2,39607E-05

y∗ σ2

q 0 y∗ σ2

a 0

r∗ σ2

q 0 r∗ σ2

a 0

π∗ σ2

q 0 π∗ σ2

a 0

y σ2

θ 7,06433E-07 y σ2

y∗ 1,52137E-06

q σ2

θ 2,58783E-08 q σ2

y∗ 7,59352E-09

r σ2

θ 3,42641E-08 r σ2

y∗ 1,82802E-08

π σ2

θ 4,9475E-08 π σ2

y∗ 5,52389E-08

πF σ2

θ 7,85346E-08 πF σ2

y∗ 1,95393E-08

πH σ2

θ 4,14972E-08 πH σ2

y∗ 9,10081E-08

s σ2

θ 9,77154E-08 s σ2

y∗ 2,24458E-07

c σ2

θ 5,00147E-09 c σ2

y∗ 1,52793E-09

y∗ σ2

θ 0 y∗ σ2

y∗ 1,72444E-05

r∗ σ2

θ 0 r∗ σ2

y∗ 0

π∗ σ2

θ 0 π∗ σ2

y∗ 0

y σ2

πF5,07172E-07 y σ2

r∗1,21542E-06

q σ2

πF1,81174E-06 q σ2

r∗1,06342E-05

r σ2

πF5,07169E-07 r σ2

r∗3,16339E-07

π σ2

πF9,09411E-07 π σ2

r∗1,5617E-07

πF σ2

πF2,13946E-06 πF σ2

r∗1,73059E-07

πH σ2

πF5,86361E-07 πH σ2

r∗4,39773E-07

s σ2

πF1,20929E-05 s σ2

r∗1,14904E-05

c σ2

πF3,75332E-07 c σ2

r∗5,72437E-06

y∗ σ2

πF0 y∗ σ2

r∗0

r∗ σ2

πF0 r∗ σ2

r∗1,52782E-06

π∗ σ2

πF0 π∗ σ2

r∗0

y σ2

πH4,1637E-06 y σ2

π∗ 1,35655E-07

q σ2

πH1,26591E-06 q σ2

π∗ 1,2728E-06

r σ2

πH3,65915E-07 r σ2

π∗ 1,3364E-08

π σ2

πH1,62832E-06 π σ2

π∗ 3,3037E-09

πF σ2

πH5,2251E-07 πF σ2

π∗ 1,38129E-08

πH σ2

πH2,75552E-06 πH σ2

π∗ 2,16619E-09

s σ2

πH9,49196E-06 s σ2

π∗ 5,2479E-08

c σ2

πH2,47721E-07 c σ2

π∗ 4,6105E-08

y∗ σ2

πH0 y∗ σ2

π∗ 0

r∗ σ2

πH0 r∗ σ2

π∗ 0

π∗ σ2

πH0 π∗ σ2

π∗ 5,42584E-06


104

F Appendix F

Appendix F contains numerical results of the posterior distributions for the model.

