Matrix Polynomial Conditions for the Existence of Rational Expectations.

Transcription

1 Matrix Polynomial Conditions for the Existence of Rational Expectations. John Hunter Department of Economics and Finance Section, School of Social Science, Brunel University, Uxbridge, Middlesex, UB8 PH, United Kingdom. Christos Ioannidis School of Management, University of Bath, Bath, BA AY, United Kingdom. Abstract In this article we derive conditions that determine the existence of Rational Expectations under minimal conditions using a Generalized Bézout Theorem. We demonstrate that as long as the matrix polynomials derived from the model is regular, then a monic polynomial factor always exists and from this result we can derive a backward forward solution. For the existence of the Multivariate Rational Expectations solution, real roots are su cient, but not necessary as has previously been suggested in the literature and its existence can be established by the presence of a simple rank order condition. This less restrictive approach allows researchers to consider a wider class of models with rational expectations. Keywords: Generalized Bézout Theorem, Polynomial Divisor, Rank Condition, Rational Expectations JEL:C0, C51, C1 1

2 1 Introduction A number of recent articles have emphasized the importance of nding a unique forward solution to multivariate rational expectations (MRE) models, McCallum (00), Binder and Pesaran (1995), (199) and (000). Binder and Pesaran (199) show that a minimum state vector (MSV) solution to multivariate rational expectations models exists when the parameter matrices in the state space representation of the MRE commute or satisfy property P (Schneider, 1955). In this article we show that it is always possible to extract from the polynomial representation of the state space form a binomial function that divides the spectrum of the solution to the RE model into backward and forward components as long as the spectral representation of the model is de ned by a regular polynomial and the polynomial divisor is monic. The forward expectations exist when the remainder to the factorization is nilpotent and the Generalized Bézout Theorem (GBT) holds. The GBT allows us to obtain a factorization of the state space representation of the MRE without the requirement that the roots of the polynomial matrix are real, thus simplifying the e ort required to verify the internal coherence of a model under rational expectations. This article is organized as follows. Firstly the state space representation of the MRE model is presented following Broze and Szafarz (1991) and Binder and Pesaran (199). Second, based on Theorem 1 and, we state the conditions required for an MRE model to exist. Third, we obtain the backward forward representation of the MRE model via the application of the spectoral division to the polynomial distributed lag representation and nally we present our conclusions. Generalized Multivariate Expectations Models The following model with future and past expectations has been considered by Broze and Szafarz (1991), Broze, Gourieroux and Szafarz (1995), and Blinder and Pesaran (1995, 199): KX KX HX A 00 y t = A ko y t k + A kh E(y t+h k j t k ) + u t (1) k=1 k=1 h=1 where y t and u t are G vectors of decision and forcing variables. A kh ; k = 0; 1; :::K; h = 0; 1; :::H; are G G dimensioned matrices of xed coe cients and t represents a non-decreasing information set at time t; containing current and lagged values of y t and u t : t = fy t ; y t 1 ; :::; u t ; u t 1 ; :::g:binder and Pesaran (1995) show that (1) has a canonical form x t = Ax t 1 + BE(x t+1 j t ) + w t () where x t = (x 0 t; x 0 t 1; :::x 0 t K+1 ); x0 t = (yt; 0 E(yt+1j 0 t ); :::E(y 0 t+h j t)) and y t is a G vector of decision variables. The remaining elements in () are de ned

3 as A = D0 1 D 1; B = D0 1 D 1; w t = D0 1 # t ; # t = (# 0 t; 0 0 n; :::0 0 n) 0 ; # t = (u 0 t; 0 0 G ; :::00 G ; with n = (H +1)G. The vectors # )0 t and x t are both of dimension m 1, m = K(H + 1)G; u t is a G vector of forcing variables, 0 0 i is an i 1 vector of zeros for i = G; n; I j is a j j identity matrix for j = G; n; m and # t is of dimension n 1: The matrices D i for i = 1; 0; 1 are de ned as: D 1 = 1 0 n 0 n 0 n 0 n 0 n 5 ; D 0 = 0 1 K 1 0 n I n 0 n 5 ; D 1 = 0 n 0 n 0 n 0 n 0 n 0 n K I n 0 n 0 n 0 n 0 n 0 n I n 0 n 0 n 0 n I n 5 with k; k = 1; 0; 1; :::K, 1 = 0 G 0 G 0 G 0 G I G 0 G 0 G 0 G 5 ; 0 = I G A 01 A 0H 0 G I G 0 G 5 and i = 0 G 0 G I G 0 G A i0 A i1 A ih 0 G 0 G 0 G 5 for i = 1; :::K: 0 G 0 G I G 0 G 0 G 0 G This representation allows for the inclusion of both forward and lagged expectations (perceptions), as is the case in Gauthier (00). Re-writing equation () in polynomial distributed lag form we obtain P (L 1 )x t 1 = (BL I m L + A)x t 1 = w t () and the spectral representation of P (L 1 ) is given by a matrix-valued complex function: P (z) = (Bz I m z + A): () In the next section, we show that the MSV solution of () requires the factorization of this matrix polynomial (). Dividing the state polynomial of the RE model In this section we show that the z-transform of (1) can be factored by a monic matrix polynomial as long as () is a regular matrix polynomial (Gohberg et al (198)). When the GBT holds the factorization divides the spectra and this polynomial divisor decomposes the solution into two independent components.

