Large sample inference for a multivariate linear model with autocorrelated errors

Journal of Statistical North-Holland

Planning

and Inference

187

41 (1994) 187-204

Large sample inference for a multivariate linear model with autocorrelated errors S. Sethuraman Augusta College

I.V. Basawa Departmenr of Statistics, University of Georgia, 204 Statistics Building, Athens, GA 30602-19.52, USA Received

27 July 1992; revised manuscript

received

22 September

1993

Abstract We consider a multivariate linear model with autocorrelated errors. The mean vector of the process is assumed to be linear in the time-trend parameter /I and the within-group variation parameter y. The least-squares estimators of b and y, and the related estimators of the autoregressive parameter 0 and the error covariance matrix C are derived and their asymptotic distributions are obtained. Large sample tests of Hi: y=O and Hz: /3=0 are derived and the limit distributions of the restricted least-squares estimators 8”, and jH1 are obtained under H, and H,, respectively. AMS Subject Classification: Primary

62M 10.

Key words: Multivariate linear model; growth asymptotic tests; limit distributions.

curves; autoregressive

processes;

least-squares

estimation;

1. Introduction Several authors have considered the problem of parameter estimation when several independent realizations of a time series are available. Anderson (1978) discussed some problems of inference based on independent realizations from autoregressive processes. More recently, Basawa et al. (1984) and Basawa and Billard (1989) considered the problems of parameter estimation and tests of homogeneity for regression models with multiple observations and autocorrelated errors. Kim and Basawa (1992) have discussed empirical Bayes estimation based on data from several independent autoregressive processes. Hwang and Basawa (1992) studied estimation and the local asymptotic normality for nonlinear autoregressive processes with multiple observations. In all the references cited above, the multiple observations at each time point are Correspondence to: IV. Basawa, Athens, GA 30602-1952, USA.

Department

0378-3758/94/$07.00 0 1994-Elsevier SSDI 0378-3758(93)E0087-W

of Statistics,

University

of Georgia,

Science B.V. All rights reserved

204 Statistics

Building,

188

S. Sethuraman, I. V. Basawa / Multivariate

linear model with autocorrelated

errors

assumed to be independent. However, in some situations, the multiple observations at a time point may not be independent. For instance, crop yields from units in the same plot observed over several time points would generally be correlated at each time point as well exhibiting time correlation. Responses of individuals belonging to the same group tend to be correlated within the group as well as along the time axis. In these situations, it would be desirable to consider a multivariate model which permits correlation within each block or group and also incorporates dependence between the observation vectors at different time points. Consider the model defined by

Xt=Lx1 a+&+

yt

(1.1)

Yt= i

t=l,...,m,

(1.2)

and OiYt_i+&,,

i=l

where X, is an (n x 1) vector of observations on n individuals at time t, l,, 1 denotes a vector of ones, C,=(C,i, . . . . C,,) denotes a (1 x q) vector of nonrandom regression coefficients in the time-trend component, /3 is a (q x 1) vector of unknown parameters, B is an (n x r) matrix of nonrandom covariates corresponding to the n individuals, y is a (r x 1) vector of unknown parameters and Y, is an (n x 1) vector of unobserved errors. In the multivariate Gauss-Markov model (see e.g. Rao, 1973, pp. 543-544), one typically assumes that the error vectors {Y,}, t = 1, . . . . m, are independent and identically distributed. Here we relax this assumption and suppose instead that ( Yt} are autocorrelated. The time dependence between (Y,}, t = 1,2, . . . , is represented by (1.2). Note that (0,}, k= 1, . . . . p, in (1.2), are unknown (scalar) parameters and {st} is a sequence of (n x 1) vector random errors which are assumed to be independent and identically distributed with mean zero and covariance matrix Z. The model specified by (1.1) and (1.2) can be useful as a growth curue (or response surface) model for the situation when a group of n individuals is monitored over several time points and observations on a characteristic of interest are recorded for these individuals. The first term on the right-hand side of (1.1) represents the time trend, and the trend component is assumed to be the same for all individuals in the group. The second term on the right-hand side of (1.1) is a covariate adjustment representing individual variations and the covariates are assumed to be free of time. The difference equation in (1.2) implies that the observations on each individual over time follow a pth-order autoregressive process. In most standard growth curve (or response surface) models, the error vectors are assumed to be normal. In this paper, we shall not impose the normality assumption. Consequently, our model in (1.1) and (1.2) generalizes the usual multivariate linear model in two directions: (i) the error vectors are assumed to be autocorrelated rather than independent and (ii) the error vectors are not necessarily normal and no specific distributional assumptions are made. The univariate version (i.e. n= 1) of (1.1) and (1.2) with y =0 is discussed, for instance, by

