Approximation of Dynamic, Stochastic, General Equilibrium Models through Persistent Excitation. M. Aqib Aslam

Luisa Corrado

Sean Holly

December 19, 2007 Abstract Over the last few decades, economists have been studying increasingly complex dynamic stochastic general equilibrium (DSGE) models which require more powerful and reliable computational methods to solve them. To this end numerous approaches have been developed. This paper applies a technique known as Persistent Excitation excites a perfect foresight version of the model with white noise processes. We can recover precisely the same decision rules as would be obtained using Uhlig’s speci…c method of log-linearisation around the steady state. We also extend the method to solving Hansen’s real business cycle model.

1

Introduction

Over the last few decades, economists have been studying increasingly complex dynamic stochastic general equilibrium (DSGE) models which require more powerful and reliable computational methods to solve them1 . To this end numerous approaches have been developed. Traditionally, dynamic programming had been the …rst choice for solving DSGE models - economists would begin by applying Value Function Iterations to the social planner’s 1

A previous generation of nonlinear, rational expectations models also provided major computational challenges; see Lipton et al (1982), Holly and Zarrop (1983), Fair and Taylor (1983), Anderson and Moore (1985).

1

problem. This approach is extremely reliable and accurate (Santos and Vigo, 1998). However, it is slow and su¤ers from the curse of dimensionality. In addition, it is very complicated to apply to non-Pareto optimal economies, eliminating its use for any monetary economics models incorporating staggered pricing. As a result the development of new solution methods became a key avenue for research over the preceding decades, with Perturbation methods and Projection methods becoming the most commonly applied. Perturbation methods (see Judd and Guu, 1993; Gaspar and Judd, 1997) involve building a Taylor series expansion of the policy functions of the agents around the steady state. Log-linearisation methods for dynamic models became the workhorse of macroeconomic analysis. However, these linear approximations ignored higher-order terms and were certainty equivalent approximations. As a result Judd and Guu (1993) extended the methods to compute the higher-order terms of the expansion, something which has been more recently explored by Schmitt-Grohé and Uribe (2004) and Lombardo and Sutherland (2007). Projection methods (see Judd, 1992,1998; Reddy, 1993; and McGrattan, 1996, for early expositions) on the other hand, take basis functions to build an approximated policy function that minimises a residual function. They derive approximate solutions to functional equations and are also known as minimum weighted residual methods2 . The Parameterised Expectations approach (PEA) is considered to be a special case of these methods, as they involve using a parameterised polynomial to approximate the policy function. However, unlike the PEA, the functions that are approximated do not have to be the conditional expectations that characterise the …rst order conditions of the model3 . 2

Assuming we have a functional we want to approximate, which is implicitly de…ned by the functional equation, e.g. Bellman equation or Intertemporal Euler, the researcher chooses a family of polynomials and uses a …nite linear combination of that family to approximate the function. The residual function is obtained by substituting this combination into the functional equation. Simple techniques include least squares, which seeks to minimise the square of the residual function; the Galerkin method, which uses the orthogonality of the residual function with the observation vectors (represented by the polynomials used in the approximation) and …nds the roots of the residual function, and the collocation method, which assumes that the residual function is equal to zero at a given set of points (idea is that is collocates with the original polynomial at these points). 3 The most well-known examples of Projection methods are the Finite Elements method, where the basis functions are nonzero only locally, and Spectral methods, where the basis functions are nonzero globally.

2

This paper proposes a new method for approximating DSGE models using persistent excitation. Persistent Excitation seeks …rst to solve the version of the model where forward expectations are held constant. Each endogenous, exogenous and expectational variable are ‘excited’ using orthogonal white noise processes. The resulting solution paths for each variable are then used in regressions, the output from which is then put through Sims’(2000) rational expectations solver to give the approximate decision rules for each of the state and non-state variables4 . To demonstrate the method of Persistent Excitation we chose three models which represented an increasing level of complexity, allowing us to highlight the method’s potential. We start with the deterministic Brock-Mirman (1972) model, followed by its stochastic variant and then proceed to solve Hansen’s (1985) Real Business Cycle model with indivisible labour supply. The structure of this paper is as follows. Section 2 provides a description of Persistent Excitation. Section 3 applies he method to the Brock-Merman version of the stochastic growth model. Section 4 then applies the method to the more complicated model of Hansen’s Real Business Cycle model. Section 5 concludes.

2

System Identi…cation and Persistent Excitation

Persistent Excitation is a technique that has been widely used in system identi…cation, involving the modelling of dynamic systems from experimental data. An identi…cation experiment is performed by exciting the system using some sort of input signal such as a step, a sinusoid or a random signal, and observing its input and output over an interval. The modeler would then try to …t a parametric model of the process to the recorded input and output sequences. The …rst step is to determine an appropriate form of the model (typically a linear di¤erence equation of a certain order). As a second step some statistical method is used to estimate the unknown parameters of the model. Identi…cation can be viewed as a form of model approximation or model reduction. A system is identi…able if the corresponding parameter estimates (for example obtained via least squares) are consistent. To guar4

We show, in fact, that we can recover precisely the same decision rules as would be obtained using Uhlig’s speci…c method of log-linearisation around the steady state.

3

antee consistency an input must be used that excites the process su¢ ciently. For a system to be suitably identi…ed requires that a signal be su¢ ciently exciting. In particular, a signal is persistently exciting if it excites all modes of the system. A signal ut is said to be persistently exciting of order n if (i) the following limit exists N 1 X ru ( ) = lim ut+ u0t N !1 N t=1

where ru ( ) is the th autocovariance of ut and (ii) the moment (variance-covariance) matrix of ut 0 ru (0) ru (1) ru (n 1) B ru ( 1) ru (0) ru (n 2) B Ru (n) = B .. .. .. . .. @ . . . ru (1 n) ru (2 n) ru (0)

(1)

1 C C C A

(2)

is positive de…nite. Condition (i) is a simple statement of the law of large numbers which ensures the consistency necessary for identi…cation. Examples of persistently exciting signals include white noise and step functions. If we let ut be white noise, of zero mean and variance 2 , i.e. E[ut ] = 0,

E[ut u0t ] =

2

,

E[ut u ] = 0 for t =

(3)

and for which ut ’s are uncorrelated across time. Therefore ru ( ) = 2 0; , and Ru (n) = 2 In which is always positive de…nite and white noise is de…ned as persistently exciting of all orders. If ut was a step function of magnitude . Then ru ( ) = 2 for all . Hence Ru (n) is nonsingular if and only if n = 1. Therefore a step function is persistently exciting of order 1 only. Consider a linear, time invariant, discrete time single-input-single-output system with input sequence ut and output sequence yt . The output is assumed to be corrupted by noise. If we consider a time invariant discrete time system which is asymptotically stable, by the Spectral Factorisation Theorem5 there exists a factorisation of (z) such that (z) = H(z)H(z 1 ) 5

See Zarrop (1079).

