solving heterogeneous agent models with aggregate ...the dimensionality of the system f is still an...

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Solving Heterogeneous Agent Models with Aggregate Uncertainty and many Idiosyncratic States in Discrete Time by Perturbation Methods

Solving Heterogeneous Agent Models withAggregate Uncertainty and many Idiosyncratic

States in Discrete Time by Perturbation Methods

AMCM, Lillehammer

C. Bayer, R. LuettickeU Bonn, CEPR & IZA UCL & CEPR

February 2019

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Solving Highly-Dimensional Heterogeneous Agent Models with Aggregate Uncertainty

Motivation

Heterogeneous agents models with aggregate uncertainty. . . have become of widespread use in macro

because heterogeneity can change the transmission mechanism

I McKay et al. (2016), Kaplan et al. (2017), Luetticke (2018)

because re-distributional policies have business cycle impact

I e.g. McKay and Reis (2016)

because business cycle policies have distributional impact

I e.g. Gornemann et al. (2012)

because heterogeneity gives rise to new drivers of the cycle

I e.g. Guerrieri and Lorenzoni (2017), Mitman (2016), Bayeret al. (2018)

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Motivation

Heterogeneous agents models with aggregate uncertainty

Yet, these models are computational demanding to solve

I The original Krusell and Smith (1997, 1998) algorithm isnotoriously slow

I Therefore, many papers use MIT shocks

I or are restricted to relatively simple household decisions

I We depart from the Reiter (2002, 2009) perturbation method

I And (try to) provide an accessible algorithm that can dealwith high-dimensional heterogeneity

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Intuition

Placing what we do in the literature

& Intuition

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Intuition

Reiter (2002): Solve by perturbation

The heterogeneous agent model:

I Can be written as a non-linear difference equation

I that is function valued and

I needs to be linearized around the stationary equilibrium (StE)

I Functions need to be approximated by finite dimensionalobjects (e.g. coefficients of polynomials, splines, etc.)

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Intuition

Reiter (2009): Reduce dimensionality ex ante

Problem:

I Dimensionality of the difference equation is large

Proposal:

I Reduce dimensionality ex ante (before solving the StE)

I e.g. (sparse) splines to represent policy functions

I Then linearize

Winberry (2016) extends this to the distribution functions

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Intuition

Ahn et al. (2017): Reduce dimensionality after linearization

Problem:

I Hard to determine where to be sparse

I StE is easy to solve

I Why not use all information/precision here?

Proposal:

I Reduce dimensionality after linearization

I Relies on SVD of the Jacobians of the difference equation

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Intuition

What we do

Problem:

I Jacobian is large before reduction

I Only sparse in continuous time

Proposal:

I Reduce dimensionality after StE, but before linearization

I Extract from the StE the important basis functions torepresent individual policies (akin to image compression)

I Perturb only those basis functions but use the StE as“reference frame” for the policies (akin to video compression)

I Similarly for distributions (details later)

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Intuition

Video compression

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Setup

Setup

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Setup

Recursive Dynamic Planning Problem

Consider a household problem in presence of aggregate andidiosyncratic risk

I St is an (exogenous) aggregate state

I sit is a partly endogenous idiosyncratic state

I µt is the distribution over s

I Bellman equation:

ν(sit ,St , µt) = maxx∈Γ(sit ,Pt )

u(sit , x) + βEν(sit+1(x , sit),St+1, µt+1)

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Setup

Recursive Dynamic Planning Problem

Consider a household problem in presence of aggregate andidiosyncratic risk

I St is an (exogenous) aggregate state

I sit is a partly endogenous idiosyncratic state

I µt is the distribution over s

I Euler equation:

u′[x(sit ,St , µt)] = βR(St , µt)Eu′[x(sit+1,St+1, µt+1)],

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Setup

No aggregate risk

Recall how to solve for a StE

I Discretize the state space (vectorized)

I Optimal policy h̄(sit ;P) induces flow utility ūh̄ and transitionprobability matrix Πh̄

I Discretized Bellman equation

ν̄ = ūh̄ + βΠh̄ν̄ (1)

holds for optimal policy (assuming a linear interpolant for thecontinuation value)

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Setup

No aggregate risk

I and for the law of motion for the distribution (histograms)

d µ̄ = d µ̄Πh̄ (2)

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Setup

No aggregate risk

Equilibrium requires

I h̄ is the optimal policy given P and ν (being a linearinterpolant)

I ν̄ and d µ̄ solve (1) and (2)

I Markets clear (some joint requirement on h̄, µ,P, denoted asΦ(h̄, µ,P) = 0)

