solving linear rational expectations models: a z-transform approach

Fall 2011 Macro Group Brown Bag

Fei TAN

Department of EconomicsIndiana University Bloomington

December 5, 2011

Introduction Minimal Tools Solution Technique Illustrative Examples Concluding Remarks

Solving Linear Rational Expectations Models:A z-Transform Approach

Fei Tan & Todd B. Walker

Department of Economics, Indiana University

Solving Linear RE Models Pres. by Fei Tan Department of Economics, Indiana Univ. 2 / 31

A Road Map...

1 Introduction

2 Minimal Tools

3 Solution TechniqueUnivariate Case: A Toy ModelA Summary of Solution TenetsMultivariate Case: A Monster ModelConnection to Gensys

4 Illustrative ExamplesExample 1: Leeper (1991)Example 2: Simple RBC ModelExample 3: Decoupled System

5 Concluding Remarks

A Road Map...

1 Introduction

2 Minimal Tools

A Road Map...

1 Introduction

2 Minimal Tools

A Road Map...

1 Introduction

2 Minimal Tools

A Road Map...

1 Introduction

2 Minimal Tools

Motivation: Call For New Solution Device

Fact: multivariate linear rational expectations models are extensivelyemployed, solved, and estimated for empirical research on cause andeffect in macroeconomy. [Sims & Sargent]

Several limitations of existing solution techniques significantly re-strict scope of available research, e.g. observational equivalence andidentification with policy regimes.

Call for new solution device.

Technical Difficulties And Problems

Infinite regress in expectations: when forming expectations, agentsforecast forecasts of others under incomplete information, makingstate vector explode over time. [Townsend (1983)]

Numerical rather than analytical solutions: explicit characterizationof equilibrium will make cross-equation restrictions transparent andfacilitate econometric implementation. [This work]

Stringent driving processes: inclusion of oversimplified stochasticprocesses for exogenous shocks poses serious question: “how generalis our macro model?” [Todd in E550 class]

Technical Difficulties And Problems (Cont’d)

Observational equivalence: two distinct policy regimes may generateidentical equilibrium time series and empirical test based on simplecorrelations in data may lead to spurious results. Such identificationchallenge may extend to more general setups. [Leeper and Walker(2011)]

Decoupling issue: existing methods, e.g. root-counting approach andwinding number criterion, give incorrect existence condition to de-coupled systems. [Sims (2007)]

Not user-friendly perhaps: conventional device, e.g. Gensys, re-quires cleverness in models containing multiple leads and lags, with-holding equations, general driving processes, etc. [Sims (2002)]

Main Results

A summary of key points in this work:

We generalized linear rational expectations solution method of White-man (1983) to multivariate case, allowing use of generic exogenousdriving process which must only satisfy covariance stationarity;

We derived multivariate cross-equation restrictions linking Wold rep-resentation of exogenous process to endogenous variables. This map-ping is multivariate version of Hansen-Sargent formula;

We connected our solution methodology to other popular approaches,e.g. Sims (2002);

We introduced several motivating examples that highlight usefulnessof our approach.

Main Results

Minimal Tools

A stochastic process {yt} is said to be covariance stationary if Eyt =µ for all t ∈ T , and cov(yt, ys) = σt,s depends only on t − s.An overview of basic tools in this work:

Wold Representation Theorem: any covariance stationary process{yt} can be represented as yt = µt +

∑∞j=0 θjεt−j with

∑∞j=0 θ

2j <∞.

Here {εt} ∼ WN(0, σ2) (our basic building blocks);Corollary of Riesz-Fischer Theorem: any square summable sequence{cn}∞−∞ has z-transform g(z) =

∑∞j=−∞ cjzj where ck can be recov-

ered from g(z) via inversion formula;Wiener-Kolmogorov Prediction Formula: expectations are formedoptimally via Etyt+k =

]+εt;

Smith Normal Form: useful decomposition technique for polynomialmatrices.

Minimal Tools

∑∞j=0 θ

2j <∞.

]+εt;

Minimal Tools

∑∞j=0 θ

2j <∞.

]+εt;

Minimal Tools

∑∞j=0 θ

2j <∞.