4

(4)

where

D(z) (5) C(z) and C and D are polynomials. H is a pulse transfer function. It is similar to the generating function of the weighting function (autocovariance) and can be interpreted as the steady state amplitude of the output. Its poles must lie inside the unit circle and its zeros inside or on the unit circle - this is equivalent to a stability condition for this particular function and ensures that the factorisation exists and is unique. If the input signal is white noise, the output signal is a weakly stationary process with spectral density (z). Spectral factorisation means that all stationary processes could be thought of as outputs of dynamic systems with white noise inputs. And given a rational spectral density function (z), then there exists an asymptotically stable linear dynamic system such that the output of the system is a stationary process with spectral density (z) if the input is discrete white noise. This gives: yt = H(z 1 )ut (6) H(z) =

where ut is a sequence of independent mean zero unit variance random variables and z is the unit forward shift operator (i.e. zxt = xt+1 ). These results can be generalised such that we can perform the same spectral factorisation for any exogenous processes, e.g. xt and t , which are stationary and stochastic to give us the following complete model yt = Z(z 1 )xt + H(z 1 )

t

(7)

where Z is a rational transfer function. It is a assumed that the model represents a stable minimum-phase system with time delay s so that B(z 1 ) Z(z ) = A(z 1 ) The …nal model can therefore be written as B(z 1 ) D(z 1 ) x + yt = t A(z 1 ) C(z 1 ) where 1

A(z) B(z) C(z) D(z)

= = = =

(8)

t

1 + a1 z + : : : + an z s b0 + b1 z + : : : + bm z m c0 + c1 z + : : : + cq z q d0 + d1 z + : : : + dr z r 5

(9)

Letting z

1

L (the backward shift operator) we get yt =

B(L) D(L) xt + A(L) C(L)

(10)

t

Assuming common factors, A(L) = C(L), allows us to rearrange the expression to obtain the Autoregressive Distributed Lag form A(L)yt = B(L)xt + D(L)

2.1

(11)

t

Model Approximation by Persistent Excitation

Consider now a non-linear, dynamic, stochastic model with forward expectations written in implicit form: Fi (yt ; yt 1 ; ::::yt s ; Et yt+1 ; ::::; Et yt+v ; xt ; xt 1 ; ::::; xt r ;

t)

=0

where F is an m 1 vector valued function (i = 1; :::; m) and t = 1; :::; T . This represents a set of m nonlinear equations. y represents the endogenous variables (both state and other) of which there are m, with a maximum lag of s and a maximum lead of v; and there are n exogenous variables, x, of maximum lag r; t is an m-dimensional vector of white noise processes. We can carry out a …rst-order expansion about some initial nominal trajectory (yot , xot , ot ) to give: s X j=0

@Fi @yt j

o

yet j +

f X j=0

@Fi @Et yt+1+j

o

Et yet+1+j +

r X j=0

@Fi @xt j

o

x et j +

@F i @ t

o

et

where the su¢ x o denotes nominal values and the perturbations to the initial path are de…ned as yet = yt

yot , x et = xt

xot , et =

t

ot

Assuming that f = 0, i.e. forward expectations beyond t + 1 are excluded from the model, then we can write this representation in vector polynomial form in the lag operator L, as A(L)t yet + C(L)t Et yet+1 + B(L)t x et = et 6

(12)

where L is lag operator de…ned by Lj xt = xt j and A(L)t , C(L)t and B(L)t are matrices of polynomials in the lag operator of order s, v and r respectively s

A(L)t = A0t + A1t L + A2t L + ::: + Ast L = 2

s X

Ajt Lj

j=0

C(L)t = C0t + C1t L + C2t L2 + ::: + Cvt Lv = B(L)t = B0t + B1t L + B2t L2 + ::: + Brt Lr =

v X

j=0 r X

Cjt Lj Bjt Lj

j=0

and the (p; q) elements of the matrices Ajt , Bjt and Cjt are given by @Fp @yq;t j

[Ajt ]pt =

where p; q = i; :::m. In general, the polynomial matrices in the …rst-order Taylor expansion are time-varying. However, we want to obtain a low-order constant coe¢ cient linear approximation to (12) of the form A(L)e yt + C(L)Et yet+1 + B(L)e xt = et

If we de…ne zet = (Et yet+1 ; x et ; ~t )0 as a stacked vector then E[e zi (s)] = 0; E[e zi (s)e zj (t)] = i ij i; j = 1; :::; p s; t = 1; :::; T

st

with p = 2m + n, i sets the size of the perturbations for the ith endogenous variable, and ij is the Kronecker delta ij

= 0 for i 6= j, 1 for i = j

Note that the entire perturbation sequence is white. we can then write the relationship between the endogenous variables, the perturbed exogenous and expectational variables and the perturbed errors as an autoregressive distributed lag (ARDL) A0 y t = A1 y t

1

+ ::: + As yt

s

+ D1 Et yt+1 + B1 xt + :::Br xt 7

1

(13)

3

The Brock-Mirman Model

We …rst consider the solution of the Brock-Mirman (1972) model where the utility function is logarithmic and there is full depreciation. This model is extremely useful as it is one of the few general equilibrium models which has an analytical solution and therefore provides a natural benchmark for the evaluation of any approximations. The deterministic Brock-Mirman economy is set up as follows. max U0 =

c0 , c1 ,...

s:t:

1 X

t

ln Ct

(14)

t=0

Kt = Yt Ct Yt = AKt 1 0 Ct 0 Kt K0 given

The …rst order conditions can be reduced to the following three equations: t t

Kt Yt

=

1 Ct

= A t+1 Kt = Yt Ct = AKt 1

(15) 1

The simple structure of the optimality conditions derives from the constant elasticity of assumption utility function which implies that the marginal rate of substitution between adjacent dates depends only on consumption at those dates. As a result, consumption in the current period only depends on the current capital stock and on future consumption, but is independent of past consumption. Reducing the FONCs further we can obtain a pair of dynamic equations which characterise the system, the …rst of which is the Euler equation 1 Ct 1 = Ct+1 AKt Kt = AKt 1 Ct

8

1

(16) (17)

Further substitution gives us the Euler equation in terms of the single state variable, Kt AKt Kt+1 = AKt 1 (18) AKt 1 Kt There is also a Transversality condition that provides a terminal condition for the solution to the Euler equation above lim

t!1

t

t Kt

(19)

=0

Using dynamic programming the asymptotic optimal policy functions will be of the form Kt =

AKt

1

Ct = (1

)AKt

1

(20)

These policy functions therefore represent the benchmark analytical solution against which we will be comparing the solutionobtained by persistent excitation. The key di¤erence between the stochastic and deterministic versions of the Brock-Mirman model is the uncertainty that is introduced by asserting that output in each period t depends not only on the amount of capital Kt 1 but also on the realisation of a stochastic variable Zt , capturing some exogenous varying factors. The equilibrium conditions can once again be summarised by two nonlinear dynamic equations 1 Ct 1 = AEt [Ct+1 Zt+1 Kt Kt = AZt Kt 1 Ct

1

]

The analytical solution to the stochastic Brock-Mirman is virtually identical to that of the deterministic case, except for the exogenous productivity shock Kt = g(Kt 1 ; Zt ) = AZt Kt 1 Ct = h(Kt 1 ; Zt ) = (1 )AZt Kt

1

Therefore we once again obtain some benchmark analytical solutions which will provide the frame of reference later on. A log-linearisation is carried out which still leads to a solution for the model. and later on in the paper we will see that these values play a crucial role in the determination of the existence of a sensible solution to the model under persistent excitation.