S. Sethuraman, I.V. Basawal

Multivariate

linear model with autocorrelated

errors

189

Fuller (1976) and it is a standard model in the econometrics literature. We remark here that the extension to the multivariate case involves estimation of C, and the related asymptotics becomes more complicated and nonroutine. As a further motivation for the assumption that the errors might be correlated over individuals, one can imagine a situation where the sample units (individuals) are chosen, for instance, according to a block design. The covariance matrix Z will then be a block diagonal perhaps allowing a constant intraclass correlation within blocks. The latter special case is discussed by Sethuraman and Basawa (1993) in a different context. One may imagine a more general and unspecified C in a situation where spatially correlated individuals are observed over a number of time points. If sufficient information is not available for modeling the spatial covariances, one may simply view C as completely unspecified. In this paper, we consider the general situation when C is unspecified. The model can be generalized in several ways. For instance, a random effects component may be added in (1.1) and a moving average component can be introduced in (1.2). Another possibility is to consider a random coefficient autoregressive model replacing the fixed coefficient model in (1.2). Furthermore, a general multivariate autoregressive moving average process can be considered to model { Yt}. The model in (1.2) is an example of a multivariate autoregressive process in its simplest form involving only p parameters. A general multivariate autoregressive process would involve many more parameters; see, for instance, Brockwell and Davis (1991). In this paper, we limit ourselves to the basic model in (1.1) and (1.2). Our first goal is to estimate c(=(pT, yT)T treating 8=(0,, . . . , 8JT and C as unknown nuisance parameters. The main result on estimation is contained in Theorem 2.5. Section 2 is concerned with the least-squares estimators and their limit distributions as m-+ co. In Section 3, we derive Wald-type test statistics for testing the hypotheses H,: y =0 (no covariate effect) and H,: j = 0 (no time tend). Section 4 is concerned with the limit distributions of the restricted least-squares estimators bH1and yH2under H, and HZ, respectively.

2. The limit distributions of the least-squares estimators 2.1. The basic model The model in (1.1) and (1.2) can be rewritten compactly as x*=/t?/+

r*,

(2.1)

where X* =(X 1r . . . . X,) is an (n x m) matrix of observations, A =(A,, . . . . A,) is an (n x mk) matrix of regression coefficients with A, = (1, x 1C,, B) being of order (n x k), k = q + r, r] is an (mk x WI)matrix of parameters defined as a block diagonal matrix with all the diagonal elements equal to the (k x 1) matrix tx=(p’, yT)T, and finally,

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S. Sethuraman, I. V. Basawa /Multivariate linear model with autocorrelated errors

y*=(Y,,

. . . . Y,) is an (n x m) matrix of errors with E(vec Y*)=O

and

Cov(vec Y*)=I’@Z,

(2.2)

where vet(M) denotes the vector obtained by stacking the columns of the matrix M, r is an (m x m) stationary covariance matrix of ( VI, . , . , U,) and (U,} follows a stationary scalar pth-order autoregressive process, viz., U,= f

6, Utmk+et,

t=O, fl,

f2 ,...,

(2.3)

k=l

and {e,} is a sequence of independent and identically distributed random variables with mean zero and variance 1; see, for instance, Brockweil and Davis (1991) for the derivation of the elements of r. Finally, C is the within-group (n x n) covariance matrix defined in Section 1, i.e. Z=Cov(eJ, and 0 in (2.2) denotes the direct product. Note that in the standard linear multivariate model, f in (2.2) is replaced by the identity matrix 2. In order to proceed further, it is convenient to use the vet notation and rewrite (2.1) as follows: vec(X*)=@cr+vec(Y*). From now on, we shall denote vec(X*) and vec(Y*) by simply X and Y, respectively, and write the model as x=LDcr+ Y,

(2.4)

with EY=O,

cov Y=T@C.

