Reduced rank regression in cointegrated models

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1 Journal of Econometrics 06 (2002) Reduced rank regression in cointegrated models.w. Anderson Department of Statistics, Stanford University, Stanford, CA , USA Received May 200; revised 30 May 200; accepted June 200 Abstract he coecient matrix of a cointegrated rst-order autoregression is estimated by reduced rank regression (RRR), depending on the larger canonical correlations and vectors of the rst dierence of the observed series and the lagged variables. In a suitable coordinate system the components of the least-squares (LS) estimator associated with the lagged nonstationary variables are of order =, where is the sample size, and are asymptotically functionals of a Brownian motion process; the components associated with the lagged stationary variables are of the order =2 and are asymptotically normal. he components of the RRR estimator associated with the stationary part are asymptotically the same as for the LS estimator. Some components of the RRR estimator associated with nonstationary regressors have zero error to order = and the other components have a more concentrated distribution than the correspondingcomponents of the LS estimator.? 2002 Elsevier Science S.A. All rights reserved. JEL classication: C32 Keywords: Nonstationary autoregressions; Least-squares estimators; Error-correction form; Asymptotic distributions. Introduction he type of cointegrated model treated in this paper is a nonstationary autoregressive process of which some linear functions are stationary. Many statistical procedures for the cointegrated models are procedures used for stationary autoregressive processes but the sampling behavior of the statistics el.: ; fax: address: twa@stat.stanford.edu (.W. Anderson) /02/$ - see front matter? 2002 Elsevier Science S.A. All rights reserved. PII: S (0)00095-

2 204.W. Anderson / Journalof Econometrics 06 (2002) reects the nonstationarity property. he linear autoregressive model of nite order is formally a kind of linear regression. he procedure studied in this paper is the reduced rank regression (RRR) estimator of the coecient matrix (Anderson, 95). his estimator is compared with the least-squares (LS) estimator of the matrix and the improvement in eciency is evaluated. he classical regression model is Y t = B t + Z t ; (.) where Z t is an unobserved random vector disturbance of p components with EZ t = 0; EZ t Z t = ZZ, and E t Z t = 0, and t is an observed random vector of q components. In this paper we specify E t = 0 in order to concentrate on the properties of the covariances; in practice sample covariances are calculated in terms of deviations from sample means. hen E t t = and EY t Y t = YY = B B + ZZ : (.2) Given a sample of observations (Y ; );:::;(Y ; ) the sample covariances are S YY = Y t Y t; S = t ; S Y = Y t t: (.3) he LS estimator of the coecient matrix B is ˆB = S Y S : (.4) Dene the residuals as Ẑ t = Y t ˆB t ;;:::;. he estimator of ZZ is SẐẐ = S YY ˆBS ˆB : (.5) In many problems the rank of B (or some submatrix of B) is specied to be k min(p; q). For example, the limited information maximum likelihood (LIML) estimator of the vector satisfying B = 0 in a simultaneous equation model requires the rank of B to be not greater than p (Anderson and Rubin, 949). he reduced rank regression estimator of B can be dened in terms of the solutions to the determinantal and vector equations ˆBS ˆB tsẑẑ =0; (.6) ˆBS ˆB f = t SẐẐ f; f SẐẐ f = : (.7) Let the solutions to (.6) be ordered t t p, and dene the vector f i as the solution to (.7) for t = t i. hen dene F =(f n+ ;:::;f p ); (.8) where n = p k. hen the RRR estimator of B is ˆB k = SẐẐ FF ˆB: (.9) Note that ˆB k is the product AC of the p k matrix A = SẐẐ F and the k p matrix C = F ˆB. A and C are normalized by A S Zˆ Zˆ A = I and CS C = =