105

F.1 Posterior results monetary policy 1

Parameters Prior mean Post. mean 90% HPD interval Prior Pstdev

σ 2.000 2.2368 2.2188 2.2554 gamm 0.2500

ψ 1.500 1.8797 1.8391 1.9159 gamm 0.3000

ǫH 1.000 0.7792 0.7648 0.7930 gamm 0.3000

θH 0.500 0.7493 0.7289 0.7631 beta 0.2500

θF 0.500 0.7288 0.7195 0.7423 beta 0.2500

ωR 0.700 0.6269 0.6142 0.6390 beta 0.2000

ωY 0.500 0.5069 0.4997 0.5125 gamm 0.1000

ωΠ 1.500 1.6875 1.6718 1.7016 gamm 0.2500

ρs 0.500 0.3683 0.3546 0.3797 beta 0.2000

ρq 0.500 0.5295 0.5107 0.5501 beta 0.2000

ρθ 0.500 0.3602 0.3414 0.3748 beta 0.2000

ρπF 0.500 0.2639 0.2517 0.2768 beta 0.2000

ρπH 0.500 0.5357 0.5199 0.5471 beta 0.2000

ρr 0.500 0.4619 0.4499 0.4733 beta 0.2000

ρa 0.500 0.9433 0.9378 0.9468 beta 0.2000

ρy∗ 0.500 0.8217 0.8149 0.8294 beta 0.2000

ρr∗ 0.500 0.7611 0.7441 0.7783 beta 0.2000

ρπ∗ 0.500 0.6203 0.6064 0.6336 beta 0.2000

Table F.1.1: Posterior results of the structural parameters, monetary policy 1

106

Shock Prior. mean Post. mean 90% HPD interval Prior Pstdev

σs 2.000 0.2679 0.2353 0.2974 invg Inf

σq 2.000 0.2397 0.2352 0.2457 invg Inf

σθ 2.000 0.2641 0.2354 0.2871 invg Inf

σπH 2.000 1.2334 1.1713 1.3240 invg Inf

σπF 2.000 1.2153 1.1698 1.2461 invg Inf

σr 2.000 1.0983 1.0853 1.1178 invg Inf

σa 2.000 0.4715 0.3741 0.5740 invg Inf

σy∗ 2.000 0.2695 0.2352 0.2989 invg Inf

σr∗ 2.000 0.7285 0.6884 0.7571 invg Inf

σπ∗ 2.000 0.2391 0.2352 0.2441 invg Inf

Table F.1.2: Posterior results of the shocks, monetary policy 1

107


Parameters Prior mean Post. mean 90% HPD interval Prior Pstdev

σ 2.000 1.9040 1.8842 1.9259 gamm 0.2500

ψ 1.500 1.5496 1.5341 1.5696 gamm 0.3000

ǫH 1.000 0.8779 0.8631 0.8945 gamm 0.3000

θH 0.500 0.7002 0.6775 0.7200 beta 0.2500

θF 0.500 0.3925 0.3832 0.4010 beta 0.2500

ωR 0.700 0.6950 0.6844 0.7061 beta 0.2000

ωY 0.500 0.4722 0.4663 0.4782 gamm 0.1000

ωΠ 1.500 1.7473 1.7353 1.7597 gamm 0.2500

ωQ10.250 0.4265 0.4194 0.4329 gamm 0.1000

ρs 0.500 0.4528 0.4486 0.4569 beta 0.2000

ρq 0.500 0.3962 0.3856 0.4122 beta 0.2000

ρθ 0.500 0.5092 0.4902 0.5288 beta 0.2000

ρπF 0.500 0.5980 0.5878 0.6089 beta 0.2000

ρπH 0.500 0.6100 0.5915 0.6233 beta 0.2000

ρr 0.500 0.5415 0.5354 0.5483 beta 0.2000

ρa 0.500 0.4901 0.4729 0.5027 beta 0.2000

ρy∗ 0.500 0.3669 0.3488 0.3823 beta 0.2000

ρr∗ 0.500 0.6498 0.6324 0.6645 beta 0.2000

ρπ∗ 0.500 0.4747 0.4647 0.4855 beta 0.2000


108

Shock Prior. mean Post. mean 90% HPD interval Prior Pstdev

σs 2.000 0.2788 0.2352 0.3148 invg Inf

σq 2.000 2.1001 2.0856 2.1143 invg Inf

σθ 2.000 0.2766 0.2360 0.3093 invg Inf

σπH 2.000 0.2491 0.2352 0.2655 invg Inf

σπF 2.000 2.1140 2.0537 2.1622 invg Inf

σr 2.000 0.2386 0.2352 0.2429 invg Inf

σa 2.000 1.2058 1.0509 1.3286 invg Inf

σy∗ 2.000 0.2638 0.2352 0.2912 invg Inf

σr∗ 2.000 1.5407 1.5163 1.5648 invg Inf

σπ∗ 2.000 1.4722 1.4423 1.4964 invg Inf


109


Parameters Prior mean Post. mean 90% HPD intervall Prior Pstdev

σ 2.000 1.9138 1.8681 1.9521 gamm 0.2500

ψ 1.500 0.6293 0.6024 0.6601 gamm 0.3000