9

3.0.1

Log-Linearisations

This section looks at the application of log-linearisations to the deterministic Brock-Mirman model. Given the steady states described in (15) we have 4 nonlinear equations for the 4 variables (Ct , Kt , Yt , t ). The steady states are: 1

(21)

K = ( A) 1 Y 1 = K C 1 = K

(22) (23)

The next step actually involves log-linearising the FOCs using a particular change of variables. These are: 0 = ^ t+1 (1 )k^t ^ t 0 = ^ t + c^t 0 = k^t 1 y^t C Y 0 = k^t c^t + y^t K K This gives us 4 linear equations for the 4 log-transformed variables.

Solving for the Policy Function As with the dynamic programming example we can solve algebraically for the policy function k^t = h(k^t 1 ) which in log-linear terms will look like k^t = k^t 1 . Therefore the …rst stage involves postulating the form of the policy function k^t = k^t

1

Next, from the four log-linearised equilibrium conditions we isolate the multiplier and the state variable k^t 1 by eliminating c^t and y^t to obtain ^ t = ^ t+1 1^ k^t+1 = kt

10

(1 1

)k^t ^t

Using the postulated decision rule and repeated substitution between the two expressions we obtain an expression for the state variable policy function 1 1

k^t =

k^t

1

(1

1

1

)

which implies 1 1

=

1

(1

1

)

This expression rearranges to yield a quadratic equation in solutions for are therefore or

= Since

1

. The two

1

> 1 and lies outside the unit circle such that the associated

solution is an unstable equilibrium, we can conclude that policy functions are therefore k^t = c^t =

k^t k^t

=

and the

1 1

In order to calculate the policy function for capital in levels-form we reverse Xt the log-transformations of the variables using x^t = log to give X log log

Kt Kt 1 = log K K Kt Kt 1 = log K K Kt Kt 1 = K K

11

Substituting in for K we get Kt 1

(

A) 1

0

= @

Kt =

Kt

1

(

A) 1 1

(

A) 1

(

A) 1 AKt

Kt =

1

1 A

Kt

1

1

This policy function is identical to that obtained from dynamic programming. Therefore the log-linearisations provide an identical policy function for Kt as a function of the state variable, Kt 1 to that given by the analytical solution of the model. Using the parameter values this means that the policy function would be Kt = 0:3135Kt0:33 1 3.0.2

Persistent Excitation

the Persistent Excitation approach involves solving for the endogenous variables of the model under two scenarios. First, the normal (unperturbed) model is solved to obtain a series for consumption and capital. Second, the exogenous variables of the model are then perturbed by a simple white noise process, and the perturbed model is then solved for to obtain a second pair of time series for both consumption and capital. Both series are then used to construct a pair of ARDLs which represent the approximation to the nonlinear model. Then the ARDLs are solved using a standard rational expectations solution programme, such as Blanchard and Kahn (1980) or Sims (1997), or AIM (Anderson and Moore, 1985) in order to obtain the policy functions. To begin with, it was necessary to provide a starting guess for the solution to the two equations. Under Persistent Excitation, all forward variables are treated as initially …xed. Therefore Ct+1 was set up as an exogenous variable while all other variables were treated as endogenous. Once the optimal time paths and steady states have been obtained for the chosen initial values, the next step is to construct the ARDL which constitutes the output of the PE approach, as these are the approximations to the nonlinear equilibrium conditions. OLS regressions corresponding to 12

the ARDLs are run to understand the extent to which the excited solutions for the control and state variables, Ct and Kt , can be explained by lagged variables and the speci…c white noise process used. The variables to be used within the ARDLs should correspond to those which appear in the familiar two equilibrium conditions of the deterministic Brock-Mirman Model Ct+1 = AKt 1 Ct Kt = AKt 1 Ct Under PE, we have solved this simultaneous system of equations twice. First, the model was solved holding Ct+1 exogenous and …xed, and subsequently the model is solved when Ct+1 has been perturbed by a normal white noise process. This generated two time series for Ct and Kt . In the case of consumption, the …rst is termed CtU where the U superscript denotes the series is from the unperturbed model, and the second series is CtP , where the P superscript denotes the solution is from the perturbed version. Now we wish to set up the ARDLs which can be used as an approximation for the two nonlinear equilibrium conditions which can then be solved to obtain the appropriate laws of motion for both endogenous variables in terms of the state variable Kt 1 , of the form Ct = h(Kt 1 ) Kt = g(Kt 1 ) As the model yields a linear law of motion when it is log-linearised we construct log-linear ARDLs with the regressors matching the RHS variables in the nonlinear equations above. The form of the variables we wish to run CtP KtP regressions on are: U and U . Therefore the two ARDLs are written as Ct Kt CtP CtU KP log tU Kt log

= =

P Ct+1 KtP + log 2 U KtU Ct+1 KtP 1 CtP log + log 1 2 KtU 1 CtU 1

log

(24) (25)

We can actually use the standard log-linearised equations to get an idea of what the coe¢ cients should equal in these equations. Following Uhlig’s transformations, which involve substituting in for Ct = C exp c^t where 13

c^t = log

Ct , and rearranging, we get the following two log-linearised exC

pressions c^t = c^t+1 k^t =

1)k^t

(

AK

1^ kt 1

C c^t K

Using the structural parameter values of = 0:33, A = 1 and would suggest coe¢ cients on the regression of 0 0

= 0; = 0;

1 1

= 0:67; = 1:05;

2 2

= 0:99, this

=1 = 2:18

The results from the regression corresponding to equation (24) is given below: Ordinary Least Squares Estimation Coe¢ cient Std. Error 0:67 9:6265e 015 1 1 9:3025e 015 2