This can be solved for efficiently

I d µ̄ is vector corresponding to the unit-eigenvalue of Πh̄I Using fast solution techniques for the DP, e.g. EGM

I Using a root-finder to solve for P

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Setup

Introducing aggregate risk

With aggregate risk

I Prices and distributions change over time

Yet, for the household

I Only prices and continuation values matter

I Distributions do not influence the decisions directly

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Setup

Redefining equilibrium (Reiter, 2002)

A sequential equilibrium with recursive individual planning

I A sequence of discretized Bellman equation, such that

νt = ūPt + βΠht νt+1 (3)

holds for optimal policy, ht (which results from νt+1 and Pt)

I and a sequence of histograms, such that

dµt+1 = dµtΠht (4)

holds given the optimal policy

I (Policy functions, ht , that are optimal given Pt , νt+1)

I Prices, distributions and policies lead to market clearing

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Setup

Compact notation (Schmitt-Grohé and Uribe, 2004)The equilibrium conditions as a non-linear difference equation

I Controls: Yt =[νt Pt Z

Yt

]and

I States: Xt =[µt St Z

Xt

]where Zt are purely aggregate states/controls

I Define

F (dµt ,St , dµt+1,St+1, νt ,Pt , νt+1,Pt+1, εt+1) (5)

=

dµt+1 − dµtΠht

νt − (ūht + βΠht νt+1)St+1 −H(St , dµt , εt+1)

Φ(ht , dµt ,Pt , St)εt+1

s.t.

ht(s) = arg maxx∈Γ(s,Pt )

u(s, x) + βEνt+1(s′) (6)

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Setup

Compact notation (Schmitt-Grohé and Uribe, 2004)

The equilibrium conditions as a non-linear difference equation

I Function-valued difference equationEF (Xt ,Xt+1,Yt ,Yt+1, εt+1) = 0

I turns real-valued when we replace the functions by theirdiscretized counterparts

I Standard techniques to solve linearized version

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Setup

So, is all solved?

The dimensionality of the system F is still an issue

I With high dimensional idiosyncratic states, discretized valuefunctions and distributions become large objects

I For example:4 income states (grid points)× 100 illiquid asset states× 100 liquid asset states=⇒ ≥ 40, 000 control variables in F

I Same number of state variables

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Our technique

Our Method

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Our technique

Our idea

1.) Apply compression techniques as in video encoding

I Apply a discrete cosine transformation to all value/policyfunctions (Chebycheff polynomials on roots grid) DCT

I Define as reference “frame”: the StE value/policy function

I Write fluctuations as differences from this reference frame

I Assume all coefficients of the DCT from the StE close to zerodo not change after shock

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Our technique

Our idea

2.) Neglect changes in the rank correlation structure of µ

I Calculate the Copula, C̄ of µ in the StE

I Perturb only the marginal distributions

I Use fixed Copula to calculate an approximate jointdistribution from marginals

I Idea follows Krusell and Smith (1998) in that some momentsof the distribution do not matter for aggregate dynamics

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Our technique

Details

1.) Apply compression techniques as in video encoding

I Let Θ̄ = dct(ν̄) be the coefficients obtained from the DCT ofthe value function in StE

I Define an index set I that contains the x percent largest (i.e.most important) elements from Θ̄

I Let θ be a sparse vector with non-zero entries only forelements i ∈ I

I Define Θ̃(θt) =

{Θ̄(i) + θt(i) i ∈ IΘ̄(i) else

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Our technique

Details

Decoding

I Now we reconstruct νt = ν(θt) = idct(Θ̃(θt))I This means that in the StE the reduction step adds no

additional approximation error as ν(0) = ν̄ by construction

I Yet, it allows to reduce the number of derivatives that need tobe calculated from the outset

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Our technique

Details

Analogously for the histogram

I We µt as C̄ (µ̄1t , . . . , µ̄nt ) for n being the dimensionality of theidiosyncratic states

I The StE distribution is obtained when µ = C̄ (µ̄1, . . . , µ̄n)I Typically prices are only influenced through the marginal

distributions

I The approach ensures that changes in the mass of onedimension, say wealth, are distributed in a sensible way acrossthe other dimensions

I The implied distributions look “similar” to the StE one(different in (Reiter, 2009))

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Our technique

Details

Too many equations

I The system

F({dµ1t , . . . , dµnt },St , {dµ1t+1, . . . , dµnt+1}, St+1, (7)

θt ,Pt , θt+1,Pt+1) =dC̄ (µ̄1t , . . . , µ̄nt )− dC̄ (µ̄1t , . . . , µ̄nt )Πht

dct[idct(Θ̃(θt))−

(ūht + βΠht idct(Θ̃(θt+1))