The matrix @ is of order (mn x k) and is defined by 6 =(L

&),

with &=COlnxl

and

$2=l,,10B,

where C=((C$i)),

t=l,

. . ..W&

i=l, ..*,q,

is an (m x 4) matrix and B=((b,j))

S=l,...,

Iz,

is an (n x Y) matrix. The parameter

j=l,.,.,

r,

vector a==(flT,yT)= is of order (k x l), where

k=q+r.

2.2. The generalized least-squares estimator of a when Z and r are known The generalized least-squares estimator of a is given by &G=(@TF-‘@)-‘@TF-‘X,

(2.5)

S. Sethuraman, 1.V. Basawal M&variate where

F = r @ C. It is assumed

exists. If F is known,

linear model with autocorrelated errors

that C and r are both positive

&o is unbiased

definite

191

so that F- ’

for c( with

COV(&)=(@~F-~@)-~.

(2.6)

We first derive the limit distribution of So assuming F is known. It will be shown later (Theorem 2.5) that when F is unknown it can be replaced by a consistent estimator without altering the limit distribution. Consider the following regularity conditions: (Cl). If 8(z)=1-e1z-02z2--..-eep zp, suppose Q(z) # 0 for all z such that \zI < 1. (C2) As m-+co, and for all i, j= 1, . . . . k,

(ii) and

for each h = 0, 1,2, . . , where CB=C,i

for i=l,...,q,

and unity for i = q + 1, . . . , q + r = k. Remarks. The condition (Cl) is used to ensure the existence of the stationary solution of (2.3), and the conditions in (C2) are standard in stabilizing time-trend coefficients (C,i); see, for instance, Fuller (1976). Note that, for 16 i
($,Cti)

(free from h and j),

( zl Ca)-1’2m-1’2+ai

and for q + 1 d i, j d q + r, (C2)(iii) simplifies

{~$~G+icF+h,j}

to

{($lc22)(~lc~2)~1'2+1.

Let D, =diag {(EYE1 C$1/2, i = 1, . . , q; Jiii, . . . , fi} matrix. We can now state the following theorem.

denotes

the (k x k) norming

Theorem 2.1. Under conditions (Cl) and (C2), as m-too, D,,,(& - LY)-f+Nk(O, G- ‘),

(2.7)

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S. Sethuraman,

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linear model with autocorrelated

errors

where G is a (k x k) positive-definite matrix dejined as

G= lim (D,‘@TF-l@D~l}. m-+cJZ

(2.8)

(The existence of the limit G is assured by (Cl) and (C2).)

The proof will be given later at the end of Section 2.3 in order to avoid unnecessary duplication. 2.3. The simple least-squares estimator of CI Note that Bo in (2.5) depends on Z and r. When C and r are unknown, we need to estimate them from the data. The estimates of C and r will be based on the residuals, and the residuals, in turn, involve CI.For this reason, one needs a preliminary estimate of CIwhich does not involve C and r and which is, nevertheless, D, consistent for cc. The simple least-squares estimator BOof tx,obtained by minimizing (X - @u)~(X- @a), is given by 60 = (@i’@)_‘QTX. In this section we shall derive the limit distribution D,-consistent.

(2.9) of 6, and show that it is

Theorem 2.2. Under conditions (Cl) and (C2), we have, as m-+a,

D,@,-c()5 N,(O, A-‘EA-‘),

(2.10)

where the matrices A and E are defined in the proof: Proof. We have D,(BO-a)=D,(@T@)-lDmD;l@TY.