3 .W. Anderson / Journalof Econometrics 06 (2002) diag(t n+ ;:::;t p ). his estimator was derived by Anderson (95) as the maximum likelihood estimator when Z ;:::;Z are normally independently distributed. Alternatively, the RRR estimator is ˆB k = S Y ˆ ˆ ; (.0) where ˆ =(ˆ n+ ;:::; ˆ p ) and ˆ i is the solution to S Y S YY S Y ˆ = r 2 S ˆ; ˆ S ˆ = (.) and ri 2 = t i =(+t i ) is the square of the ith canonical correlation between Y and. In the form (.0) the matrices S Y ˆ and ˆ are normalized by ˆ S ˆ = I and (S Y ˆ ) S YY (S ˆ Y )= ˆ S Y S YY S ˆ Y = ˆR 2 = diag(rn+ 2 ;:::;r2 p). he asymptotic distribution of ˆB k in this model has been given and discussed by Anderson (999). An autoregressive process Y t = BY t + Z t (.2) has the form of (.) with t replaced by Y t. he process is stationary if the roots of B I = 0 (.3) satisfy i. In this case (.2) can be solved repeatedly so that Y t = B s Z t s (.4) s=0 with covariance matrix YY = B s ZZ B s : (.5) s=0 he LS estimator is (.4) and the RRR estimator is (.9) and (.0) with the t replaced by Y t. In econometric models some components of t can be exogenous variables and some can be components of Y t [as noted by Anderson (95). he likelihood ratio criterion (LRC) for testingthe null hypothesis that the rank of B is k is n 2 loglrc = log( ri 2 ); (.6) i= when n = p k. he limitingdistribution of (.6) is 2 with (p k)(q k) degrees of freedom (Anderson, 95) when the null hypothesis is true. (q = p for the AR model.)

4 206.W. Anderson / Journalof Econometrics 06 (2002) Nonstationary models; cointegration 2.. he cointegrated rst-order autoregression he AR process is nonstationary if i for at least one i. A special case is B = I; then i =;i=;:::;p; and the model is Y t = Y t +Z t. For Y 0 = 0 we obtain Y t = t s=0 Z t s and EY t Y t = t ZZ : We say {Y t } is integrated of order ; in symbols {Y t } I(). Note that the rst dierence Y t = Y t Y t = Z t is integrated of order 0; {Y t } I(0). A model is cointegrated of order n if i =;i=;:::;n; and i ; i= n +;:::;p. he error-correction form of the model is Y t = Y t + Z t ; (2.) where = B I. hen n of the eigenvalues of are 0. Note that (2.) is of the form of (.) with Y t replaced by Y t and t replaced by Y t. he LS estimator of and the RRR estimator have the forms of those given in Section. Dene S Y;Y = S Y Y = Y t Y t; S Y; Y = Y t Y t ; Y t Y t : (2.2) he LS estimator of is ˆ = S Y; Y S Y Y : (2.3) o nd the RRR estimator we solve SY;Y S Y;Y S Y; Y r2 SY Y = 0 (2.4) for r 2 r2 p. hen ˆ i is the solution to SY;Y S Y;Y S Y; Y ˆ = r 2 SY Y ˆ; hen the RRR estimator of is ˆ S Y Y ˆ =: (2.5) ˆ k = S Y; Y ˆ ˆ : (2.6) 2.2. ransformation In order to study the behavior of ˆ and ˆ k we want to change coordinates so as to distinguish the nonstationary and stationary dimensions. We shall assume the followingcondition.

5 .W. Anderson / Journalof Econometrics 06 (2002) Condition A. here are n linearly independent solutions to! = 0; (2.7) where n is the multiplicity of = as a root of the characteristic equation B I =0. Let the solutions of (2.7) be assembled into the matrix =(! ;:::;! n ); then = 0 and the rank of is n. Note that (2.7) is equivalent to! B =!. his assumption implies that {Y t } is I(). Condition A implies that the rank of is k = p n, and there exists a p k matrix 2 such that 2 = 22 2; (2.8) 22 is nonsingular, and =( ; 2 ) is nonsingular. Dene [ [ [ [ t Y t Wt Z t t = = 2t ; W t = = 2Y t W 2t : (2.9) 2Z t hen ( t ; 2t ) constitutes a stationary process. hese results have been shown in Anderson (2000, 200b), see also Johansen (995). he transformed process t satises the autoregressive model t = t + W t ; (2.0) t = t + W t ; (2.) where = B( ) ; (2.2) [ 0 0 = ( ) = I = : (2.3) 0 22 In terms of subvectors the process { t } is generated by t = ;t + W t ; (2.4) 2t = 22 2;t + W 2t : (2.5) Let 0 = ; = = 0 and W 0 = W ; = = 0. hen t = t s=0 W ;t s constitutes a random walk, and { 2t } is a stationary process. 3. Asymptotic distribution of the LS estimator 3.. Asymptotic distribution of the sample covariances he LS estimator of in (2.) is ˆ = S ; ˆ S ; (3.)