]+εt;

Minimal Tools

∑∞j=0 θ

2j <∞.

]+εt;

Minimal Tools

∑∞j=0 θ

2j <∞.

]+εt;

Univariate Case: A Toy Model

Consider a simple asset valuation model

Pt = βEt(Pt+1) + dt, 0 < β < 1 (1)

Dividend process dt has Wold representation

∞∑j=0

ρjεt−j =1

1− ρLεt = A(L)εt, 0 < ρ < 1 (2)

“Guess” a solution form for Pt with square-summable coefficients

∞∑j=0

Cjεt−j = C(L)εt (3)

Information set at t for forecasting Pt+1: {dt, dt−1, . . .} = {εt, εt−1, . . .}.

Pt = βEt(Pt+1) + dt, 0 < β < 1 (1)

∞∑j=0

ρjεt−j =1

1− ρLεt = A(L)εt, 0 < ρ < 1 (2)

∞∑j=0

Pt = βEt(Pt+1) + dt, 0 < β < 1 (1)

∞∑j=0

ρjεt−j =1

1− ρLεt = A(L)εt, 0 < ρ < 1 (2)

∞∑j=0

Pt = βEt(Pt+1) + dt, 0 < β < 1 (1)

∞∑j=0

ρjεt−j =1

1− ρLεt = A(L)εt, 0 < ρ < 1 (2)

∞∑j=0

Univariate Case: A Toy Model (Cont’d)

Wiener-Komolgorov optimal prediction formula

EtPt+1 = L−1[C(L)− C0]εt =

εt (4)

Plugging in expressions for Pt, EtPt+1, and dt gives

C(L)εt = βL−1[C(L)− C0]εt + A(L)εt (5)

Above equation holds for all realizations of {εt}, implying z-transformsof LHS and RHS are identical as analytic functions on open unit disk,i.e.

C(z) = βz−1[C(z)− C0] + A(z) (6)

EtPt+1 = L−1[C(L)− C0]εt =

εt (4)

C(z) = βz−1[C(z)− C0] + A(z) (6)

EtPt+1 = L−1[C(L)− C0]εt =

εt (4)

C(z) = βz−1[C(z)− C0] + A(z) (6)

Solving for C(z) gives z-transform of Pt

C(z) =zA(z)− βC0

z− β(7)

How to pin down C0? Square summability in time domain is tanta-mount to z-transform being analytic in frequency domain

0 < β < 1⇒ C0 = A(β) =1

1− ρβ(8)

Unique solution coincides with solution by iterating forward

(1− ρβ)(1− ρL)εt =

∞∑k=0

(ρβ)kdt = Et

∞∑k=0

βkdt+k (9)

C(z) =zA(z)− βC0

z− β(7)

0 < β < 1⇒ C0 = A(β) =1

1− ρβ(8)

(1− ρβ)(1− ρL)εt =

∞∑k=0

(ρβ)kdt = Et

∞∑k=0

βkdt+k (9)

C(z) =zA(z)− βC0

z− β(7)

0 < β < 1⇒ C0 = A(β) =1

1− ρβ(8)

(1− ρβ)(1− ρL)εt =

∞∑k=0

(ρβ)kdt = Et

∞∑k=0

βkdt+k (9)

More insights about selection of C0...

Consistency of solution: any candidate for C0 satisfying existencecondition leads to a C(z), whose “C0” must be identical to that can-didate so that solution is consistent with rational expectations restric-tions imposed by model.

A rational expectations equilibrium is a fixed point of mapping be-tween perceived law of motion and law of motion generated by thosebeliefs.

Each candidate for C0 designates a perceived law of motion and asolution generated by such C0 forms a new law of motion with equi-librium conditions imposed.

Consistency of solution requires that two laws of motion be identicaland thus z-transform approach reduces to solving fixed point prob-lem.

A comparison of determinacy conditions: z-transform v.s. “usual” approach

By conventional approach, e.g. Sims (2002), one forecast error re-quires one unstable eigenvalue (1/β > 1 ⇒ β < 1) so that ex-ogenous shocks pin down all error terms influenced by endogenousshocks. This is tantamount to pinning down C0 by making C(z) ana-lytic at pole inside unit circle, z = β < 1.

