The Linearisation and Optimal Control of Large Non-linear Rational Expectations Models by Persistent Excitation. Luisa Corrado University of Rome and University of Cambridge Sean Holly University of Cambridge September 27, 2006 Abstract Since the onset of the rational expectations revolution in macroeconomics some 20 or more years ago, a variety of techniques have evolved for the solution of rational expectations models. The …rst generation of methods were for linear models starting with the method of Blanchard and Kahn (1981). Because the models are large and usually non-linear, methods for solving (and optimising) such models have evolved in parallel (Zarrop et al, 1979, Finan and Tetlow, 1999, Fair, 2003). Di¤erently from recent methods that apply secon-order approximation (Schmitt-Grohè and Uribe (2004), Sims, 2002b) in this paper we describe some computationally simple methods for linearising a non-linear model with rational expectations using persistent excitation. Each instrument, exogenous variable and expectational term is excited with a white noise process. Given superimposition, each input process is orthogonal so the reduced form can be estimate by OLS. Once the linear form is obtained and the time-consistent optimal feedback rule computed by dynamic programming, we apply the rational expectations solution of Anderson and Moore (1985) which is particularly suited when the leading structural matrix is singular. We apply the method to a nonlinear model of the UK Economy and report a series of impulse responses for output, in‡ation, the exchange rate and the short term interest rate.

1

Introduction

Although the models that dominate current macroeconomic theoretical discussion are relatively small analytical constructs, there still remains a tradition of developing and estimating structural models especially in Finance Ministries, International Organisations and Central Banks. Increasingly, these models are much more theory driven, though they continue to engage closely with the data (Kapetanios, Pagan and Scott, 2005) Recent examples of best practice include work at the New Zealand Reserve Bank (Hunt et al, 2000), the IMF (Laxton, et al, 1998), the Federal Rseserve Board [1]

(Brayton and Tinsley, 1996), the Bank of Canada (Black et al, 1994) and the Bank of England (Harrison et al, 2005). As well as being much more shaped by theoretical considerations than previous generations they also seek to incorporate rational expectations as a matter of course. Because the models are large and usually non-linear, methods for solving (and optimising) such models have evolved in parallel (Zarrop et al, 1979, Finan and Tetlow, 1999, Fair, 2003). A more recent method for solving dynamic non-linear general equilibrium models is that proposed by Schmitt-Grohè and Uribe (2004) which in turn generalizes the methods of Blanchard and Kahn (1980), King and Watson (1998), and Klein (2000). They derive a second-order approximation to the solution of a general class of discrete-time rational expectation models. To obtain an accurate second-order approximation they use a perturbation method that incorporates a scale parameter for the standard deviations of the exogenous shocks as an argument of the policy function and then take a second-order Taylor expansion with respect to the state variables as well this scale parameter. They are able to show that up to second order the coe¢ cients linear and quadratic in the Taylor expansion are independent of the volatility of the exogenous shocks. Sims (2002b) also derives a second-order approximation to the policy function for a wide class of discrete-time models by assuming in a similar way that uncertainty a¤ects only the constant term of the decision rule not the coe¢ cients of the rule. However, these methods face the curse of dimensionality and cannot be applied easily to the large nonlinear models commonly used in policy analysis. In this paper we adopt a di¤erent approach. We combine the persistent excitation method of Zarrop et al (1979) to generate a linearisation of a nonlinear model and the method of Christodolakis (1989) in order to take into account the presence of forwardlooking expectations. We then estimate a reduced form using OLS, convert it into state space, compute the optimal feedback rule under commitment using dynamic programming and solve the resulting model by means of the Anderson-Moore (1985) method. Other possible methods for solving a linear model with rational expectations are those of Blanchard and Kahn (1980) and Sims (2002a). Sims (2002a) by generalising the approach by Blanchard and Kahn (1980) provides a computationally robust solution method for linear rational expectations models, based on the QZ matrix decomposition. It requires that the model be cast into …rst-order form, but it does not require that the number of states matches the number of equations. It also avoids the arti…cial requirement that variables be designated as jump variables or not as in Blanchard and Kahn (1980). The method proposed by Anderson and Moore (1985) is one of the most general methods for a wide class of discrete-time models. First it does give a solution even if the leading structural coe¢ cient matrix is singular. In fact not every scalar equation will have the maximum lead so that some rows of the leading coe¢ cient matrix will be zero. Second, it is applied to models with more than one lead and one lag and …nally it does not need the speci…cation of the system in companion form. Given the presence in our full linearised model of a singular leading matrix (the expected exchange which appears only in the UIP equation) we apply the AIM method to derive a solution for the model while satisfying the saddlepoint

[2]

requirements of the rational expectations solution.