(2.11)

Now consider AT(D; ‘QTY) where A=(E,i, . . . . AJT is a vector of real numbers:

(2.12)

S. Sethuraman, I.V. Basawaj Multivariate linear

for i=l,...,q and unity for i=q+l,...,q+r . . ..q+r. We now have i=l, . ..) q and b2=b,i for i=q+l, where

193

model with autocorrelated errors

and b,*iis unity for

Cz=Cti

(2.13) where Z*i(n)= f

b,*iYt,.

s=l

From (2.13), and Theorem 9.1.1 of Fuller (1976, p. 36), we have, as m-+co, (2.14)

f h=-w

where

Note that Cov(Z~,,(n), z,+,,j(n))=

i b,ibsjo,,+ i S=l

f

b,ib,,joss’ y(h),

i

s=ls’=l SZS’

where ass and ass, are the (s, s)th and (s, s’)th elements of C, and y(h)=Cov(U,, {U,} being the pth-order autoregressive process defined in (2.3). Note that Cov(Y1,, Yl +h,

3=y(h)~,

From (2.14) and the Cramer-Wold

and

Cov(Yls,

YI

+h,

U,+J,

,~)=r(hb,~.

device, we have

D, ’ QTY : Nk(O, E),

(2.15)

where the (i, j)th element of E is given by

Now, using condition (C2) we have 0,

‘QT@D,1+A=((aij(O))).

(2.16)

From (2.1 l), (2.15), (2.16) and the Slutsky’s theorem, we finally obtain D,(B,,-&Nk(O,A-‘EA-‘).

Remark.

0

Theorem 2.1 implies that B,, is D,-consistent,

i.e. BO-a=O,(D,

‘).

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Multivariate

linear model with autocorrelated errors

Proof of Theorem 2.1. We have D,(Bo_tl)=(D,(~TF_1Qi)_lD,)

(D,i@‘F-‘Y).

(2.17)

Writing Yr = F - i Y, we have ITD,‘@-‘F-‘Y=ATD,l@TY1

:N(O,

(2.18)

iTGA),

following the same arguments as in the proof of Theorem 2.1, and G is as defined in (2.8). From (2.18) and the Cramer-Wold device, we have D, ‘QTF - ’ Y : Nk(O,G).

(2.19)

From (2.17) (2.19), (2.8) and Slutsky’s theorem, we finally obtain Theorem 2.1.

0

Remark. The proofs of Theorems 2.1 and 2.2 are multivariate analogues of Theorem 9.1.1 of Fuller (1976); also, see Hannan (1970), Theorem 10, and related results in Chapter VII for a more general treatment. 2.4. Estimation

of r

Recall that r is the covariance matrix of ( U1, . . . , U,) where {U,} is the pth-order scalar autoregressive process defined in (2.3). The elements of r are known functions of the autoregressive parameter 0 = (f?,, . . , Bp)T.It therefore suffices to estimate 8. We first suppose that a =(pT, yT)T is known, and use Y,= X,- 1, x 1 C,p -By, t = 1, . . . , m, as the ‘observed’ sample. A simple least-squares estimator 8 of 6’is then obtained by minimizing C,“=1(Y, - Cf= lei Y, - i)T (Y, - Cf= lei Y, - i). We have (2.20)

6=B;;A,,,

is a (px where A,, = (CT=I Y~Y,_i, i=l,...,p)T i,j=l , . . . , p)) is a (p x p) matrix. We then have Jm(&e)=(d~,,)-l(~2-1W

1)

Theorem 2.3. Under condition (Cl), and as m-co,

where

(6

e) :

N,(O, TV,

vector, and B,“=((Cy=,YT-iYt-j,

(2.21)

mn) 3

where C,, =(C,“=,&~Y~_i,i= 1, . . . . p)’ is a (p X

Jm

1)

‘),

vector. we have

(2.22)

S. Sethuraman, 1.V. Basawal

linear model with autocorrelated

matrix of (U 1, . . . , Up), (U,} being the pth-order

and I’, is the covariance ressive process

Multivariate

195

errors

scalar autoreg-

defined in (2.3), and ((oSSS)),s, s’ = 1, . . . , n, are the elements of C.

Proof. We will show that m -lB,,s

(2.23)

and m-1’2C,,AN

0,

P

i

c;+$&

s=1

cc

rp 1 1 SfS’

(2.24)

.