What if β ≥ 1? There is no unstable eigenvalue (0 < 1/β ≤ 1) im-posing restrictions between exogenous and endogenous shocks andwe have infinite solutions, each of which is indexed by a sunspot.This is tantamount to pole of C(z) being outside unit circle and C0can be set in arbitrary manner, each of which indexes an equilibrium.

Solution Principles

Four tenets embedded in previous toy model [Whiteman (1983)]:Driving process is taken to be zero-mean linearly regular covariancestationary process with known Wold representation;Expectations are formed rationally and are computed using Wiener-Kolmogorov formula;Solutions are sought in space spanned by time-independent square-summable linear combinations of process fundamental for driving pro-cess;Rational expectations restrictions are required to hold for all realiza-tions of driving process.

Solution Principles

Multivariate Case: A Monster Model

Let’s be ambitious and solve a monster model (user-friendly)[

n∑k=1

FkL−k +m∑

ΦkL−k +l∑

n∑k=1

ykL−kηt − ΦkΠ

xkL−kνt

Wold representations for vector exogenous and endogenous processes

xt = A(L)εt and yt = C(L)εt (11)

Further assumptions:Fn is of full rank (can be relaxed though...);Roots of det[zn(F(z−1) + G(z))] =

∑hk=0 fkzk are distinct, with rp of

them lie inside unit circle while h− rp of them lie outside unit circle.

n∑k=1

FkL−k +m∑

ΦkL−k +l∑

n∑k=1

xkL−kνt

n∑k=1

FkL−k +m∑

ΦkL−k +l∑

n∑k=1

xkL−kνt

Multivariate Case: A Monster Model (Cont’d)

Wiener-Komolgorov formula for endogenous & exogenous forecasterrors

ηt+k = yt+k − Etyt+k = L−k

(k−1∑i=0

)εt (12)

νt+k = xt+k − Etxt+k = L−k

(k−1∑i=0

)εt (13)

z-transform of equilibrium conditions become

(F(z−1) + G(z))C(z) = (Φ(z−1) + Ψ(z))A(z) +

n∑t=1

n∑s=t

(FsΠysCt−1 − ΦsΠ

xsAt−1)z−s+t−1

Applying Smith canonical form gives

zn(F(z−1) + G(z)) = U(z)−1P1(z)︸︷︷︸S(z)

P2(z)V(z)−1︸︷︷︸T(z)

(k−1∑i=0

)εt (12)

(k−1∑i=0

)εt (13)

(F(z−1) + G(z))C(z) = (Φ(z−1) + Ψ(z))A(z) +

n∑t=1

n∑s=t

zn(F(z−1) + G(z)) = U(z)−1P1(z)︸︷︷︸S(z)

P2(z)V(z)−1︸︷︷︸T(z)

(k−1∑i=0

)εt (12)

(k−1∑i=0

)εt (13)

(F(z−1) + G(z))C(z) = (Φ(z−1) + Ψ(z))A(z) +

n∑t=1

n∑s=t

zn(F(z−1) + G(z)) = U(z)−1P1(z)︸︷︷︸S(z)

P2(z)V(z)−1︸︷︷︸T(z)

Inverting S(z) gives z-transform identity

T(z)C(z) =znU(z)∏k

j=1(z− zj)

{(Φ(z−1) + Ψ(z))A(z) +

n∑t=1

n∑s=t

How to pin down C0,C1, . . . ,Cn−1? Square summability in time do-main is tantamount to z-transform being analytic in frequency do-main

(z− zj)T(z)C(z)|z=zj = 0, j = 1, . . . , k (17)

Stacking restrictions yield a compact expression

A<[k×q]

= − R<[k×np]

C[np×q]

j=1(z− zj)

{(Φ(z−1) + Ψ(z))A(z) +

n∑t=1

n∑s=t

(z− zj)T(z)C(z)|z=zj = 0, j = 1, . . . , k (17)

A<[k×q]

= − R<[k×np]

C[np×q]

j=1(z− zj)

{(Φ(z−1) + Ψ(z))A(z) +

n∑t=1

n∑s=t

(z− zj)T(z)C(z)|z=zj = 0, j = 1, . . . , k (17)