The Method

2

In this section we describe the steps we go through in order to (1) linearise a non-linear rational expectations model, (2) estimate a reduced form, (3) convert it into state space form, (4) compute the optimal control solution (5) solve the full linearised model while satisfying the saddlepoint requirements of the rational expectations solution.

2.1

Linearization by Persistent Excitation.

Assume we can write our non-linear, stochastic model as: Fi (yt ; yt

1 ; ::::yt s ; Et yt+1 ; ::::; Et yt+v; xt ; xt 1 ; ::::; xt r ; t )

=0

(2.1)

i = 1; :::; m t = 1; :::; T There are m endogenous variables, y, with a maximum lag of s, a maximum lead of v and a maximum of r lagged values of n exogenous variables, x; t is m-dimensional vector of white noise processes. We can expand (2.1) about some initial path to give: s X j=0

@Fi @yt j

o

yet

j+

v X j=0

@Fi @Et yt+1+j

o

Et yet+1+j +

r X j=0

@Fi @xt j

o

x et

j+

@Fi @ t

o

e (2.2) t

i = 1; :::; m t = 1; :::; T The perturbations to the initial path are de…ned as: yet = yt

yot ; x et = xt

xot ; et =

t

ot

(2.3)

In general (2.2) is time-varying. However, we want to obtain an approximation to this time-varying representation. We will also only be interested in some subset of the endogenous variables, the targets and a subset of the exogenous variables, among which will be the policy instruments. Assume v = 0, then we can write this representation in vector polynomial form in the lag operator L, as: A(L)e yt + C(L)Et yet+1 + B(L)e xt = et

(2.4)

In order to derive a constant coe¢ cient, linear representation we perturb a subset of the instrument vector and the expectational variables1 . The perturbations are a sequence of orthogonal white noise processes. The use of white noise perturbations is one way of meeting a basic identi…ability condition (Hannan, 1971, Zarrop, 1979) 1 It is also possible to perturb the error processes. This may help to capture aspects of the nonlinear model missed by the direct linearisation.

[3]

that the perturbations have an absolutely continuous spectrum with spectral density non-zero on a set of positive measures in ( ; ). The white noise sequence, since it contains all frequencies, will excite all of the dynamic modes of the non-linear model. De…ne zet = (Et yet+1 ; x et ; et )0 as a stacked vector then: E[e zi (s)] = 0; E[e zi (s)e zj (t)] = i; j = 1; :::; p;

i

ij

(2.5)

st

s; t = 1; :::; T

with p = 2m + n; ij is the Kronecker delta and i sets the size of the perturbations. We can then write the relationship between the target variables, the perturbed instruments, the expectational terms and the errors as an autoregressive distributed lag model: A0 y t = A1 y t

1

+ ::: + As yt

s

+ D1 Et yt+1 + B0 xt + ::: + Br xt

(2.6)

r

Since the perturbations are orthogonal by construction, A0 is an identity matrix and the matrices A1 to As are diagonal, so each equation can be estimated separately by ordinary least squares. This superimposition simpli…es the construction of the linearisation. It is also possible to improve the e¢ ciency of the estimates by running a sequence of linearisations by persistent excitation and then calculate the mean group estimator across the panel of linearisations following the approach of Pesaran and Smith (1995).