)

)

Then, from (2.21), (2.23), (2.24) and Slutsky’s theorem, the result in (2.22) will follow. Condition (Cl) ensures that the process {Y,}, f =O, + 1, f2, . . . . is stationary and ergodic. The (i, j)th element of m-‘B,, is m-‘~$lY~_iY~_j~E(Y~ by the ergodic

theorem.

Y*= f

Yi-j),

The stationary

(2.25)

solution

of (1.2) is given by

Yi&*_i,

(2.26)

i=O where {Yi} are certain functions of B, with Y,= Davis (1991). From (2.26), we have Cov(Y,)=

EY,=O,

1; see, for instance,

and

Cov(Y,,

Brockwell

and

Yt+J=

Consequently E(Y;f K-j)=(

$oYIYl+h)tr(Z) (2.27)

=y(h)tr(C), where

h=(i-j)

and

y(h)= E(U, U,+h)=(~l~OYIYl+,,).

Therefore,

(2.23) follows.

In

order to verify (2.24), we first note that for each i = 1, . . . , p, (I;= 1ETY,_i}, m = 1,2, . . . , is a zero mean martingale. Using the Cramer-Wold device and the central limit theorem for martingales (Billinglsley, 1961, the result in (2.24) follows readily. When

01is unknown,

we replace

CIby B. and define

Y:=X,-11,.1Ctfio-B90. The least-squares given by

estimator

B=B*-~ A* Inn Inn,

of 8 when c1 is unknown

will be based on {Y:> and is

(2.28)

196

S. Sethuraman, I. V. Basawal Multivariate linear model with autocorrelated errors

where AZ,, B$, are defined in the same way as A,, and B,, with { Yt}replaced by { r;*}. Now, observe that Y* = Y- @(I?,- a), where Y*=(YFT, . . . . YzT)= is an (mn) x 1 vector, etc. Using similar arguments as those in the proof of Theorem 9.3.1 of Fuller (1976), we can show that, as m+co, 5

r:=yt*_1=

5

1=1

YTY,_l+O,(l).

t=1

and f r=1

r;“-‘i~_j=

f

YT-iYT_j+O,(l).

f=l

Consequently, q

,~(e-e)=Jm(8-8)+0,(1).

(2.29)

Theorem 2.4. Under Conditions (Cl) and (C2), and as m+co,

&?a - 0) 1:N,(O,

t2 r, ‘).

(2.30)

Proof. The result follows readily from (2.29) and Theorem 2.3.

0

Remark. It may be noted that the estimates 6 and e”may not be fully efficient in the non-Gaussian case. A Gaussian ML estimate of 0 is considered by Risager (1980). However, in the context of our paper, we only need &-consistent estimates of 8 and Theorems 2.3 and 2.4 provide the results for this purpose. 2.5. Estimation

of C

First, suppose that c1and 6’are known. Then, a natural estimator of C is given by f=m-‘f

(2.31)

EtET, t=1

C,p-By. Note that E(f) = C. In order to whereE,=Y,-~~S1&Yt_iand Yt=Xf-ll,xl derive the limit distribution of the upper triangular elements of C, consider, for simplicity, the case n = 2. The three upper triangular elements of C are then given by m R,=

We have E(R,)=

&

(

m-l

f t=1

&, m-l

5 Etl.ztZ,m-l 1=1

T

’

Etz i=l c

’ >

R = (ol 1, o12, oz2)=. We will show that

UL - R)

zN3(0, WI,

(2.32)

S. Sethuraman,

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Multivariate

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197

where the elements of W are given by

In order to verify the result in (2.32) consider

where AT= (&, &, A,). Note that (&> are independent and identically distributed zero mean random variables. Applying the ordinary central limit theorem, we find that

~Tfi(R,-

R) : N(0,ATWA),

and hence, by the Cramer-Wold device, we obtain the desired result in (2.32). Note that for the above result to hold, we have assumed that the moments E(E:& E(E~ 1E~~), E(E;& E(E:~F~~), E(E<~E:~), E(E~~E:~) all exist and are finite. In the general case, one needs to assume the existence of the fourth-order moments E(E~~E~~E~~E~Jfor all i, j, k, I= 1,2, . . . , n. When 8 and c( are unknown, replace 8 and a by &and BO,and define

where {Y:] are as defined in Section 2.4 and e”,is the ith element of & An estimate of C when 9 and CIare unknown is then given by (2.33) Using the same arguments as those in Theorem 9.3.1 of Fuller (1976), one can show that, as m-+co,

WT+ O,(l), t=1

and hence &(2-C)

will have the same limit distribution as J&(c^-C).