T-Statistic 6:96e + 013 1:075e + 014

Corrected R-Squared No. of Observations Degrees of Freedom Sum of Squared Residuals Durbin-Watson Statistic Std. Error of Regression

1 286 284 8:4078e 20:7386 1:7332e

14

028 015

The results from the regression corresponding to equation (25) is given below: Ordinary Least Squares Estimation Coe¢ cient Std. Error 1:052 0:0015433 1 2:181 0:0029946 2

T-Statistic 681:84996 728:39118

Corrected R-Squared: No. of Observations: Degrees of Freedom: Sum of Squared Residuals: Durbin’s h-Statistic: p-value: Std. Error of Regression

0:99955 286 284 7:8892e 005 0:052985 0:47887 0:00052892

We can see immediately that the results from the OLS regressions match the coe¢ cients from the log-linearised equations. The …t is in fact almost perfect as can be seen from the corrected R-Squared. The reason we obtained these values is due to the fact that CtU = C and KtU = K, which occurs due to the appropriate truncation of the time series for CtU and KtU such that the initial convergence path to the analytical steady state is removed. Therefore the variables in both Uhlig’s approach and Persistent Excitation are identical. As such Persistent Excitation does what the log-linearisations are doing using a more direct and numerical approach, without the tedious job of manually log-linearising the equilibrium conditions. Having obtained the following expressions ^ P E + C^ P E C^tP E = 0:67K t t+1 PE PE ^ ^ Kt = 1:05Kt 1 2:18C^tP E P

C where C^tP E = log tU , we plug the two approximations into Sims’ solver. Ct ^ tP E = K ~ t and C^ P E = C~t . Therefore, the two For ease of notation let K t ARDLs which are approximating the nonlinear equilibrium conditions for the deterministic Brock-Mirman model can be expressed with parametric coe¢ cients as We now proceed to solve these equations for the decision rules using Sims (2002) method for solving forward-looking models.This yields precisely the 15

same rule as under Uhlig’s approach. C^tP E = ^ PE = K t

^ P E = 0:33K ^ PE K t 1 t 1 P E ^ ^ PE K t 1 = 0:33Kt 1

In order to calculate the policy functions in levels-form we reverse the logtransformations of the variables, as we did in the log-linearisation case above, P ^ tP E = log Xt , where X U = X using X t XtU log

KtP KtU

= 0:33 log

KtP KtU

=

KtP 1 KtU 1

KtP 1 KtU 1 0:33

Given that KtP is like any normal series of Kt and after substituting in for KtU = K = 0:1771, Kt = K 0:67 Kt0:33 1 0:33 Kt = 0:3135Kt 1 This policy function is identical to that obtained from dynamic programming. Therefore the log-linearised ARDLs provide an identical policy function for Kt as a function of the state variable, Kt 1 to that given by the analytical solution of the model.

3.1

Stochastic Brock-Mirman Model

We now move onto the stochastic version of the Brock-Mirman model. The common parameter values to be used remain as = 0:33; = 1;

= 0:99 A=1

In addition we need to specify the value for the coe¢ cient in the exogenous AR(1) process: = 0:95.

16

3.1.1

Log-Linearisation: Application of Uhlig’s Toolkit

This section looks at the application of log-linearisations to the stochastic Brock-Mirman model. The starting point to implementing this approach is once again the relevant FONCs with respect to Ct , Kt and t that were derived earlier.In addition now we require an expression for the exogenous process as well as the production function Yt = AZt Kt 1 ln Zt = ln Zt 1 + "t Therefore there are at least 5 nonlinear equations for the 5 variables (Ct , Kt , Yt , Zt , t ). As before, having determined the linearised conditions which characterise the dynamics of the system (FONCs, constraints, exogenous process) we the proceed to …nd the steady states of all variables. The choice of retaining the multiplier in the analysis ensures a richer examination of the dynamics of the model. The the steady states are Z = 1 1

K = ( A) 1 Y 1 = K C 1 = K Log-linearising the Equilibrium Conditions The log-linearised equilibrium equations in the stochastic case are very similar to those in the deterministic case except for the appearance of the exogenous shock term. 0 = Et [ (1 )k^t + ^ t+1 ^ t + z^t+1 ] 0 = ^ t + c^t 0 = k^t 1 y^t + z^t 1 1 0 = k^t + c^t y^t z^t =

z^t

1

+ "t 17

Impulse responses to a shock in technology 1.5 capital output consumption

Percent deviation from steady state

1

technology

0.5

0

-0.5

-1 multiplier -1.5 -1

0

1

2

3 4 Years after shock

5

6

7

8

Figure 1: Impulse Responses to a Shock in Technology

This gives us 5 linear equations in 5 variables. The solution provided by Uhlig’s toolkit is as follows P = 0:33; 0:33 R=

0:33

0:33

0

Q = 1:00 1:00 ; S=

1:00

1:00

0

For comparison we graph the impulse responses. The results fromUhlig’s programme are plotted in Figures 1 and 2, showing how the variables respond to a percentage deviation in technology and capital respectively.We can see that capital and consumption track one another perfectly and are mirrored by the impulse response for the Lagrange multiplier. Substituting for P , Q, R and S into the policy function provided by Uhlig’s toolkit, we get the following decision rules k^t = 0:33k^t c^t = 0:33k^t

+ z^t ^t 1+z 1

(26) (27)

One can immediately see that the coe¢ cient on k^t 1 is the parameter = 0:33. Once again we can calculate the policy function for both variables in 18

Impulse responses to a one percent deviation in capital 1

Percent deviation from steady state

0.8

0.6

0.4

capital output consumption

0.2

0

technology

-0.2

-0.4 -1

multiplier 0

1

2

3 4 Years after shock

5

6

7

8

Figure 2: Impulse Responses to a 1% deviation in Capital

levels-form by reversing the log-transformations of the variables using x^t = Xt log . In the case of the decision rule for capital, we get X log log

Kt Kt 1 = log K K Kt Kt 1 = log K K Kt Kt 1 = Zt K K

19

+ log Zt

Zt Z

Substituting in for K we get Kt 1

(

A) 1

0

= Zt @

Kt = Z t

(

Kt

1

(

A) 1 1

A) 1

( A) 1 AZt Kt

Kt =

1

1 A

Kt

1

1

This policy function is identical to the analytical solution, which was obtained via dynamic programming. Therefore the log-linearisations provide an identical policy function for Kt as a function of the endogenous and exogenous state variables, Kt 1 and Zt respectively. Once again evaluating the policy function at the structural parameters Kt = 0:3135Zt Kt0:33 1

The entire time path can be generated using this policy function once we have initial values for K0 and Z0 . The impulse responses are given in Figures 1 and 2. 3.1.2

Persistent Excitation

In the case where exogenous productivity shocks, Zt , were added to the model the two nonlinear equations to be solved were 1 Ct 1 = AEt [Ct+1 Zt+1 Kt Kt = AZt Kt 1 Ct