2.2

State Space form

For the case of

t

= 0, the state vector is simply de…ned as: zt0 = (yt ; ::::; yt

s 1 ; xt 1 ; ::::; xt r 1 )

(2.7)

and the state transition matrix: zt = Azt

1

+ Bxt + DEt zt+1

(2.8)

where: 2

6 I 6 6 0 6 A=6 6 : 6 6 4 0

1

2

0 I 0 : 0

::: ::: I 0 : 0

s

2

:::

r

0 : I 0 0

I :::

I

0

3

2

7 6 0 6 7 6 : 7 7 6 7; B = 6 6 7 7 6 7 6 5 4 0

1

3

2

7 6 0 6 7 6 : 7 7 6 7;D = 6 6 7 7 6 7 6 5 4 0

1

3 7 7 7 7 7 7 7 7 5

z is a f = n (s + r 1) dimensional vector, x is an m-dimensional vector. The transition matrix A is f f , B is f m, D is f f . We also de…ne j = A0 1 Aj for j = 1; ::; s; i = A0 1 Bi for i = 1; ::; r; 1 = A0 1 D1 . [4]

Since the state space form contains forward-looking expectations we could follow the approach of Blanchard and Kahn (1980), Sims (2002a), Amman and Kendrick (1998), Anderson and Moore (1985), among others, and solve explicitly for the rational expectations solution. Instead, on the assumption that there is a su¢ ciently strong commitment technology in place to rule out time inconsistent behaviour, we derive, …rst, an optimal rule via dynamic programming. For the present let us hold the expectation …xed. We rewrite the state transition equation as: zt = Azt

1

+ Bxt + Cet

(2.9)

the vector e subsumes the expected values, and in general it could also include any other exogenous variables as well as stochastic error terms.

2.3

The Optimal Control Solution

To solve for the optimal control rule we de…ne a loss function in terms of the state variables z and the control or instrument variables, x. n

Lt =

1X (zt 2

ztd )0 Q(zt

ztd ) + (xt

xdt )0 N (xt

xdt )

(2.10)

t=0

where the superscript de…nes desired values for the state variables and the policy instruments, Q is a symmetric, semi-positive de…nite f f matrix, and N is a symmetric m m positive de…nite matrix. To minimize (2.10) subject to the state transition equation (2.9) we can apply the well-known method of dynamic programming to compute an optimal control rule of the form: xt = Kt zt

1

+ kt

(2.11)

where Kt (t = 1; ; T ) are a sequence of feedback control matrices and kt (t = 1; :::; T ) represents what is known as the tracking gain in the control literature. These are solved for recursively by …rst solving the period T problem to obtain a solution for xT conditional on xT 1 . This is used to write a value function for period T which depends on xT 1 and which in turn forms part of the objective function for the period T 1 problem. Using this procedure, along with the terminal conditions HT = Q and kT = hT = QzTd we can solve for the sequence of feedback control matrices and tracking gains as: KT

=

(N + B 0 HT B)

1

(B 0 HT A)

kT

=

(N + B 0 HT B)

1

B 0 (HT CeT

HT

1

= Q + (A + BKT )0 HT (A + BKT )

hT

1

= kT

1

+ (A + BKT )0 (hT

hT

N xdT )

HT CeT + N xdT )

These are solved recursively to obtain the control rule. Note that the feedback gains, Ki , for i = 1; T , depend only on the (constant) matrices of the transition equation [5]

and the loss function. The feedback part of the control rule then feeds only o¤ the lagged state vector zt 1 . By contrast, the tracking gains ki , for i = 1; T , vary over time depending upon the current and future values of the exogenous variables and expectations in the vector e. This is the feedforward part of the control rule.

3

Incorporating Rational Expectations

We now consider the linearised model (3.1) and the optimal feedback rule (2.11) and solve it by derving a restriced VAR respresentation using the Anderson-Moore (1985) AIM algorithm. AIM is a rational expectations algorithm for computing the vector autoregressive reduced-form of the forward-looking linear structural model2 : v X

i= r

ei A

yt+i xt+i

= "t

(3.2)

p 1

ei ; for i = r; ::::; v represent the structural coe¢ cients of the where p = m + n and A model composed by (3.1) and (2.11) for di¤erent lags and lead of variables. We start rewriting the model in a more compact form: e t = "t AW

(3.3)

where A is the vector of structural coe¢ cients for di¤erent orders of leads and lags: e A

(p p)(r+v+1)

Wt

e r :::::::::; A e0 ; ::::::::::; A ev ] [A [wt

0 r ::::::wt ::::::::wt+v ]