Remark. See also Hannan (1970, p. 228, Theorem 14) for a more general treatment.

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S. Sethuraman, I. V. Basawal Multivariate linear model with autocorrelated errors

2.6. The generalized least-squares estimator of dowhen C and r are unknown

Now, returning to Section 2.2, we recall that &o depends on Z and r. When 2 and r are unknown, proceed as follows: (i) First, find the simple least-squares estimator B, which does not depend on C and r. (ii) Using BO,find the residuals Yj+=x,-l,.&)-BY,. (iii) Compute 8 using (2.28). (iv) Using { YF} and 3 found in (ii) and (iii), compute the residuals &p= Yl*-

i

“elY;"_i.

i=l

(v) Compute 2 using (2.32). (vi) Finally, plug in t? and c” in Bo to get the desired estimate of &. Theorem 2.5. Under Conditions (Cl) and (C2), the limit distribution of D,(& same as that of D,(& - LX),and it is given by Theorem 2.1.

- a) is the

Proof. The result follows by repeated use of the arguments of Theorem 9.3.1 of Fuller

(1976) and the results derived in Sections 2.1-2.5. The details are omitted. Remark.

See also Hannan (1970, p. 449), for a more general result.

3. Large sample tests

In this section, we discuss Wald-type tests for testing the composite hypotheses Hi: y =0 and H,: /?=O, treating all other model parameters as unknown nuisance parameters; see Basawa (1991) for a general background on asymptotics for testing of composite hypotheses. 3.1. A test of HI: y=O

From Theorem 2.5, we have Dm(EG-~) 5 N,JO, G-l),

k=q+r,

(3.1)

S. Sethuraman,

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Multivariate

linear model with autocorrelated

errors

199

where G is defined in (2.8). Partition a = VT, YT)T, D,,,=[:

J,-,]

ki = (Bk zi)‘, where Cg=diag{(

$rC:y”,

i=l, . ...4 ’ ) j

I is an Yx r identity matrix and

It follows immediately from (3.1) that -Y) 5 IWO, Gz2),

(3.2)

G22=(G22-G21G;~G12)-1.

(3.3)

&(k where

The test statistic for H,: y=O, suggested by (3.9) is T,=mjj;(G22)-1j3G,

(3.4)

where (?22 is G2* with all the parameters replaced by their estimates. It follows readily from (3.2) and (3.4) that Tr-+?(r),

under Hr.

(3.5)

Consider a sequence of contiguous alternatives, Kr: y = hlm-“*, where hr is an (r x 1) vector of real numbers. Using standard theory, one can show that T1 *x** (r, Z), where x** denotes a noncentral chi-square. The noncentrality to be A: =hT(G;f)-‘h,,

(3.6) parameter Af is seen (3.7)

where G$ is G22 with y =O. 3.2. A test of HZ: p=O Using the notation of Section 3.1, it follows from (3.1) that

G&i - 8) 5 N,(O, G”),

(3.8)

where G” =(Gll -G12G&iGZ1)-1. Consider the test statistic Tz =~oc:(c”“)-‘c;13;,,

(3.9)

200

S. Sethuraman,

I. V. Basawa / Multivariate

linear model with autocorrelated

errors

where cl1 represents G” with all the parameters estimated. We then obtain (3.10) where Kz: p = C; ‘h2, h2 being a q x 1 vector of real numbers, and (3.11)

4. Restricted least-squares estimation This section is concerned with least-squares estimates /?nI and $& obtained under the hypotheses H,: y =O, Hz: j?=O, respectively. 4.1. The limit distribution

of bH,

Under HI, the model in (1.1) and (1.2) reduces to X,=L.IC,P+Y,, (4.1) Yf= f Bi Yf-i+E,,

t=l,...,m.

i=l

The model in (4.1) can be rewritten compactly as X=&B+

(4.2)

y,

where the (mn x 1) column vectors X and Y and the (mn x q) matrix e51are as defined in (2.4). Note that E(Y)=0 and Cov(Y)= F =T@C. The generalized least-squares estimator of /? obtained from (4.2) is seen to be jjul =(c$;F-I&)-’

qS;F-‘X.