)]St+1 −H(St , dµt)Φ(ht , dµt ,Pt , St)

has too many equations

I Use only difference in marginals and the differences on I

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Krusell-Smith example

A first working example

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A simple KS economy

Incomplete Markets and TFP

I Household productivity can be high or low

I No contingent claims

I Households save in capital goods (which they rent out)

I Households supply labor (disutility) and consume (utility)

I Aggregate productivity (TFP) follows a log AR-1 process

I Cobb-Douglas production function

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A simple KS economy

Numerical setup

I Asset grid has 100 points ( =⇒ a total grid size of 200)I Policies solved by EGM (instead of VFI)

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Different levels of “compression”Individual consumption policies

0 5 10 15 20 25 30 35Capital

0

0.5

1

1.5C

onsu

mpt

ion

Full grid, low incomeFull grid, high incomeApproximation, low incomeApproximation, high income

Figure: Policy and 10 most important basis functions

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Different levels of “compression”Individual consumption policies

0 5 10 15 20 25 30 35Capital

0

0.5

1

1.5C

onsu

mpt

ion

Full grid, low incomeFull grid, high incomeApproximation, low incomeApproximation, high income

Figure: Policy and 50 most important basis functions

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Different levels of “compression”Individual policy response to a 20%TFP shock

0 5 10 15 20 25 30 35Capital

0

0.5

1

1.5C

onsu

mpt

ion

Full grid, low incomeFull grid, high incomelow income, +20% TFP-Shockhigh income, +20% TFP-Shock

Figure: Perturbing 10 most important basis functions

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Different levels of “compression”Individual policy response to a 20%TFP shock

0 5 10 15 20 25 30 35Capital

0

0.5

1

1.5C

onsu

mpt

ion

Full grid, low incomeFull grid, high incomelow income, +20% TFP-Shockhigh income, +20% TFP-Shock

Figure: Perturbing all 200 basis functions

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Different levels of “compression”Aggregate response to a 20%TFP shock

0 10 20 30 40Quarter

-10

0

10

20

30

40

50

60P

erce

nt

InvestmentCapitalOutputConsumption

Figure: Perturbing 10 most important basis functions

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Different levels of “compression”Aggregate response to a 20%TFP shock

0 10 20 30 40Quarter

-10

0

10

20

30

40

50

60P

erce

nt

InvestmentCapitalOutputConsumption

Figure: Perturbing all 200 basis functions

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Taking stock

I When looking only at the StE policy function one concludesthat roughly 50% of the information is needed to reconstructthe policies well

I This is roughly level of state reduction a Reiter (2009)approach would achieve

I Using the StE as reference one can achieve much higherreduction

I For the aggregate dynamics maintaining only 3-6% of thebasis functions suffices

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Simulation performance

Figure: Simulations of Krusell & Smith model

(a) Simulation (b) Simulation close-up

0 200 400 600 800 1000Quarter

34.2

34.4

34.6

34.8

35

35.2

35.4

Cap

ital

Reiter-ReductionReiter-FullKrusell-Smith

0 20 40 60 80 100Quarter

0.985

0.99

0.995

1

1.005

1.01

Per

cent

dev

iatio

ns

Reiter-ReductionReiter-FullKrusell-Smith

Notes: Both panels show simulations of the Krusell & Smith (1998)model with TFP shocks solved with (1) the Reiter method with ourproposed state-space reduction, (2) the original Reiter method with-out state-space reduction, (3) the original Krusell & Smith algorithm

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Error Statistics

Table: Den Haan errors

Absolute error (in %) for capital Kt

Reiter-Reduction Reiter-Full K-S

Mean 0.0119 0.0119 0.1237Max 0.0152 0.0152 0.3491

Notes: Differences in percent between the simulation of thelinearized solutions of the model and simulations in which wesolve for the intratemporal equilibrium prices in every periodand track the full histogram over time for t = {1, ..., 1000};see Den Haan (2010)

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Computing time

Table: Run time for Krusell & Smith model

StE K & S Reiter-Reduction Reiter-Full

in seconds 6.28 49.85 0.43 0.91

Notes: Run time in seconds on a Dell laptop with an Intel i7-7500U CPU @ 2.70GHz 4. Model calibration and numberof grid points as in Den Haan et al. (2010). Code in Matlab.