(3.4)

(3.5)

and W collects the state vectors of the endogenous and exogenous variables w = [y; x] for various leads and lags:

3.1

Impulse Response Analysis

To derive the restricted Vector Autoregressive Representation of the structural model AIM constructs a matrix P which is a vp (v + r)p matrix that comprises a set of auxiliary conditions (Q) which are imposed to ensure the non-singularity of the 0 leading matrix Hv plus a set of stability conditions Vu on the forward-looking part of the system which provide the additional equations that close the system under saddle-path stability. 2

By combining the matrix coe¢ cients of the structural model (3.1) and of the optimal feedback P Ai Bi yt+i y x rule (2.11) we get vi= r = "t where KT;i and KT;i denotes the (constant) Kiy Kix xt+i coe¢ cients in the feedback rule for y and x respectively for various lags of the endogenous and exogenous variables. So in our general notation: Ai Bi ei = A Kiy Kix

[6]

z }| { P

vp

2

wt r Q 4 :::: 0 Vu wt+v (v+r)p

1

3

5=0

(3.6)

The matrix P can then be partitioned into left and a square right blocks so that P = (P1 j P2 ): P

Q 0 Vu

=

P1 vp rp

P2 vp vp

The conditioning equation can be rewritten as: 2 3 2 wt r wt P1 4 :::: 5 + P2 4 :::: wt 1 wt+v

1

(3.7)

3

5=0

(3.8)

where wt now depends on its past values through the coe¢ cient matrix P1 and on its future through the coe¢ cient matrix P2 : Given det(P2P1 ) 6= 0 and premultiplying by P2 1 yields the auto-regressive representation wt = ri=1 Gi wt i where G is given by the …rst p rows of P2 1 P1 : The reduced form VAR system can also be expressed as a …rst-order VAR model:

where

Wt = GWt 2 Gr 6 I 6 6 0 G 6 4 : 0

1

+ ut Gr 1 0 I : 0

(3.9)

3

::: G1 ::: 0 7 7 ::: 0 7 7 ::: : 5 ::: I

where G is r r: However we are interested in the e¤ect of the structural shocks "t in the structural representation: S0 Wt = SWt

1

+ "t

(3.10) 1

1

By comparing (3.1) and (3.10) we can see that ut S0 "t and G S0 S: Hence the …rst-order VAR (3.1) and (3.10) can be also written in the standard MA(1) form as: 1 1 1 X X X Wt = Gi u t i = Gi S0 1 " t i = Ct i "t i i=0

i=0

i=0

Therefore the responses of Wt to shocks in "t Ct i :

[7]

i

are determined by the rows of

4

An Application

In this section we use a nonlinear model of the UK economy (Arden et al, 2000) which in the version we use contains forward looking components in an uncovered interest parity condition for the exchange rate. The model comprises 94 nonlinear equations and 8 exogenous variables. The linearization was obtained by passing white noise, with a standard deviation set to the sample average, through both the short term interest rate and the (exogenised) exchange rate for 92 periods and storing the e¤ect of these stochastic perturbations on in‡ation; , and output growth, g. In order to smooth the use of the interest rate, r, as an instrument we also included as an endogenous variable, the …rst di¤erence of the interest rate, r. The vector 0 y in (2.4) is now y = ( ; g; eer; r) while x = r. We then estimated a distributed lag model of in‡ation and output growth on the interest rate and the exchange rate, eer, with …ve lags in in‡ation and output growth, and eight in the interest rate and the exchange rate3 . We make use of an uncovered interest parity condition that the expected change in the exchange rate is equal to the interest rate di¤erential between the domestic interest rate and the overseas interest rate, rw. We also want to allow for the possibility of independent shocks to in‡ation and output growth. So the vector 0 of exogenous variables, is now de…ned as e = (Et eert+1 ; rwt ; ;t ; g;t ), where the last two elements are designed to allow for shocks to in‡ation and output growth, and Et eert+1 is the expected exchange rate at time t. This means our linearization takes the structural form: A0 yt = A1 yt

1

+ ::: + As yt

5

+ D1 et + B1 xt + ::: + B8 xt

(3.1)