(4.3)

An alternative form of $n, is given by ~~,=(C~~-~C)-~(l~xlC-~lnxl)-~(C~r-~O1;T~~C-~)X.

(4.4)

We have EH,(j&r,)=~

and

COV~~(~~,)=(C~~-~C)-~(~~,~C-~~,.~)-~.

When F is unknown, we plug in a reasonable estimate of F in & to obtain pm. First assume F is known and denote bn, in (4.4) by &(F). Denoting, as before, C,$=diag we have, under H1, as m-co,

CW%I,(F)-~)~ N,(QU,Tx ,C-‘Lx I)-~A;~),

(4.5)

S. Sethuraman, I.V. Basawal Multivariate linear model with autocorrelated errors

201

where A1 = lim Cz-’ (CTT-’ C)Cz-‘. ??!-+a3

(4.6)

The limiting matrix A1 exists under conditions (Cl) and (C2). When F is unknown, we proceed as follows: (i) First, compute the simple least-squares estimator &nl of /I, given by /J&n1=(c’c)-

‘CTX,

(4.7)

where X=(X1, . . . ,X,)

with Zi=n-’

i Xi+ j=

1

Using similar arguments as before, one can show that, under conditions (Cl) and (C2), we have, under Hi, CSL,

-&N,(O,A-%A-‘),

where A is as defined in (2.16) and K=(K,J Kij=a,**

~

(4.8) with

aij(h)y(h).

j=-_3c

(ii) Assuming /I is known, the least-squares estimator of 8 is given by @I) = K:;;,‘A,,, where A,,,,, and B,, are as defined in (2.20) with Y,=X,- 1, x lC,p. The limiting distributions of the least-squares estimators @I) and ewhen j3 is replaced by &,u, are as given in (2.22) and (2.30), respectively. (iii) If p and 19are known, an estimate of C is given by (2.31). Replacing p by BOH,and 8 by 8, we obtain an estimate of C, given by

where &:=Y:-

it&Y:+, i=l

with Y,*=X,-I,&ficl”,. The limiting distribution of c” can then be derived as in Section 2.5, under the additional condition of the finiteness of the fourth-order moments: E(EliEljElkE11) for all i, j, k, I= 1, . . . . II. (iv) Finally, plug in F^= f @ c^ in fiu, in (4.3) to obtain the desired estimate B. One can show that Cg(p-/I) has the same limit distribution as that in (4.5).

202

S. Sethuraman, I. V. Basawa / Multivariate linear model with autocorrelated errors

4.2. The limit distribution Under

of pH2

Hz: y =0 the model (1.1) reduces X,=By+

to the linear model

Y,, (4.9)

~ Bi Y,_i+&r,

Y,=

t=l , . . ..m.

i=l

The model in (4.9) can be rewritten X=&y+

compactly

as (4.10)

Y,

where &=lmxlOB. The generalized

least-squares

j&i2=(&F-1f$2)-1 Note that fH2 can be rewritten

estimator

of Y under

H2 is given by

&-lx.

(4.11)

as

Y*H2=(BT~-1B)-l(1T,xlr-1l;r,x1)-1BTC-1(1~xlr-1OInxn)X.

(4.12)

E(y*n,)=y

(4.13)

We have

Assuming

and

F is known,

Cov(y*,,)=(BTC~lB)~‘(l;f,.l~~‘l,.l)~’.

we shall first find the limiting

&(yAH2-y)=(BTC-1B)-1 x BrZ-‘(l;

m(lLx lr-‘l,, xmr-

01

1X

nxn

distribution 1 )YL

of $uH2.We have

’ 1

‘2.