ǫH 1.000 0.5373 0.5087 0.5668 gamm 0.3000

θH 0.500 0.4003 0.3832 0.4151 beta 0.2500

θF 0.500 0.9704 0.9527 0.9986 beta 0.2500

ωR 0.700 0.6352 0.6181 0.6508 beta 0.2000

ωY 0.500 0.3670 0.3587 0.3751 gamm 0.1000

ωΠ 1.500 1.3469 1.3303 1.3605 gamm 0.2500

ωQ10.250 0.2284 0.2150 0.2409 gamm 0.1000

ωQ20.250 0.1476 0.1329 0.1612 gamm 0.1000

ρs 0.500 0.5459 0.5195 0.5647 beta 0.2000

ρq 0.500 0.2466 0.2039 0.2892 beta 0.2000

ρθ 0.500 0.9265 0.8990 0.9533 beta 0.2000

ρπF 0.500 0.2480 0.2114 0.2761 beta 0.2000

ρπH 0.500 0.2913 0.2608 0.3248 beta 0.2000

ρr 0.500 0.0482 0.0153 0.0816 beta 0.2000

ρa 0.500 0.5376 0.5121 0.5595 beta 0.2000

ρy∗ 0.500 0.9890 0.9803 0.9984 beta 0.2000

ρr∗ 0.500 0.8235 0.8072 0.8452 beta 0.2000

ρπ∗ 0.500 0.4693 0.4554 0.4817 beta 0.2000


110

Shock Prior mean Post. mean 90% HPD interval Prior Pstdev

σs 2.000 0.2780 0.2368 0.3120 invg Inf

σq 2.000 0.7443 0.7062 0.7810 invg Inf

σθ 2.000 0.2959 0.2569 0.3391 invg Inf

σπH 2.000 0.2426 0.2352 0.2555 invg Inf

σπF 2.000 0.2443 0.2352 0.2575 invg Inf

σr 2.000 0.6194 0.5733 0.6899 invg Inf

σa 2.000 0.3735 0.3003 0.4514 invg Inf

σy∗ 2.000 0.2480 0.2352 0.2638 invg Inf

σr∗ 2.000 1.6792 1.6298 1.7176 invg Inf

σπ∗ 2.000 0.2385 0.2352 0.2428 invg Inf


111

G Appendix G

Appendix G contains description as well as plots of the data used in the model.

112

G.1 Data description

Variable Code Description

Yt yNorway, Gross Domestic Product (Mainland),

Total, Current Prices, SA, Market Prices, NOK,

Rt r Norway, Interbank Rates, NIBOR, 3 Month,

Πt pi Norway, Consumer Price Index, Total, Index

ΠFt pif Norway, External trade in commodities, Price Index, Imports, SA

Qt q i-44 index

Ct cFinal consumption expenditure of households, Current prices,

seasonally adjusted (NOK million)

Π∗t pistar United States, Consumer Price Index, All Items, SA, Index

R∗t rstar

Euro Area, Interbank Rates, LIBOR, 3 Month, Fixing

United States, Interbank Rates, LIBOR, 3 Month, Fixing

Y ∗t ystar

Euro Area, Gross Domestic Product,

Total, Calendar Adjusted, Current Prices, SA, Market Prices, EUR

United States, Gross Domestic Product,

Total, Current Prices, SA, AR, USD

Table G.1.1: Data description

113

G.2 Plots of the data−

.02

−.0

10

.01

.02

pi

1995q1 2000q1 2005q1 2010q1 2015q1 2020q1Quarter

−.0

6−

.04

−.0

20

.02

.04

pif

1995q1 2000q1 2005q1 2010q1 2015q1 2020q1Quarter

−.0

3−

.02

−.0

10

.01

.02

pis

tar

1995q1 2000q1 2005q1 2010q1 2015q1 2020q1Quarter

−.0

20

.02

.04

r

1995q1 2000q1 2005q1 2010q1 2015q1 2020q1Quarter

−.0

20

.02

.04

rsta

r

1995q1 2000q1 2005q1 2010q1 2015q1 2020q1Quarter

−.1

−.0

50

.05

.1.1

5q

1995q1 2000q1 2005q1 2010q1 2015q1 2020q1Quarter

−.0

50

.05

c

1995q1 2000q1 2005q1 2010q1 2015q1 2020q1Quarter

−.1

−.0

50

.05

.1y

1995q1 2000q1 2005q1 2010q1 2015q1 2020q1Quarter

−.0

50

.05

.1ysta

r

1995q1 2000q1 2005q1 2010q1 2015q1 2020q1Quarter

Figure G.2.1: Plots of the data used in the model

115

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