1

]

As we can see from this equation, there are now two forward-looking variables in the Euler equation, Ct+1 and Zt+1 . Zt is already an exogenous shock process, but it too will be perturbed using an orthogonal white noise process in the same way that Ct+1 was perturbed. As we noted earlier the log of these shocks follow an AR(1) process. Once again both the unperturbed and perturbed models were solved. We continued to treat the forward-looking variable Ct+1 as exogenous such that the perturbations remained consistent with the deterministic case. 20

Once again we proceed with obtaining the ARDLs which can be used as an approximation for the two nonlinear equilibrium conditions which can then be solved to obtain the appropriate laws of motion for both endogenous variables in terms of the endogenous state variable Kt 1 and the exogenous state variable Zt , of the form Ct = h(Kt 1 ; Zt ) Kt = g(Kt 1 ; Zt ) As the model yields a linear law of motion when it is log-linearised it would be sensible to construct log-linear ARDLs with the regressors matching the RHS variables in the nonlinear equations above. The form of the variables CP KP we wish to run regressions on are tU and tU . Therefore the two ARDLs Ct Kt are written as CtP CtU KP log tU Kt log

P P Ct+1 Zt+1 KtP + log + log 2 3 U U Ct+1 Zt+1 KtU KtP 1 CtP ZtP + log + log 1 log 2 3 CtU ZtU KtU 1

=

1 log

=

We can actually use the standard log-linearised equations to get an idea of what the coe¢ cients should equal in these equations. Following Uhlig’s transformations, which involve substituting in for Ct = C exp c^t where Ct c^t = log , and rearranging, we get the following two log-linearised exC pressions c^t = Et c^t+1 k^t =

A Kt

Et Zt+1 1^ 1 k

( 1

+ AK

z^t

1)k^t C c^t K

Using the structural parameter values of = 0:33, A = 1 and would suggest coe¢ cients on the regression of =1 = 1:05

1 1

2 2

= 1 = 3:18

= 0:95, this

= 0:67 = 2:18

3 3

The results from the OLS regressions provide coe¢ cients which are very close 21

to these suggested coe¢ cients Ordinary Least Squares Estimation Coe¢ cient 1:0000 1 1:0000 2 0:6700 3

Std. Error 9:6265e 015 9:3025e 015 2:7754e 015

Corrected R-Squared No. of Observations Degrees of Freedom Sum of Squared Residuals Durbin-Watson Statistic Std. Error of Regression

T-Statistic 1:6518e + 014 1:0455e + 014 2:414e + 014 1 286 284 1:3556e 17:5251 2:1964e

027 015

Ordinary Least Squares Estimation Coe¢ cient 2:2257 1 3:2270 2 1:0672 3

Std. Error 0:010278 0:0096675 0:0037355

Corrected R-Squared: No. of Observations: Degrees of Freedom: Sum of Squared Residuals: Durbin’s h-Statistic: p-value: Std. Error of Regression

T-Statistic 216:55530 333:79945 285:68684 0:99965 286 284 1:1037e 003 5:6798 6:7416e 009 0:00019818

The …t is exact. The reason we obtained these values is due to the fact that CtU = C and KtU = K, which occurs due to the appropriate truncation of the time series for CtU and KtU such that the initial convergence path to the analytical steady state is removed. Therefore the variables in both Uhlig’s approach and Persistent Excitation are identical. As such Persistent Excitation does what the log-linearisations are doing using a more direct and numerical approach. 22

Having obtained these expressions we plug the two approximations into Sims’solver PE PE ^ P E + ut Et Z^t+1 + 0:67K C^tP E = Et C^t+1 t PE PE PE ^ ^ ^ ^ P E + vt Kt = 2:21Ct + 3:22Zt + 1:05K t 1

CtP ^ tP E ~ t , C^tP E log U . For ease of notation let K K C~t Ct and Z^tP E Z~t . Therefore, the two ARDLs which are approximating the nonlinear equilibrium conditions for the deterministic Brock-Mirman model can be expressed with parametric coe¢ cients as

where C^tP E

C~t = ~t = K

~

~ t + vt + 2 Et Z~t+1 + 3 K ~ ~ ~ 1 Ct + 2 Zt + 3 Kt 1 + ut

1 Et Ct+1

Once the ARDL has been obtained, it is necessary to solve the ‘linearised’ model for the policy function. There is no obvious method of solving for the model other than by substitution from one equation into another until one of the variables has been eliminated. as was done in the case of loglinearisations. In this case the obvious variable to eliminate is C~t+1 in an ~ t and C~t in terms of the lagged attempt to obtain policy functions for K endogenous variables. Therefore to obtain a solution to these equations, Sims (2002) method for solving forward-looking models is used.The policy function is C~t+1 ~ t+1 K

=

0:3301 0:3300

h

i ~t + K

1:0001 0:9999

h i Z~t

This yields precisely the same rule as under Uhlig’s approach (ignoring the forward shift on the Z~t variable and rounding the coe¢ cients appropriately), again because Uhlig’s Matlab codes use precisely those QZ decompositions that were programmed by Sims: C^tP E = ^ tP E = K

^ tP E1 + Z^tP E = 0:33K ^ tP E1 + Z^tP E K ^ tP E1 + Z^tP E = 0:33K ^ tP E1 + Z^tP E K

In order to calculate the policy functions in levels-form we reverse the logtransformations of the variables, as we did in the log-linearisation case above, 23

1.4 Consumption Capital Technology

1.2

1

0.8

0.6

0.4

0.2

0

0

10

20

30

40

50

60

70

80

90

100

Figure 3: Impulse Responses to a Shock to Technology

P

^ tP E = log Xt , where XtU = X, to give after substituting in for the using X XtU analytical expression of K Kt =

AZt Kt

1

Using the parameter values this means that the policy function would be Kt = 0:3135Zt Kt0:33 1 This policy function is identical to that obtained from dynamic programming. Therefore the log-linearised ARDLs provide an identical policy function for Kt as a function of the state variables, Kt 1 and Zt , to that given by the analytical solution of the model. The impulse responses were also calculated for the model and are shown in Figures 3 and 4. We can see that they match perfectly those from both the Log-linearisations. Therefore we have successfully shown that Persistent Excitation can solve for the Stochastic Brock-Mirman Model, providing solutions which match those derived analytically. 24