6 208.W. Anderson / Journalof Econometrics 06 (2002) where S ; = S = t t; S ; = t t ; t t : (3.2) he deviation of the estimator from the process parameter is = = S W : (3.3) o study the behavior of this statistic we need to distinguish the random walk dimensions from the stationary process dimensions. When t is partitioned into subvectors of n and k components t =( t ; 2t ), respectively, we may use the notation [ S S W =(S W ; S 2 W ) W S 2 W : (3.4) S S 2 W S 22 W Since { t } is a random walk, we need to dene a Brownian motion in order to describe the asymptotic distribution of some covariance submatrices. Consider a sequence of random vectors {W t } with EW t = 0 and EW t W t = WW = ZZ. Dene (u) as the weak limit of [u W t = u [u W t ; 0 6 u 6 : (3.5) u For xed u the limitingdistribution of (3.5) is N(0;u WW ) and increments are independent. We partition the vector (u) into n and k components as (u)=( (u); 2 (u)) = (u). hen S = 2 ;t ;t = d 0 t 2 t 2 W ;t r r=0 s=0 W ;t s (u) (u)du = I : (3.6) See Billingsley (968) or Johansen (995), for example. he second subvector 2t = 2 2;t + W t ;t=:::; ; 0; ;::: generates a stationary process 2t = s s=0 2W 2;t s. Hence S 22 = 2;t p 2;t 22 s = WW s=0 s 22: (3.7)

7 S ; =.W. Anderson / Journalof Econometrics 06 (2002) Before treating S 2 we consider [ S ; S 2 [ ; We have S W = d 0 S 2 W = S 2 ; S 22 ; W t ;t = d(u) (u)=j = W t 2;t S 2 [ J J 2 22 S 22 + [ S W S 2 W S 2 W S 22 W : (3.8) ; (3.9) p 2 W = 0: (3.0) Since {W t ; 2t } is stationary d S 2 W = S 2 W N(0; ): (3.) We use summation by parts to evaluate S 2 ; = 2t ;t Note that [ = 2 [ = 2 2t t ( 2 2;t + W 2t )W t p 2 WW : (3.2) tr( 2 )( 2 ) =( 3=2 )(tr =2 2 2 ) p 0: (3.3) hen from (3.8), 22 = 0, and S 2 S 2 = 22 (S 2 ; S2 W p W J 2, we conclude that ) (3.4) d 22 (J WW ): (3.5)

8 20.W. Anderson / Journalof Econometrics 06 (2002) p Hence =2 S 2 0. We summarize as S S 2 [ d I 0 : (3.6) S S 22 More details are given in Anderson (2000) Asymptotic distribution of the LS estimator It will be convenient to use the notation vec A =(a ;:::;a p) for A = (a ;:::;a p ). A useful relation is vec ABC =(C A)vec B. heorem. Suppose W ; W 2 ;::: are independently distributed with EW t = 0; EW t W t = WW ; and EW t t s = 0; s=; 2;::: : hen [ ( ˆ ); 2 2 p 0; (3.7) and [ ( ˆ ); ( ˆ 2 2 ) d [J : I d W ; S 2 W (22 ) (3.8) vec S 2 N(0; 22 WW ): (3.9) Proof. o demonstrate (3.7) write [ [ ( ˆ ); ˆ 2 2 = S W ; S 2 S W S 2 p p S 2 S 22 (3.20) and use =2S W 0; S 2 W 0, and (3.6). he limit in distribution (3.8) follows from (3.20), (3.9), (3.0), and (3.6). he limit in distribution (3.9) follows from vec S 2 W = =2 ( 2;t W t ) and (3.). he conditions on {W t } could be weakened. he main point here is that W t does not have to be normal. o convert (3.8) to the original terms of Y t and Z t we dene S j Z = Z t j;t. [ hen ˆB B = ( ) J I + S 2 Z (22 ) 2 +[o( );o( 2 ): (3.2)