8

where: 2

1 6 0 A0 = 6 4 0 0 5 X

2

6 0 Ai = 6 4 0 i=1 0

i11

0 13 1 23 0 1 0 0

0

i13 i22

0 0

i23

0 0

3 2 0 0 0 6 0 0 0 7 7 ; D1 = 6 4 1 0 5 1 1 0 0 3 2 0 0 8 X 6 0 0 7 7; Ai = 6 4 0 0 5 i=6 0 0

1 0 0 0

3 0 1 7 7; 0 5 0

0 i13 0 i23 0 0 0 0

2

1

3

6 2 7 7 B1 = 6 4 1 5 1 3 0 0 7 7; 0 5 0

8 X

2

j1

6 j2 7 7 Bj = 6 4 0 5 j=2 0

and B2 (4) = 1. This produced a 39 1 state vector. Some insight can be gained into the linearisation method if we examine the relationship between in‡ation and output growth revealed by the persistent excitation of the interest rate. For example, a simple regression of in‡ation on output resulting from the perturbations to the interest rate gives: 3

The relatively high lag order was necessary to fully capture the delay between changes in interest rates and the exchange rate and their full e¤ects to be re‡ected in in‡ation and output growth.

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3

Effect on Inflation and Output of Stochastic Perturbations to Interest Rate 1.5 Output 1.0 0.5 0.0 -0.5 Inflation

-1.0 -1.5

98 99 00 01 02 03 04 05 06 07 08 09 10 DYR

DINFR

Figure 1: Stochastic Perturbations to Interest Rate

Effect on Inflation and Output of Stochastic Perturbations to Exchange Rate 1.0 Output 0.5

0.0

-0.5

-1.0

Inflation 98 99 00 01 02 03 04 05 06 07 08 09 10 DY

DINF

Figure 2: Stochastic Perturbations to Exchange Rate

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Dependent Variable: Method: Least Squares Variable Coe¢ cient t-Statistic C -0.008102 -0.559238 0.428850 4.693387 t 1 -0.544334 -4.835404 t 2 0.259059 2.250824 t 3 -0.704680 -6.439743 t 4 0.004484 0.073881 t 5 gt 1 0.226832 6.375036 gt 2 -0.095983 -2.328444 gt 3 0.098812 2.170162 gt 4 -0.033774 -0.728274 gt 5 0.106045 2.534038 gt 6 0.067147 2.078174 gt 7 0.032141 1.098118 gt 8 0.074717 2.538922 R-squared 0.766780 Durbin-Watson 2.305172 Note that the relationship is speci…ed in terms of the change in the in‡ation rate (this restriction is easily accepted: a Wald test gave a p-value of 0.65). This con…rms that the implicit trade o¤ between in‡ation and output growth is vertical, a property of the underlying nonlinear model. In Figures 1 and 2 we have plotted the paths for in‡ation and output as an outcome of one particular stochastic realisation. In the …rst …gure we show the e¤ect on in‡ation and output of just the realisation for the interest rate. In Figure 2 we show the e¤ect of just the realisation for the exchange rate. The Figures con…rm some of the stylised facts of in‡ation-output relationships. Output tends to lead, responding to interest rates …rst. The e¤ect on in‡ation comes through later. There is also considerable persistence in in‡ation. In Figure 2 we show the e¤ect of the exchange rate on output and in‡ation. Now the e¤ect on in‡ation and output is more or less immediate. Holding monetary policy constant, negative shocks to the exchange rate raise both output and in‡ation.

4.1

Impulse Response Analysis

In this section we turn to an analysis of the linearised model’s impulse responses using the AIM algorithm to solve for the saddlepath solution. For simplicity as we are now calibrating the model we set the exogenous world interest rate, rw, to 0:05. We also use in the calibration exercise the estimates of the coe¢ cients of the linearised equations of in‡ation and output and the interest rate optimal feedback 0 rule derived from the UK model. We denote our state vector as w = [ ; g; eer; r; r]. 0 Finally " = [" ; "g ; "eer ; "r ; 0] is the vector of shocks to the endogenous and exgenous variables.The results are shown in Figures 3 to 6. In all cases we are using a loss function with a weight of 100 on in‡ation, 10 on output growth and 5 on the change in [10]