Impulse responses to a one percent deviation in Capital 1

Percent deviation from steady state

0.9 0.8 0.7 0.6 0.5 0.4 0.3

CapitalConsumption

0.2 0.1 0 -2

Technology 0 2

4 Years after shock

6

8

10

Figure 4: Impulse Responses to a 1% Deviation in Capital

25

4

Hansen’s Real Business Cycle Model

The other model to which Persistent Excitation will be applied is an extension of Hansen’s (1985) real business cycle model. The problem is one of a decentralised economy made up of identical households, …rms and the government. An Euler equation approach is used to obtain the equilibrium conditions for this model. As there are taxes, the solution to the optimisation problem faced by the social planner is no longer the same as in the decentralised economy where both …rms and households are price-takers. In addition this sub-optimality means that we cannot easily …nd a solution using standard dynamic programming, so we will be comparing the Persistent Excitation solution to those obtained under Uhlig’s log-linearisations. The household problem is to maximise max

Ct ;Nt ;Kt ;It

E0

subject to the budget constraint Ct + Kt

Kt

1

1 X

t

(log Ct

ANt )

t=0

= (1

)(Rt

)Kt

1

+ (1

)Wt Nt

where the evolution of capital is determined by Kt = (1

)Kt

1

+ It

(28)

Rt and Wt denote the rental price of capital and the wage rate respectively while the parameters and denote the discount factor and the depreciation rate of capital respectively. Furthermore, ; 2 [0; 1) denote tax rates on capital and labour income respectively. Taxes on capital are after depreciation allowances. To obtain the optimality conditions we use a stochastic Lagrangian L = E0

1 X

t

[log Ct ANt + t ((1

)(Rt

)Kt 1 +(1

)Wt Nt Ct Kt +Kt 1 )]

t=0

where

t

is the Lagrange multiplier. The FONCs with respect to Ct , Nt , Kt

26

and

t

@L @Ct @L @Nt @L @Kt @L @ t

are as follows6 = 0,

1 Ct

=

= 0,A= = 0,

t

t t (1

)Wt

= Et [

t+1 ((1

= 0 () Ct + Kt

Kt

)(Rt+1

1

= (1

) + 1)]

)(Rt

)Kt

1

+ (1

)Wt Nt

The representative …rm produces output Yt according to a Cobb-Douglas production technology with constant returns-to-scale in capital, Kt , and hours worked, Nt Yt = Zt Kt 1 Nt1 (29) Uncertainty comes from productivity disturbances, Zt . log Zt = log Zt

1

+ "t ,

"t

IID(0;

2

)

(30)

as was the case in the stochastic Brock-Mirman model. Factor demands come from …rms which, renting capital at rate Rt , and hiring e¢ ciency units of labour at a rate Wt ; maximise pro…ts as follows = max Zt Kt 1 Nt1

R t Kt

Kt ;Nt

1

W t Nt

The …rst order conditions for the …rm are Rt =

Z t Kt

Wt = (1

1 1 1 Nt

=

)Zt Kt 1 Nt

Yt Kt 1

= (1

(31) ) NYtt

(32)

The government in this RBC model has a balanced budget constraint given by Gt = (Rt )Kt 1 + Wt Nt From the preceding equations the equilibrium conditions that will characterise the dynamics of the model are selected: t is eliminated to give us the Euler equation, and the household budget constraint is transformed into the 6

To ensure that we are on the unique optimal path we need initial conditions K0 , Z0 as well as the TVC: limt!1 t t Kt+1 = 0

27

more general form of the feasibility constraint thereby eliminating It . Both …rm …rst ordr conditionss are retained, and we close with the production function and the exogenous AR(1) stochastic technological process h i 1 1 = E ((1 )(R ) + 1) (33) t Ct+1 t+1 Ct ACt Ct Gt Wt

(34) (35) (36) (37)

= (1 )Wt = Yt Kt + (1 )Kt 1 Gt = (Rt )Kt 1 + Wt Nt Yt = (1 ) Nt

Rt =

Yt Kt 1

(38)

Yt = Zt Kt 1 Nt1 log Zt = log Zt 1 + "t

(39) (40)

Therefore there are at least 8 equations for the 8 variables (Ct , Nt , Kt , Yt , Wt , Rt , Zt , Gt ). The parameter values to be used are = 0:36; = 1%;

= 0:025; A = 2:5846;

= 1=1:01; = 0:4;

= 0:95 = 0:25

These equations provide the fundamental inputs for the various approximation methods we will be following. We now move on to see the application of each of the approximation methods to the more complex Hansen model. The common parameter values to be used are listed again for convenience = 0:36; = 1%;

= 0:025; A = 2:5846;

= 1=1:01; = 0:4;

= 0:95 = 0:25

First the solution under log-linearisations will be reviewed, followed by parameterised expectations. The subsection concludes with the results for Persistent Excitation.

4.0.3

Log-Linearisation: Application of Uhlig’s Toolkit

This section looks at the application of log-linearisations to Hansen’s model. The starting point to implementing this approach is once again the relevant …rst order conditions. 28

Therefore there are at least 8 nonlinear equations for the 8 variables (Ct , Nt , Kt , Yt , Wt , Rt , Zt , Gt ). As before, having determined the linearised conditions which characterise the dynamics of the system (FONCs, constraints, exogenous process) we the proceed to …nd the steady states of all variables. The steady states are:

Labour

1 (1 A N =K R

Capital

K=

Technology Output Interest Investment

Z=1 Y = RK R = 1 +(1 (1) ) I = (1 )K + K = K

Wages

W = (1 ) h G = K (R

Consumption C =

Government

)(1

1

R

)

= 0:62465

1 1

(1 A[R

= 0:32789 )( R ) 1

)(1 (R

)

(1

= 9:5286

)R ]

=1 = 1:1028 = 0:041667 = 0:23821

1

R

) + (1

)

R

The log-linearised equilibrium equations are: 0 0 0 0 0 0 0 z^t+1

= = = = = = = =

= 2:1526

i

= 0:23998

Et [^ ct c^t+1 + (1 + (1 ))^ rt+1 ] z^t + k^t 1 n ^ t w^t ^ z^t + ( 1)kt 1 + (1 )^ nt r^t w^t c^t (R )K k^t 1 + W N n ^ t + RK r^t + W N w^t ^ z^t + kt 1 + (1 )^ nt y^t C^ c K k^t + (1 )K k^t 1 G^ gt + Y y^t z^t + "t

G^ gt

This gives us 8 equations in 8 variables. Implementing Uhlig’s Toolkit The log-linearisation approximation is carried out using Uhlig’s toolkit. Let vt denote the vector of endogenous non-state (‘other’) variables vt =

c^t

y^t

n ^t 29

r^t

g^t

w^t

0

The endogenous state variable is kt while the exogenous state variable is zt . Uhlig’s input equations are 0 = AAk^t + BB k^t 1 + CCvt + DD^ zt 0 = Et [F F k^t+1 + GGk^t + HH k^t 1 + JJvt+1 + KKvt + LL^ zt+1 + M M z^t z^t+1 = N N z^t + "t+1 From 8 nonlinear di¤erence equations in 8 unknowns we end up with 8 linear di¤erence equations in 8 unknowns. The aim is to solve them jointly, so as to determine the recursive equilibrium laws of motion for all variables. The linear policy functions will be k^t = P P k^t 1 + QQ^ zt ^ vt = RRkt 1 + SS z^t z^t = N N z^t 1 + "t From Uhlig’s toolkit the implied policy functions for each variable are: c^t n ^t r^t w^t g^t y^t k^t