4 The following formulation represents the division of any l th order matrix polynomial P (z) : P (z) = P o (z)p 1 (z) + R(z): If the divisor is monic or P 1l = I m ; then the factorization is unique and the following theorem applies: Theorem 1 If P (z) is an l th order regular matrix polynomial or det(p (z)) 0, then P 1 (z) is a monic right divisor of P (z) of order l < l: Proof. See Gohberg et al (198), p It follows from the rst order structure of () that P 1 (z) = (I m z C) is a binomial and as is shown below, the remainder R(z) is a constant matrix that is not time dependent. Therefore: Corollary 1 In the linear case where P 1 (z) = (I m z C) and: R(z) = P l C l + ::: + P 0 P 1l i = P l i+1 + P l i+ C + ::: + P l C i for i = 0; 1; ; ::l 1: Proof See Gohberg et al (198), p It follows from corollary 1 for the case considered here that R(z) = P l C l + ::: + P 0 = P (C) and the spectral division is exact when the GBT holds or P (C) = 0 (Gantmacher, 190). Theorem It follows from theorem 1 and Corollary 1 that in equation (): P (z) = P o (z)p 1 (z), P (C) = 0: Proof. Following Gantmacher (190) dividing P (z) by the binomial (I m z C) : P (z) = Bz(I m z C) + (BC I m )z + A = (Bz + (BC I m ))(I m z C) + (BC C + A) = P o (z)p 1 (z) + P (C) where P (C) = (BC C + A) and P o (z) = (Bz + (BC I m )) and it follows, that P (z) = P o (z)p 1 (z) when P (C) = 0: By application of the formulae in Corollary 1: P 00 = I m BC and P 01 = B: Therefore for Rational Expectations Model to exist, the matrix polynomial () must have a right side divisor that partitions the spectrum and from Theorem 1 and Corollary this is obtained when the following conditions hold:

5 det(p (z)) 0 B = 0 P (C) = 0: The above conditions imply, the regularity of the matrix polynomial, the existence of forward looking expectations and that the GBT holds. This approach, allows us to consider a wider class of models to those obtained when AB = BA or more generally to show that A and B must satisfy Property P. 1 Binder and Pesaran (1995) show that C is a solution to P (C) = 0 provided the following rank condition holds: rankf(j i ) 0 B J 0 i I m + I m Ag = m 1: This can be shown by the following argument. Let C have a canonical form SS 1 ; then: P (C) = BS S 1 SS 1 + A = 0 = BS S + AS = 0 where = diag[ 1 ::: n ] is a matrix containing roots of the dynamic system that underlies (), and comes from the solution to the eigenvalue problem ( i ) = det(b i I m i + A) = 0: 1 It follows from the application of Frobenius Theorem that when AB = BA the matrix P (C) = BC C + A = 0 has roots: = b + a = 0; where a; b and are eigen roots of the matrices A, B and C respectively (Binder and Pesaran, 1995). However, in (), AB = BA only when B = A and the model has the rst order form: y t = Ay t 1 + AE(y t+1 ji t) + u t: Otherwise, the GBT holds when P (C) = 0 and C BC = A or b = a. Or as is explained by Motzkin and Taussky (195), any matrix pair (C; B) with roots (; b ) is said to satisfy property P when F (C; B) has as its roots f(; b ): More generally, Schneider (1955) shows that property P follows for every pair of matrices (A i ; A j ) in an ordered matrix polynomial when Hence, C BC has roots b when (A i A j A j A i )R i = 0: (BC CB)R i = 0; so Property P will be satis ed when BC = CB or AB = BA (Binder and Pesaran, 1995). Excepting when a matrix pair commute, Property P is not useful without knowledge of R i : However, when the Jordan form C = SJ i S 1 satis es P (C) = 0; then = a and based on the partition of the spectrum a = b. Hence, conditions (ii)-(iv) required by Binder Pesaran (199) for an RE solution follow from the results presented here without the need to show that Property P holds. 5