A<[k×q]

= − R<[k×np]

C[np×q]

Existence and Uniqueness Conditions

Existence problems arise if endogenous shocks η cannot adjust tooffset exogenous shocks x. Solution exists iff

span(A<) ⊆ span(R<) (19)

Analytical solution for yt is given by

yt = yPt︸︷︷︸

Perfect foresight solution

+ yRt︸︷︷︸

Remainder term yt−yPt

In order for solution to be unique, one must be able to determineremainder term from knowledge of R<C. Solution is unique iff

span(R′>) ⊆ span(R′<) (21)

+ yRt︸︷︷︸

span(R′>) ⊆ span(R′<) (21)

+ yRt︸︷︷︸

span(R′>) ⊆ span(R′<) (21)

Connection to Gensys

THEOREM Consider a special case of monster model

[F0L0 + F1L−1]yt = Φ1L−1xt + F1Π1L−1ηt (22)

where F1 is of full rank. Assume both eigenvalues of −F−11 F0 and roots of

det[zF0 + F1] = 0 are nonzero and distinct. Then...1 Factorization equivalence: eigenvalues of −F−1

1 F0 are exactly in-verse of corresponding roots of det[zF0 + F1] = 0, or, those roots ofdeterminant of Smith Normal Form for zF0 + F1;

2 Existence equivalence: restrictions imposed by unstable eigenvaluesin Gensys are exactly those imposed by roots inside unit circle in z-transform approach;

3 Uniqueness equivalence: Gensys and z-transform approach yieldidentical uniqueness condition.

Example 1: Leeper (1991)

The following bivariate system in (πt, bt) delivers a cashless versionof Leeper (1991) model:

Etπt+1 = απt + θt (23)

bt + β−1πt = [β−1 − γ(β−1 − 1)]bt−1 + αβ−1πt−1 − (β−1 − 1)ψt (24)

Decoupling problem: merely counting ] of unstable eigenvalues doesnot necessarily deliver existence conditions!Casting above system into monster model gives

(1 00 0

)︸︷︷︸

L−1 +

(−α 01β

)︸︷︷︸

−αβ−[

1β− γ( 1

β− 1)

])︸︷︷︸

(πtbt

)︸︷︷︸

(1 00 −( 1

β− 1)

)︸︷︷︸

(θtψt

)︸︷︷︸

(1 00 0

)︸︷︷︸

(1 00 0

)︸︷︷︸

L−1(ηπtηb

)︸︷︷︸ηt

(1 00 0

)︸︷︷︸

L−1 +

(−α 01β

)︸︷︷︸

−αβ−[

1β− γ( 1

β− 1)

])︸︷︷︸

(πtbt

)︸︷︷︸

(1 00 −( 1

β− 1)

)︸︷︷︸

(θtψt

)︸︷︷︸

(1 00 0

)︸︷︷︸

(1 00 0

)︸︷︷︸

L−1(ηπtηb

)︸︷︷︸ηt

(1 00 0

)︸︷︷︸

L−1 +

(−α 01β

)︸︷︷︸

−αβ−[

1β− γ( 1

β− 1)

])︸︷︷︸

(πtbt

)︸︷︷︸

(1 00 −( 1

β− 1)

)︸︷︷︸

(θtψt

)︸︷︷︸

(1 00 0

)︸︷︷︸

(1 00 0

)︸︷︷︸

L−1(ηπtηb

)︸︷︷︸ηt

Roots Inside & Outside Unit Circle

Applying Smith Form decomposition gives

z(F(z−1) + G(z)) = U(z)−1

0 z(z− 1

)(z− 1

1β−γ(

1β−1)

)︸︷︷︸

V(z)−1 (25)

det[zn(F(z−1) + G(z))] has three distinct roots: z1 = 0, z2 = 1α , and

z3 = 11β−γ( 1

β−1)

Existence and unique conditions depend on whether these roots areinside or outside unit circle.

z(F(z−1) + G(z)) = U(z)−1

0 z(z− 1

)(z− 1

1β−γ(

1β−1)