= = = = = = =

0:5159k^t 0:4331k^t 0:9172k^t 0:5159k^t 0:2814k^t 0:0828k^t 0:9579k^t

+ 0:4576^ zt zt 1 + 1:5067^ zt 1 + 1:9643^ zt 1 + 0:4576^ zt 1 + 2:7442^ zt 1 + 1:9643^ zt 1 + 0:1282^

1

For completeness the impulse responses from Uhlig’s programme are plotted in Figures 5 and 6, which show how the …ve variables respond to a percentage deviation in technology and capital respectively. Substituting for P P and QQ into the policy function provided by Uhlig’s toolkit, we get a solution of k^t = 0:9579k^t

1

+ 0:1282^ zt

(41)

The entire time path can be generated using this policy function once we have initial values for k^0 and z^0 . Alternatively the series can be obtained in levels-form either by transforming the policy function back and using K0 and Z0 or by transforming the log-data series generated using rule (41).

30

Impulse responses to a shock in technology 3 government

Percent deviation from steady state

2.5

2

1.5

output interest labor

capital

1

wages consumption 0.5

0

-0.5

-1 -5

0

5

10 Years after shock

15

20

25

Figure 5: Impulse Responses to a Shock in Technology for Uhlig’s Toolkit

31

Impulse responses to a one percent deviation in capital 1

capital

0.8

Percent deviation from steady state

0.6 consumption wages 0.4 0.2 output 0 -0.2 government -0.4

labor

-0.6 -0.8 interest -1 -5

0

5

10 Years after shock

15

20

25

Figure 6: Impulse Responses to a 1% Deviation in Capital for Uhlig’s Toolkit

32

4.0.4

Persistent Excitation

Hansen’s model involves two key di¤erences from the Brock-Mirman model. There are now two forward-looking variables in the consumption Euler, Ct+1 and Rt+1 , which are perturbed along with the exogenous process, Zt . Moreover, while shocks to the capital stock and the technology process are su¢ cient to excite all of the model, for Hansen’s model we need also to perturb each error term in order to fully identify the model. Therefore, once we have obtained the solution paths for each of the seven endogenous variables, we proceed to construct the ARDLs which will be used to approximate the seven nonlinear equilibrium conditions. These can then be solved, as before, to yield the appropriate laws of motion in terms of the endogenous state variable Kt 1 and the exogenous state variable Zt , as follows Ct Kt Nt Rt Wt Gt Yt

= = = = = = =

h(Kt g(Kt g(Kt g(Kt g(Kt g(Kt g(Kt

1 ; Zt ) 1 ; Zt ) 1 ; Zt ) 1 ; Zt ) 1 ; Zt ) 1 ; Zt ) 1 ; Zt )

As the model yields a linear law of motion when it is log-linearised it would be sensible to construct log-linear ARDLs with the regressors matching the RHS variables in the nonlinear equations above. The form of the variables KP CP we wish to run regressions on are once again tU and tU , which represent Ct Kt the perturbed series normalised by the steady state. Therefore the seven

33

ARDLs are written as CtP CtU NP log tU Nt RP log tU Rt WP log tU Wt GP log Ut Gt YP log tU Yt KP log tU Kt log

= = = = = = =

P P Ct+1 Rt+1 + log + uC 2 t U U Ct+1 Rt+1 KtP 1 WtP ZtP + log + log + uN 1 log 2 3 t KtU 1 WtU ZtU KtP 1 NtP ZtP log + log + log + uR 1 2 3 t KtU 1 NtU ZtU CtP + uW 1 log t CtU KtP 1 NtP RtP WtP log + log + log + log + uG 1 2 3 4 t KtU 1 NtU RtU WtU KtP 1 NtP ZtP + log + log + uYt 1 log 2 3 U U U Kt 1 Nt Zt P P Kt 1 Gt YtP CtP log + log + log + log + uK 1 2 3 4 t KtU 1 GUt YtU CtU 1 log

XtP . Again we use the standard XtU log-linearised equations to get an idea of what the coe¢ cients should equal in these equations. Under Uhlig’s transformations, which involve substituting Ct in for Ct = C exp c^t where c^t = log , and rearranging, we arrive at the C following log-linearised expressions as before ~ Once again for ease of notation, let X

0 0 0 0 0 0 0 z^t+1

= = = = = = = =

log

Et [^ ct c^t+1 + (1 + (1 ))^ rt+1 ] z^t + k^t 1 n ^ t w^t z^t + ( 1)k^t 1 + (1 )^ nt r^t w^t c^t (R )K k^t 1 + W N n ^ t + RK r^t + W N w^t z^t + k^t 1 + (1 )^ nt y^t ^ C^ ct K kt + (1 )K k^t 1 G^ gt + Y y^t z^t + "t

Rearranging them we get the following system 34

G^ gt

c^t = Et c^t+1 n ^ t = k^t

(1 1

1

w^t +

+ 1

(1

))Et r^t+1

z^t

rt = ( 1)k^t 1 + (1 )^ nt + z^t w^t = c^t (R )K ^ WN RK WN g^t = kt 1 + n ^t + r^t + w^t G G G G y^t = k^t 1 + (1 )^ nt + z^t G Y C k^t = (1 )k^t 1 g^t + y^t c^t K K K z^t+1 = z^t + "t We can evaluate the coe¢ cients on each of the log-deviation variables to obtain the system expressed as follows c^t n ^t r^t w^t g^t y^t k^t z^t+1

= = = = = = = =

Et c^t+1 0:02485Et r^t+1 k^t 1 2:7778w^t + 2:7778^ zt 0:64k^t 1 + 0:64^ nt + z^t c^t 0:2647k^t 1 + 0:7353^ nt + 0:6618^ rt + 0:7353w^t 0:36k^t 1 + 0:64^ nt + z^t 0:975k^t 1 0:0252^ gt + 0:1157^ yt 0:0656^ ct z^t + "t

Using this system we can see, as before, what corresponding parameter values we would expect from the OLS regressions: = 1; = 1; 1 0:64; 1 = = 1; 1 1 = 0:2647; 1 = 0:36; 1 = 0:975; 1

= 0:02485; = 2:7778; 2 2 = 0:64;

3

= 0:7353; 2 = 0:64; 0:0252; 2 =

= 0:6618; 3 = 1; 3 = 0:1157;