6 Binder and Pesaran (1995) show that there are nitely many Jordan matrices J i for i = 1; ; :::l that solve (C i ) = det(bc i C i + A) = 0; where C i = SJ i S 1 for some non-singular matrix S. Therefore any matrix J i that solves (C i ) = 0 also satis es: BS i SJ i + AS = 0: (5) Vectorizing (5): ((J i ) 0 B J 0 i I m + I m A)vecS = 0: When rankf(j i )0 B J 0 i I m + I m Ag = m 1; then S is a non-singular matrix, and setting J i = ; by construction C is a solution to (C) = 0 that also solves: P (C) = BC C + A = 0: If rank((j i ) 0 B J 0 i I m + I m A) = m ; then P (C) = 0 has a unique solution when m = m 1; there are multiple solution when m < m 1 and the solution is trivial when m = 0: In conclusion, when m = m 1 and B = 0, Theorem 1 holds and a division of the spectrum of () exists. In the next section we will examine the implications of the factorization for the backward-forward solution to RE models. The backward-forward solution In deriving the backward-forward solution to the MRE, we wish to express all state variables in terms of the forcing variables. It follows from Theorem, that when P (C) = 0 : P (z) = P 0 (z)(i m z C): () Now mapping () back from the spectral dimension into the time domain: P (L 1 ) = P 0 (L 1 )(L 1 I m C) = (I m BC)F (L 1 )(L 1 I m C) where F (L 1 ) = (F L 1 I m ) and F = (BC I m )B: Re-normalizing P (L 1 ) to the t th time period, it follows from the spectral division that () decomposes into backward and forward looking components: P (L 1 )x t 1 = (I m BC)(F L 1 I m )(L 1 I m C)Lx t () Replacing P (L 1 )x t 1 by the rhs of (8): = (BC I m )(I m F L 1 )(I m CL)x t : (8) P (L 1 )x t 1 = (BC I m )(I m F L 1 )(I m CL)x t = w t ; (9)

7 where w t = D0 1 # t : If (I m BC) is non-singular, then inverting (I m F L 1 ), solves the explosive roots forward and this inversion gives rise to the usual convolution: x t Cx t 1 = 1X s=0 F s (I m BC) 1 D 1 0 E( # t+s j t ): (10) A requirement for the existence of the forward looking model is B = 0: Alternatively, when A = 0 we still can have forward looking behaviour, but there is no persistence. Therefore: P (C) = BC C = (BC I m )C = 0: This is solved, when either BC = I m or C = 0: If BC = I m then (I m BC) is by de nition singular and the problem is not well de ned, while for C = 0 the dynamic solution is purely forward looking: 1X x t = B s D0 1 E( # t+s j t ): (11) s=0 This encompasses the case examined by McCallum (00). 5 Conclusions We require for the existence of MRE models of the form, x t = Ax t 1 + BE(x t+1 j t )+w t ; a matrix C along with a Jordan matrix containing the stable roots of the system that solves the matrix polynomial P (C) = BC C+A = 0: From Theorem 1, when P (z) is a regular polynomial, P (C) = 0 and B = 0; then the spectrum of P (z) divides into components related to backward looking and forward looking behaviour. In the time domain the forward looking representation exists when BC I m has an inverse. The division of the spectrum of the matrix polynomial P (z) does not rely on the roots being real or that state matrices, A and B commute. While, in practice the proposition that A and B satisfy property P can be replaced by a more straightforward rank condition that holds when the conditions required for the division of the spectrum of P (z) are satis ed: The resultant models can be estimated using the recursive procedures developed by Binder and Pesaran (1995, 199 and 000), Dunne and Hunter (1998) and in Chapter of Burke and Hunter (005). References Binder, M. and M.H. Pesaran (1995) Multivariate Rational Expectations Models & Macroeconomic Modelling: A Review and Some New Results. In M.H. Pesaran & M. Wickens (eds), Handbook of Applied Econometrics : M acroeconomics, pp Oxford, Basil Blackwell. Without loss of generality, the forcing process can be decomposed into a white noise and a deterministic component.

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