)︸︷︷︸

V(z)−1 (25)

z3 = 11β−γ( 1

β−1)

z(F(z−1) + G(z)) = U(z)−1

0 z(z− 1

)(z− 1

1β−γ(

1β−1)

)︸︷︷︸

V(z)−1 (25)

z3 = 11β−γ( 1

β−1)

Case 1: Passive Monetary (α < 1) & Passive Fiscal (γ > 1)

One root inside unit circle, z = 0, and two roots outside unit circle,z = 1

α > 1 and z = 11β−γ( 1

β−1)

Existence requires span(A<) ⊆ span(R<) (satisfied). However, allrestrictions on C0 are redundant and C0 can be set in arbitrary way.

Uniqueness requires span(R′>) ⊆ span(R′<) (not satisfied). Thus,there are infinite solutions.

Characterization of set of sunspot equilibria. [Lubik and Schorfheide(2003)]

α > 1 and z = 11β−γ( 1

β−1)

α > 1 and z = 11β−γ( 1

β−1)

α > 1 and z = 11β−γ( 1

β−1)

Case 2: Active Monetary (α > 1) & Passive Fiscal (γ > 1)

Two roots inside unit circle, z = 0 and z = 1α < 1, and one root

outside unit circle, z = 11β−γ( 1

β−1)

Existence requires span(A<) ⊆ span(R<) (satisfied). Restrictionsimply C11

0 = − 1α and C12

0 = 0. However, C210 and C22

0 can be set inarbitrary way.Uniqueness requires span(R′>) ⊆ span(R′<) (satisfied). Thus, solu-tion is unique and is given by(

(− 1

α 01αβ

1−β1−γ+γβ

)︸︷︷︸

∞∑k=1

αβ −ρk−1

β1−β

1−γ+γβ ρk

)︸︷︷︸

(θt−k

ψt−k

β−1)

0 = − 1α and C12

(− 1

α 01αβ

1−β1−γ+γβ

)︸︷︷︸

∞∑k=1

αβ −ρk−1

β1−β

1−γ+γβ ρk

)︸︷︷︸

(θt−k

ψt−k

β−1)

0 = − 1α and C12

(− 1

α 01αβ

1−β1−γ+γβ

)︸︷︷︸

∞∑k=1

αβ −ρk−1

β1−β

1−γ+γβ ρk

)︸︷︷︸

(θt−k

ψt−k

A Comparison Of Impulse Responses (Case 2)

0 5 10 15 20 25 30−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

(a) period (+M shock)

Gensysz−Transform

0 5 10 15 20 25 30−0.4

−0.2

(b) period (+M shock)

Gensysz−Transform

0 5 10 15 20 25 30−1

−0.5

(c) period (+F shock)

Gensysz−Transform

0 5 10 15 20 25 30−20

(d) period (+F shock)

Gensysz−Transform

Case 3: Passive Monetary (α < 1) & Active Fiscal (γ < 1)

Two roots inside unit circle, z = 0 and z = 11β−γ( 1

β−1)

> 1, and one

root outside unit circle, z = 1α < 1.

0 = − 11β−γ( 1

β−1)

and C120 = β − 1. However, C21

0 and C220

can be set in arbitrary way.Uniqueness requires span(R′>) ⊆ span(R′<) (satisfied). Thus, solu-tion is unique and is given by(πt

(−ρ β − 1

11−γ+γβ 0

)︸︷︷︸

∞∑k=1

(αk−1 − αkρ (β − 1)αk

)︸︷︷︸

(θt−k

ψt−k

β−1)

> 1, and one

0 = − 11β−γ( 1

β−1)

0 and C220

(−ρ β − 1

11−γ+γβ 0

)︸︷︷︸

∞∑k=1

(αk−1 − αkρ (β − 1)αk

)︸︷︷︸

(θt−k

ψt−k

β−1)

> 1, and one

0 = − 11β−γ( 1

β−1)

0 and C220

(−ρ β − 1

11−γ+γβ 0

)︸︷︷︸

∞∑k=1

(αk−1 − αkρ (β − 1)αk

)︸︷︷︸

(θt−k

ψt−k

A Comparison Of Impulse Responses (Case 3)