2

2

3

3

35

= 2:7778; = 1; 4

= 0:7353

4

=

0:0656

Using the PE method the (2002). They are 2 3 2 C~t+1 6 N 7 6 ~t+1 7 6 6 R 7 6 6 ~ t+1 7 6 6 ~ 7 6 6 Wt+1 7 = 6 6 ~ 7 6 6 Gt+1 7 6 6 ~ 7 6 4 Yt+1 5 4 ~ t+1 K

policy functions are then recovered using Sims

0:5010 0:4217 0:9153 0:5010 0:2895 0:0872 0:9590

3

2

7 6 7 6 7h i 6 7 6 ~t + 6 7 K 7 6 7 6 7 6 5 4

0:4365 1:5373 1:9764 0:4365 2:7807 1:9815 0:1293

3

7 7 7h i 7 7 Z~t 7 7 7 5

Comparing these to the results iabove, we can see that PE method yields decision rules very close to those under Uhlig’s approach. The corresponding impulse responses, plotted in Figure ??, are extremely close to those above.

5

Conclusion

In this paper we have proposed a new approach to solving DSGE models. Persistent excitation is a method widely used in various …elds in order to identify linear systems and to approximate nonlinear systems. For the comparison, the approximation under Persistent Excitation used ARDLs which were chosen to mirror the form of Uhlig’s log-linearisations. This approach was taken in order to verify how good Persistent Excitation was in relation to Uhlig’s toolkit, which in spite of the increased interest in higher-order approximations, remains a widely used popular package. Therefore we were able to check that the coe¢ cients estimated via PE were equivalent to Uhlig’s analytical counterparts, while highlighting its greater power in terms of operating directly on the equilibrium conditions at the outset, and not being restricted by log-linearised inputs in the …rst instance. This is a relatively untried method in the context of DSGE models but it appears to have some promise. Its extension to larger models is essential. The possibility of also using it to calculate second order approximations is under examination.

36

6

References

References [1] Anderson, G. and G. Moore (1985). “A Linear Algebraic Procedure for Solving Linear Perfect Foresight Models”, Economics Letters, 17, 247252. [2] Anderson, G. (1998). “A Reliable and Computationally E¢ cient Algorithm for Imposing the Saddlepoint Property in Dynamic Models“, Mimeo, Federal Reserve Board. [3] Blanchard, O.J. and Kahn, C.M. (1980). “The Solution of Linear Difference Models under Rational Expectations”, Econometrica, 48, pp. 1305-1311. [4] Åström, K. J. (1970), Introduction to Stochastic Control Theory, Academic Press, London. [5] Boucekkine, R. (1995), “An alternative methodology for solving nonlinear forward-looking models”, Journal of Economic Dynamics and Control, Vol. 19, pp. 711–734. [6] Brock, W.A. and L. Mirman (1972), “Optimal economic growth and uncertainty: the discounted case”, Journal of Economic Theory, Vol. 4, pp. 479–513. [7] Christiano, L. J. (2002), “Solving Dynamic Equilibrium Models by a Method of Undetermined Coe¢ cients”, http://www.faculty.econ.northwestern.edu/faculty/christiano/research/Solve/papernew.pdf [8] Corrado, L. and Holly, S. (2006) "The Linearisation and Optimal Control of Large Non-linear Rational Expectations Models by Persistent Excitation", Computational Economics. 28, 2, pp. 139-153. [9] Den Haan, W. and A. Marcet (1990), “Solving the Stochastic Growth Model by Parameterising Expectations”, Journal of Business and Economic Statistics, Vol. 8, pp.31-34. [10] Den Haan, W. and A. Marcet (1994), “Accuracy in Simulations”, The Review of Economic Studies, Vol. 61, No. 1, pp.3-17. 37

[11] Fair, Ray, & Taylor, John. 1983. Solution and Maximum Likelihood Estimation of Dynamic Rational Expectations Models. Econometrica, 51. [12] Gagnon, J.E. (1990), “Solving the stochastic growth model by deterministic extended path”, Journal of Business and Economic Statistics, Vol. 8, pp. 35–36. [13] Hannan, E.J. (1971). “The Identi…cation Problem for Multiple Equation Systems with Moving Average Errors”, Econometrica, Vol. 39, 751-65. [14] Heer, R. and A, Maussner (2005), Dynamic General Equilibrium Modelling: Computational Methods and Applications, Springer. [15] Holly S. and Zarrop, M. (1983), "On Optimality and time consistency when expectations are rational", European Economic Review, 20(1-3), pp. 23-40, [16] Judd, K. L. (2003), “Perturbation Methods and Change of Variable Transformations”, Computing in Economics and Finance, Vol. 239, Society for Computational Economics. [17] Judd, K.L. (1998), Numerical Methods in Economics, MIT Press. [18] King, R., Watson, M., (1998). “The Solution of Singular Linear Difference Systems under Rational Expectations”, International Economic Review, Vol. 39, pp. 1015-1026. [19] Lipton D, Poterba J., Sachs J., Summers, L. (1982) “Multiple Shooting in Rational Expectations Models," Econometrica, 50, pp.1329-33. [20] Lombardo, G. and Sutherland, A (2007). "Computing second-orderaccurate solutions for rational expectation models using linear solution methods," Journal of Economic Dynamics and Control, Elsevier, vol. 31(2), pages 515-530, February. [21] Marcet, A. and G. Lorenzoni (1999), “The Parameterised Expectations Approach: Some Practical Issues”, in Computational Methods for the Study of Dynamic Economics, edited by R. Marimon and A. Scott, Oxford University Press, Oxford.

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[22] Schmitt-Grohe, S. and M. Uribe (2004), “Solving dynamic general equilibrium models using a second-order approximation to the policy function”, Journal of Economic Dynamics & Control, Vol. 28, pp. 755-775. [23] Sims, C. A. (1996). “Solving Linear Rational Expectations Models”, Mimeo, Yale University. [24] Söderström, S. and P. Stoica (1989), System Identi…cation, Prentice Hall International. [25] Taylor, J.B. and H. Uhlig (1990), “Solving nonlinear stochastic growth models: a comparison of alternative solution methods”, Journal of Business Economics and Statistics, Vol. 8, No. 1, pp. 1–17. [26] Uhlig, H. (1999), “"A toolkit for analysing nonlinear dynamic stochastic models easily," in Computational Methods for the Study of Dynamic Economics, edited by R. Marimon and A. Scott, Oxford University Press, Oxford. [27] Zarrop, M., Holly, S., Rustem, B. and Westcott, J.H (1979), “The Design of Economic Stabilisation Policies with Large, Nonlinear Econometric Models: Two Possible Approaches”, in P. Ormerod (Editor) Economic Modelling, London.Heinemann.

39