9 .W. Anderson / Journalof Econometrics 06 (2002) his result is interpreted to mean that the discrepancy ˆB B in the random walk direction multiplied by is approximately ( ) J I and in the stationary direction multiplied by is approximately normal with covariances given by elements of ( 22 ) ZZ =( 2 YY 2 ) ZZ. From (3.2) we derive ( ˆB B) d S 2 Z (22 ) 2 = S ZY 2( 2 YY 2 ) 2: (3.22) 4. Asymptotic distribution of the RRR estimator In the -coordinate system the RRR estimator of is ˆ k = S ; G 2 G 2; (4.) where G 2 =(g n+ ;:::;g p ); g i satises S; S ; S ; g = r 2 S g; g S g = (4.2) for r = r i ; and r i (r r p ) satises S; S ; S ; r2 S =0: (4.3) heorem 2. Under the conditions of heorem ( ˆ k ) ( ˆ 2 [ k 2 ) ( ˆ 2 d 0 S 2 W (22 ) k 2 ) ( ˆ 22 k 22 ) J 2 I S 22 W (22 ) (4.4) and J 2 = J 2 2 WW ( WW ) J = d[ 2 (u) 2 WW ( WW ) (u) (u): (4.5) 0 Proof. he equations (4.2) for g = g n+ ;:::;g p are summarized in QG 2 = S G 2 ˆR 2 2; G 2S G 2 = I; (4.6)

10 22.W. Anderson / Journalof Econometrics 06 (2002) where Q = S; S ; S ; and ˆR 2 = diag( n+ ;:::; n ). Let G 2 = (G 2 ; G 22 ). hen the Eqs. (4.6) can be written as ( ) Q G 2 + Q 2 G 22 = S G 2 + S 22 G 22 ˆR 2 2; (4.7) ( ) Q 2G 2 + Q 22 G 22 = S2 G 2 + S 22 G 22 ˆR 2 2; (4.8) [ (G 2; G 22) 2 S S2 G2 = I: (4.9) S2 S 22 Dene H by G 2 = HG 22. Since p Q 0, p Q 2 0, and 2 S p 0, (4.7), (4.8), and (4.9) are asymptotically equivalent to Q 2 G 22 =(I H + S 2 )G ˆR ; (4.0) G 22 Q 22 G 22 = S 22 G 22 ˆR 2 2; (4.) G 22S 22 G 22 = I: (4.2) We can solve (4.0) for H = I (Q 2G 22 ˆR 2 2 G 22 S2 ) = I (Q 2Q 22 S22 S2 ): (4.3) From [ J 0 d S ; 2 WW (4.4) and [ p WW 2 WW S ; ; = ; (4.5) 2 WW 22 WW we calculate [ [ Q2 d [J ( ; ) 2 2 WW ( ; ) Q ( ; ) : (4.6) Hence Q 2 Q 22 S22 d [J ( ; ) 2( ; ) 22 2 WW ( 22) and = [J ( WW ) 2 WW 2 WW ( 22) (4.7) H d I J 2 ( 22) ; (4.8)

11 .W. Anderson / Journalof Econometrics 06 (2002) where J 2 is given by (4.5). Now we calculate ˆ k = S ; G 2 G 2, where [ [ [ G2 0 G 2 = = + HG 22 ; (4.9) S ; = G 22 [ hen ˆ k is [ 0 0 S ; G 2 G 2 = WW 2 22 G 22 [ + S W S 2 ; + 2 WW 0 S 2 W (22 ) G 22H 2 WW HG S 2 W ( 22 S 22 + S 22 W ) {S 22 W (22 ) : (4.20) + 2 [S 22 (22 ) G } [ 0 o p ( =2 ) + : (4.2) o p ( ) o p ( =2 ) From (4.2) we have I = ( 22S G22 + G ) implying +o p ( =2 ); (4.22) 0 =( 22 ) S 22 (22 ) + G G 22 + o p (): (4.23) When (4.23) is used in (4.2), we obtain [ 0 S S ; G 2 G 2 = + W (22 ) H S 22 W (22 ) 0 o + p ( 2 ) : (4.24) o p ( ) o p ( 2 ) his gives us (4.4).