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Another Example

What is it good for?(an economically interesting example)

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Another Example

A two asset economy with nominal friction(Bayer et al., 2018)

K&S model plus

I a trading cost for capital

I a perfectly liquid bond

I a price setting friction for firms (Phillips curve)

I a central bank and a fiscal authority

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Another Example

A two asset economy with nominal friction(Bayer et al., 2018)

Analyze an increase in income shock variance

I requires both assets (100 points each)

I Here only 4 income states (allows to use only MATLAB’sbuild in routines and compute on a laptop in reasonable time)

I uses both value functions and consumption policies as controls

I Full set would have > 120, 000 variables

I Reduction to 204 distribution states and 635 controls forvalue functions and policies

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Another Example

Error statistics

Table: Run times and accuracy for two-asset model

Running times Stationary Equilibrium Reiter-Reduction

In seconds 388.14 80.38

Absolute error (in %) For capital Kt For bonds Bt

Mean 0.0418 0.0828Max 0.1425 0.4706

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Another Example

An uncertainty shock

Figure: Aggregate response to idiosyncratic uncertainty shock

Krusell & Smith vs. Two-asset HANK modelOutput Yt Consumption Ct Investment It

0 10 20 30 40Quarter

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Per

cent

Krusell-Smith economyHANK with illquid capital

0 10 20 30 40Quarter

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1P

erce

nt


0 10 20 30 40Quarter

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Per

cent


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Conclusion

Conclusion

No excuse!

I Even when heterogeneity is high dimensional,

I our algorithm is an easy approach to these models

I It is a fast and simple to code

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Conclusion

Conclusion

No excuse!I It requires knowledge of only two standard tools of macro:

1. Solving a recursive het. agent model for a StE2. Linearizing a rep. agent model3. (and a little twist in between)

I The fixed design for dimensionality reduction allows to employthe method to estimate models with standard techniques

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Conclusion

Discrete Cosine Transform

I A DCT expresses a finite sequence of data points in terms ofsum of cosine functions at different frequencies

I Linear, invertible function f =

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References

Ahn, S., Kaplan, G., Moll, B., Winberry, T., and Wolf, C. (2017).When inequality matters for macro and macro matters forinequality. NBER Macroeconomics Annual, Volume 32.

Bayer, C., Luetticke, R., Pham-Dao, L., and Tjaden, V. (2018).Precautionary savings, illiquid assets, and the aggregateconsequences of shocks to household income risk. Econometrica.

Den Haan, W. J. (2010). Assessing the accuracy of the aggregatelaw of motion in models with heterogeneous agents. Journal ofEconomic Dynamics and Control, 34(1):79–99.

Den Haan, W. J., Judd, K. L., and Juillard, M. (2010).Computational suite of models with heterogeneous agents:Incomplete markets and aggregate uncertainty. Journal ofEconomic Dynamics and Control, 34(1):1–3.

Gornemann, N., Kuester, K., and Nakajima, M. (2012). Monetarypolicy with heterogeneous agents. Federal Reserve Bank ofPhiladelphia, WP No. 12-21.

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References

Guerrieri, V. and Lorenzoni, G. (2017). Credit crises, precautionarysavings, and the liquidity trap. Quarterly Journal of Economics,132(3):1427–1467.

Kaplan, G., Moll, B., and Violante, G. L. (2017). Monetary policyaccording to HANK. American Economic Review, forthcoming.

Krusell, P. and Smith, A. A. (1997). Income and wealthheterogeneity, portfolio choice, and equilibrium asset returns.Macroeconomic Dynamics, 1(02):387–422.

Krusell, P. and Smith, A. A. (1998). Income and wealthheterogeneity in the macroeconomy. Journal of PoliticalEconomy, 106(5):867–896.

Luetticke, R. (2018). Transmission of monetary policy withheterogeneity in household portfolios. CFM Discussion PaperSeries, (CFM-DP2018-19).

McKay, A., Nakamura, E., and Steinsson, J. (2016). The power offorward guidance revisited. American Economic Review,106(10):3133–3158.

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Conclusion

McKay, A. and Reis, R. (2016). The role of automatic stabilizersin the US business cycle. Econometrica, 84(1):141–194.

Mitman, K. (2016). Macroeconomic effects of bankruptcy andforeclosure policies. American Economic Review,106(8):2219–55.

Reiter, M. (2002). Recursive computation of heterogeneous agentmodels. mimeo, Universitat Pompeu Fabra.

Reiter, M. (2009). Solving heterogeneous-agent models byprojection and perturbation. Journal of Economic Dynamics andControl, 33(3):649–665.

Schmitt-Grohé, S. and Uribe, M. (2004). Solving dynamic generalequilibrium models using a second-order approximation to thepolicy function. Journal of Economic Dynamics and Control,28(4):755–775.

Winberry, T. (2016). A toolbox for solving and estimatingheterogeneous agent macro models. mimeo, Chicago Booth.

MotivationIntuitionSetupOur techniqueKrusell-Smith exampleAnother ExampleConclusion

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