(4.14)

Note that B and C do not depend on m. Assume that lTx lr-ll,, Jm-+ W as m-+co, where W is some function of 8. From standard theory (see, for instance, Brockwell and Davis (1991)) using the Cramer-Wold device we can show that, under HZ, as m+co, m -“2(l;fixlr-101nxn)Y~NNn(0, Consequently,

as m-co,

WC).

we have

~(~~*-y)~N,(O,(BTT-lB)-l W-l). When F is unknown, we proceed to estimate y as follows: estimator of y under (i) First, compute the simple least-squares $0 = (BTB)_ 1Fr?, where Z=(&,

(4.15)

. . ..r?.),

(4.16)

Hz, given by (4.17)

S. Sethuraman,

1. V. Basawa 1 Multivariate

linear model with autocorrelated

203

errors

with

j=

Note that E(yO)=y

1

and Cov(9,)=v~(BTB)-‘(BTCB)(BTB)-‘,

where

II-1 v2 Em-2 m

v(O)+2

( and y(h)=Cov(U,, co, we have

C h=l

T

(m-fMN 1

U,+,J, {U,} satisfies (2.3). It may be easily seen that, under

H2, as

m+

(4.18) Note that in deriving (4.18), we have used the fact that, under (Cl), mvi -C,“= _ a, y(h) as m-co, and that B and C do not depend on m. (ii) Assuming y is known, the least-squares estimator of 8 is given by 8(y) = B;i&,, where A,, and B,, are as defined in (2.20) with Y, = X, - By. The limiting distributions of the least-squares estimators t?(y) and e’,when y is replaced by QO,are as given in (2.22) and (2.30) respectively. (iii) If y and ~9are known, an estimate of C is given by (2.31). Replacing y by y0 and 8 by & we obtain

an estimate

.f=m-'

of C, given by

5 &fEfT,

where

i=l

with r: = x, - By*@ The limit distribution of .J? can then be derived as in Section 2.5. (iv) Finally, we plug in F”= r”@ c” in y^n2 to obtain the desired estimate before, one can show that J%($j - y) has the same distribution limit distribution (under H,) is given by (4.16).

as fi($n2

y”.Arguing -

as

y), and the

Acknowledgements I.V. Basawa’s work was partially Research. We thank the referees comments.

supported by a grant from the Office of Naval for a careful reading and many constructive

204

S. Sethuraman,

I.V. Basawal

Multivariate

linear model with autocorrelated

errors

References Anderson, T.W. (1978). Repeated measurements on autoregressive processes. J. Amer. Statist. Assoc. 73, 371-378. Basawa, I.V. (1991). Generalized score tests for composite hypotheses. In: V.P. Godambe, Ed., Estimating Functions. Oxford Univ. Press, Oxford, pp. 121-132. Basawa, I.V. and L. Billard (1989). Large sample inference for a regression model with autocorrelated errors. Biometrika 76, 283-288. Basawa, I.V.. L. Billard and R. Srinivasan (1984). Large sample tests for homogeneity for time series models. Biometrika 71, 203-206. Billingsley, P. (1961). The Lindeberg-Levy theorem for martingales. Proc. Amer. Math. Sot. 12, 788-792. Brockwell, P.J. and R.A. Davis (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York. Fuller, W.A. (1976). Introduction to Statistical Time Series. Wiley, New York. Hannan, E.J. (1970). Multiple Time Series. Wiley, New York. Hwang, S.Y. and LV. Basawa (1992). Large sample inference based on multiple observations from nonlinear autoregressive processes. Stochastic Process. Appl. (to appear). Kim, Y.W. and I.V. Basawa (1992). Empirical Bayes estimation for first-order autoregressive processes. Austral. J. Statist. 34, 105-l 14. Rao, CR. (1973). Linear Statistical Inference and Its Applications. Wiley, New York. Risager, F. (1980). Model checking of simple correlated autoregressive process. Stand. J. Statist. 8,137-153. Sethuraman, S. and I.V. Basawa (1993). Parameter estimation in a stationary autoregressive process with correlated multiple observations. J. Statist. Plann. lnj (to appear).