0 5 10 15 20 25 30−1

−0.8

−0.6

−0.4

−0.2

(a) period (+M shock)

Gensysz−Transform

0 5 10 15 20 25 300

(b) period (+M shock)

Gensysz−Transform

0 5 10 15 20 25 30−0.02

−0.015

−0.01

−0.005

(c) period (+F shock)

Gensysz−Transform

0 5 10 15 20 25 30

−1.5

−0.5

(d) period (+F shock)

Gensysz−Transform

Case 4: Active Monetary (α > 1) & Active Fiscal (γ < 1)

All roots are inside unit circle, z = 0, z = 1α < 1, z = 1

1β−γ( 1

β−1)

Existence requires span(A<) ⊆ span(R<) (not satisfied). Thus, modelis too restrictive and has no stationary solutions in space we are in-terested in. However, unbounded solutions or bounded solutions insome other spaces may exist!

Case 4: Active Monetary (α > 1) & Active Fiscal (γ < 1)

All roots are inside unit circle, z = 0, z = 1α < 1, z = 1

1β−γ( 1

β−1)

Existence requires span(A<) ⊆ span(R<) (not satisfied). Thus, modelis too restrictive and has no stationary solutions in space we are in-terested in. However, unbounded solutions or bounded solutions insome other spaces may exist!

Example 2: Simple RBC Model

The following bivariate system in (ct, kt) delivers a simple RBC model:

Etct+1 = ct + (α− 1)kt + Etat+1 (28)1− αβαβ

ct + kt =1αβ

at +1β

kt−1 (29)

Assume at = A(L)εt satisfies covariance stationarity. Conventionalapproaches are not readily available since A(L) is unknown here.

Hold on! z-transform method gives solution(ct

)= C(L)

1−αLεtA(L)

1−αLεt

ct + kt =1αβ

at +1β

kt−1 (29)

)= C(L)

1−αLεtA(L)

1−αLεt

ct + kt =1αβ

at +1β

kt−1 (29)

)= C(L)

1−αLεtA(L)

1−αLεt

Example 3: Decoupled System

Consider a decoupled linear rational expectations model in (xt, yt)that has no stable solution:

xt = αxt−1 + εt (30)

Etyt+1 = βyt + vt (31)

|α| > 1 and |β| < 1⇒ root-counting approach and winding numbercriterion fail to give correct existence condition.Unstable root occurs in a part of system that is decoupled from ex-pectational equation. [Sims (2007)]Hold on! z-transform method says no solution since existence condi-tion is violated(

β−αα2β

α 0C21

0 C220 + 1

)=(0 0

)⇒ α = β

xt = αxt−1 + εt (30)

β−αα2β

α 0C21

0 C220 + 1

)=(0 0

)⇒ α = β

xt = αxt−1 + εt (30)

β−αα2β

α 0C21

0 C220 + 1

)=(0 0

)⇒ α = β

xt = αxt−1 + εt (30)

β−αα2β

α 0C21

0 C220 + 1

)=(0 0

)⇒ α = β

Wrapping up

A summary of what we did in this work:

We generalized linear rational expectations solution method of White-man (1983) to multivariate case. This permits use of generic drivingprocess which must only satisfy covariance stationarity;

We derived multivariate cross-equation restrictions linking Wold rep-resentation of driving process to endogenous variables;

We connected our solution methodology to Gensys and showed anequivalence relationship between two techniques;

Wrapping up

Research Agenda

Two applications of this work:

Incomplete information: one may solve and estimate incompleteinformation models where agents have heterogeneous beliefs.Observational equivalence: two distinct parameter regions may gen-erate empirically indistinguishable equilibrium time series. Identifi-cation problem becomes less severe as model complexity grows. In-vestigation of this point requires a tractable tool.

Monster model completely separates endogenous variables (LHS) andexogenous variables (RHS) (decomposition of parameter space).Factorization of polynomial matrix depends on private parameters, in-dependent of selection of driving processes and model complexity.“Semi-analytic” approach: (1) fix numerical private parameters; (2)pick numerical policy parameters for distinct regimes; (3) employsymbolic representations for general driving processes.

Research Agenda

solving linear rational expectations models: a z-transform approach

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