the interest rate. Figure 3 shows the responses of in‡ation, output, the exchange rate and the interest rate to a unit shock to in‡ation. This supply shock simultaneously raises in‡ation and lowers output. The monetary policy response (given the loss function) is to raise interest rates. However, the increase in interest rates is not as large as the increase in in‡ation. this would appear to violate the so-called Taylor Principle that in order to stabilise the economy nominal interest rates need to be changed by more than the shock to in‡ation in order to alter real interest rates in the right direction. This does not happen here for two reasons. First, in the linearised model, nominal as well as real interest rates a¤ect output because the nominal interest rate is proxying for liquidity constraints. Secondly, the Taylor Principle has been derived for a closed economy. In the open economy case, the tightening of monetary policy induces an immediate upwards jump in the exchange rate which re-inforces the e¤ect of interest rates on both in‡ation and output so the change in nominal interest rates does not need to be more than proportional to the in‡ation shock. In Figure 4 we show the impulse responses to an output shock. A positive shock to output induces an increase in interest rates, and an appreciation in the exchange rate with the consequence that in‡ation actually falls slightly. In Figure 5 we show the responses to an interest rate shock. Given that the linearised model incorporates an optimal feedback rule, we have to interpret this shock as a form of policy mistake, inappropriately tightening monetary policy in response to mistaken readings of the economy. Again output falls and the exchange rate jumps. Once the ’policy mistake’ is recognised in the next period there is a small relaxation of policy in order to o¤set some of the consequences of the policy, so the exchange rate takes two periods to return to equilibrium, output returns to equilibrium next and …nally in‡ation. For the last shock - to the exchange rate - the impulse responses in Figure 6 show a similar pattern, monetary policy is eased to o¤set the e¤ect on in‡ation, output initially bene…ts from the lower interest rate, and eventually returns to equilibrium.

5

Conclusions

We have described a method for linearising and controlling a nonlinear rational expectations model using persistent excitation. This has the advantage that it is easy to implement with any standard modelling package capable of generating random shocks. Each instrument, exogenous variable and expectational term is excited with a white noise process and the e¤ects on the target variables is stored. Given superimposition, each input process is orthogonal so the reduced form can be estimate by OLS. Once the linear form is obtained and the optimal feedback rule computed by dynamic programming, we apply the rational expectations solution of Anderson and Moore which is particularly suited when the leading structural matrix is singular. We apply the method to a nonlinear model of the UK Economy and report a series of impulse responses for output, in‡ation, the exchange rate and the short term interest rate.

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1

x 10

5

-3

Output % Deviation

% Deviation

Inflation 0.5 0 -0.5

0

2

4

6

0 -5 -10 -15

8

0.6

0

2

4

Exchange Rate 1

% Deviation

0.4

% Deviation

8

1.5 Interest Rate

0.2 0 -0.2

6

0

2

4

6

0.5 0 -0.5

8

0

2

4

6

8

Figure 3: In‡ation Shock

2

x 10

-4

1 Output

0

% Deviation

% Deviation

Inflation

-2 -4 -6

0

2

4

6

0.5 0 -0.5 -1

8

0.02

10

0 x 10

2

4

0.01 0

0

2

4

6

8

Exchange Rate

% Deviation

% Deviation

Interest Rate

-0.01

6

-3

5 0 -5

8

0

Figure 4: Output Shock

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2

4

6

8

0.1

0.1 Output

0.05

% Deviation

% Deviation

Inflation

0 -0.05 -0.1

0

2

4

6

0.05 0 -0.05 -0.1

8

1

0

2

4

Exchange Rate % Deviation

% Deviation

8

1 Interest Rate

0.5 0 -0.5

6

0

2

4

6

0.5 0 -0.5

8

0

2

4

6

8

Figure 5: Interest Rate Shock

0.1

0.02 Output

0.05

% Deviation

% Deviation

Inflation

0 -0.05 -0.1

0

2

4

6

0.01 0 -0.01

8

0.02

0

2

4

Exchange Rate

0

% Deviation

% Deviation

8

1 Interest Rate

-0.02 -0.04 -0.06

6

0

2

4

6

0.5 0 -0.5

8

0

Figure 6: Exchange Rate Shock

[13]