12 24.W. Anderson / Journalof Econometrics 06 (2002) Discussion Let us compare the asymptotic distributions of the LS and RRR estimators. he asymptotic distribution of the LS estimator can be expressed as [ [ ( ˆ ) ( ˆ 2 2 ) d J I S 2 W (22 ) : (5.) ( ˆ 2 2 ) ( ˆ ) J 2 I S 22 W (22 ) he limitingdistribution of ( ˆ ) is the distribution of J I, while the limitingdistribution of ( ˆ k )is0. he limitingdistribution of ( ˆ 2 2 ) is that of J 2 I while the limitingdistribution of ( ˆ 2 k 2 ) is J 2 I. he distribution of J 2 is more concentrated around 0 than that of J 2. For each method the estimator of is superecient. In the original coordinate system we have ˆB k B = [ 0 ( ) and [ 2 WW ( WW ) ; IJ I + S Z (22 ) 2 +[o( );o( =2 ) (5.2) ( ˆB k B) d S Z (22 ) 2 = S ZY 2 ( 2 YY 2 ) 2: (5.3) Note that the limitingdistribution of ( ˆB k B) is the same as the limiting distribution of ( ˆB B), but the terms of higher order are dierent. he asymptotic distributions developed here do not require normality of the disturbances. In this regard the RRR estimator does not require stronger conditions than the LS estimator. his is in contrast to the fact that the asymptotic distribution of G 2, a factor of ˆ k, does depend on normality of the disturbances. Johansen (995) gave (5.3) in heorem 3:7, but he did not evaluate the term of order = in (5.2). In the usual regression model (.) the two matrices ( ˆ k ) and ( ˆ ˆ ) agree entirely except for the upper left-hand corner of ( ˆ k k) being 0 (Anderson, 999). However, in the usual regression model the transformation to canonical variables U = A Y, V = yields an error vector W = A Z = U V with uncorrelated components whereas here the transformation t = Y t, t = Y t yields an error vector W = Z t = t t in which 2 WW may not be 0.

13 .W. Anderson / Journalof Econometrics 06 (2002) Higher-order processes Now we consider an AR(m) process m Y t = B j Y t j + Z t : (6.) j= If the roots ;:::; pm of m I m B B m = 0 (6.2) satisfy j, j =;:::;pm, (6.) denes a stationary process. If j = for one or more values of j, the process is nonstationary. he model (6.) can be put in an error-correction form m Y t = Y t + j Y t j + Z t ; (6.3) j= where = m j= B j I and i = m j=i+ B j, i =;:::;m. Let the multiplicity of the root j = be n, and dene k = p n, so = = n = and j, j = n +;:::;pm. Assume Condition A. hen Y t and 2Y t can be given initial distributions so that (Y t ; 2Y t ) is stationary (Anderson, 2000), and the rank of is k. See also Johansen (995). he model (6.3) is of the form Y = A V + A 2 V 2 + Z with Y replaced by Y t, A by, V by Y t, A 2 by ( ;:::; m )= and V 2 by (Y t ;:::;Y t m+ ) = Y t. hen the parameters can be estimated by reduced rank regression (Anderson, 95). In the denitions of the LS estimator of (2.3) and the RRR estimator (2.6) Y t and Y t are replaced by Y t S Y; Y S Y Y; Y t ; (6.4) Y t SY; Y S Y Y; Y t : (6.5) he asymptotic distribution of the canonical correlations and vectors has been given by Anderson (200a) and of the correlations by Hansen and Johansen (999). References Anderson,.W., 95. Estimatinglinear restrictions on regression coecients for multivariate normal distributions. Annals of Mathematical Statistics 22, [Correction, Annals of Statistics 8, 980, p Anderson,.W., 999. Asymptotic distribution of the reduced rank regression estimator under general conditions. Annals of Statistics 70, 29. Anderson,.W., he asymptotic distribution of canonical correlations and variates in cointegrated models. Proceedings of the National Academy of Sciences 97,

14 26.W. Anderson / Journalof Econometrics 06 (2002) Anderson,.W., 200a. he asymptotic distribution of canonical correlations and variates in higher-order cointegrated models. Proceedings of the National Academy of Sciences 98, Anderson,.W., 200b. A condition for cointegration, unpublished. Anderson,.W., Rubin, H., 949. Estimation of the parameters of a single equation in a complete system of stochastic equations. Annals of Mathematical Statistics 20, Billingsley, P., 968. Convergence of Probability Measures. Wiley, New York. Hansen, H., Johansen, S., 999. Some tests for parameter constancy in cointegrated VAR-models. Econometrics Journal 3, Johansen, S., 995. Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press, Oxford.