2

4

6

8

References [1] Amman, H. and Kendrick, D. (1998). “Linear Quadratic Optimisation for Models with Rational Expectations”, Mimeo, University of Amsterdam. [2] Anderson, G. and G. Moore (1985). “A Linear Algebraic Procedure for Solving Linear Perfect Foresight Models”, Economics Letters, 17, 247-252. [3] Black, R, Laxton, D, Rose, D and Tetlow, R (1994). “The Steady-State Model: SSQPM, The Bank of Canada’s New Quarterly Projection Model: Part 1”, Bank of Canada Technical Report no. 72. [4] Blanchard, O.J. and Kahn, C.M. (1980). “The Solution of Linear Di¤erence Models under Rational Expectations”, Econometrica, 48, pp. 1305-1311. [5] Brayton, F and Tinsley, P (1996). “A guide to FRB/US: A Macroeconomic Model of the United States”, Federal Reserve Board Finance and Economics Discussion Series no. 1996-42. [6] Christodoulakis, N. (1989). “Extensions of Linearisation to Large Econometric Models with Rational Expectations”, System-Theoretic Methods in Economic Modelling II, S. Mittnik (eds), Elsevier. [7] Fair, R.C. (2003). “Optimal Control and Stochastic Simulation of Large Nonlinear Models with Rational Expectations”, Computational Economics, 21, 245-56. [8] Finan, F.S. and Tetlow, R. (1999). “Optimal Control of Large, forward-looking Models”, Board of Governors of the Federal Reserve System, October. [9] Hannan, E.J. (1971). “The Identi…cation Problem for Multiple Equation Systems with Moving Average Errors”, Econometrica, 39, 751-65. [10] Harrison, R., Nikolov, K., Quinn, M., Ramsey, G., Scott, A. and Thomas, R. (2005). “The Bank of England Quarterly Model”, Bank of England. [11] Hunt, B, Rose, D and Scott, A (2000). “The core model of the Reserve Bank of New Zealand’s Forecasting and Policy System”, Economic Modelling, 17(2), 247-74. [12] Kapetanios, G. , Pagan, A. and Scott, A. (2005). “Making A Match: Combining Theory And Evidence In Policy-Oriented Macroeconomic Modelling”, CAMA Working Papers 2005-01, Australian National University, Centre for Applied Macroeconomic Analysis. [13] King, R., Watson, M., (1998). “The Solution of Singular Linear Di¤erence Systems under Rational Expectations”, International Economic Review, 39, 10151026.

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[14] Klein, P., (2000). “Using the Generalised Schur Form to Solve Multivariate Linear Rational Expectations Models”, Journal of Economic Dynamics and Control, 24, 1405-1423. [15] Laxton, D. M., Isard, P., Faruqee, H., Prasad, E. S. and Turtleboom, B. (1998). “MULTIMOD Mark III: The Core Dynamic and Steady-State Models“, International Monetary Fund Occassional Paper no. 164. [16] Pesaran, H.M. and R. Smith (1995), “Estimating Long-Run Relationships from Dynamic Heterogeneous Panels”, Journal of Econometrics, 1995, 109, 79-113. [17] Schmitt-Grohè S. and M. Uribe (2004). “Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function”, Journal of Economic Dynamics and Control, 28, 755-775. [18] Sims, C.A. (2002a). “Solving Linear Rational Expectations Models”, Computational Economics, 20(1-2), 1-20. [19] Sims, C. A (2002b). “Second Order Accurate Solution of Discrete Time Dynamic Equilibrium Models”, Manuscript, Princeton University, Princeton, December. [20] Zarrop, M.B. (1979). “Optimal Experimental Design for Dynamic System Identi…cation”, New York: Springer-Verlag. [21] Zarrop, M. B., Holly, S., Rustem, B., and Westcott, J. H. (1979). “The Design of Economic Stabilisation Policies with Large Non-Linear Econometric Models: Two Possible Approaches” in Economic Modelling, edited by P. Ormerod, Heineman, London.

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The Linearisation and Optimal Control of Large Non ...

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