Testing cointegration in infinite order vector autoregressive processes

JOURNAL

OF

Econometa'ics ELSEVIER

Journal of Econometrics 81 (|997) 93-126

Testing cointegration in infinite order vector autoregressive processes Pentti S a i k k o n e n .,a, Ritva L u u k k o n e n b aDepartnwnt o f Statistics. SF-O0014 Unit'ersity o f HelsinkL P.O. Box 54. Finland bitts'titute of Occupational Health. Topeliuksenkatu 41a A. 00250. HelsinkL Finland

Abstract This paper studies test procedures which can be used to determine the cointegrating rank in infinite order vector autoregressive processes. The considered tests are analogs or close versions o f previous likelihood ratio tests obtained for finite-order Gaussian vector autoregressive processes, it is shown that the use o f the likelihood ratio tests is justified even when the data are generated by an infinite order non-Gaussian vector autoregressive process. New tests are also developed for cases where intercept terms are included in the cointegrating relations. These tests are based on a new approach o f estimating the intercept terms. They have the property that, under the null hypothesis, the same asymptotic distribution theory applies as in the case where the values o f the intercept terms are a priori known and not estimated. A limited simulation study indicates that the new tests can be considerably ntore powerful than their previous counterparts. © 1997 Elsevier Science S.A.

Keyi.,ords. Cointegration; Infinite order vector autoregression; Likelihood ratio test J E L ¢!assiJication: C32

1. Introduction The literature on statistical analysis of cointegrated systems has grown enorm o u s l y s i n c e t h e s e m i n a l p a p e r o f E n g l e a n d G r a n g e r ( 1 9 8 7 ) . In p r a c t i c e , t h e d e t e r m i n a t i o n o f t h e n u m b e r o f c o i n t e g r a t i n g v e c t o r s , t h a t is, t h e c o i n t e g r a t i n g r a n k is o f t e n an i m p o r t a n t p a r t o f t h i s a n a l y s i s a n d , t h e r e f o r e , a n u m b e r o f test p r o c e d u r e s have been d e v e l o p e d for this purpose. A m o n g the b e s t - k n o w n tests are the residual-based tests initiated by Engle and G r a n g e r ( 1 9 8 7 ) and further

* Corresponding author. The authors wish to thank Peter Boswijk, In Choi, Atsushi Nishio, Pierre Perron and two anonymous referees for useful comments on a previous version of this paper. 0304-4076/97/$17.00 ©

1997 Elsevier Science S.A. All rights reserved

Pii S0304-4076(97)00036-5

94

P. Saikkonen, R. LuukkonenlJournal o f Ecom,melric.~" 81 (1997) 93-126

developed by Phillips and Ouliaris (1990), the principal components tests o f Stock and Watson (1988), and the likelihood ratio tests obtained by Johansen (1988, 1991 ) in the context o f finite-order Gaussian vector autoregressive ( V A R ) processes. The likelihood ratio tests, which are based on more specific assumptions about the data generating process than some o f their alternatives, have become very popular in applications. In a recent paper Saikkonen (1992) showed that closely related versions o f (some o f ) the likelihood ratio tests can also be derived under conditions much weaker than those originally used by Johansen (1988, 1991 ). Specifically, Saikkonen (1992) only assumed an infinite-order V A R process and approximated it by a finite-order autoregression, whose order was required to tend to infinity with the sample size at a suitable rate. It is shown in this paper that the original versions o f the likelihood ratio tests can similarly be extended to infinite order V A R processes. Another, and actually more important, contribution o f the paper is that new and more powerful tests than the previous ones are developed for models with intercept terms included in the eointegrating relations. Finally, using extensions o f recent results o f Ng and Perron (1995), it is shown that all these results can be obtained by weakening previously used assumptions o f the relation between the sample size and selected order o f the V A R process. The new tests o f the paper are based on a new approach o f estimating the intercept terms o f the c,~integrating relations. The idea is similar to that already used by Elliot et al. (1996) and Pantula et al. (1994) for univariate autoregressive unit-root tests. It requires that the intercept terms are included in the model in a suitable w a y and estimated by using appropriate generalized least squares. The resulting test statistics have the interesting property that under the null hypothesis their limiting distributions are the same as in the case where the model contains no intercept terms or when the values of the intercept terms are a priori known. Since the limiting distributions o f the previous test statistics are affected by the estimation o f the intercept terms it is intuitively expected that the new tests have good power properties compared with their previous counterparts. The simulation results o f the paper confirm this intuition. The paper is organized as follows. S~¢tion 2 contains preliminary considerations about the model and testing problem. Section 3 presents the considered test procedures and the asymptotic distributions o f the test statistics under the null hypothesis. Section 4 studies finite sample properties o f the tests by Monte Carlo simulation. Section 5 concludes. Proofs o f the theorems are given in a mathematical appendix.

2. Preliminaries Following Saikkonen (1992), we consider an n-dimensional time series y t , t -- 1. . . . , T, pai~itioned as

P. Saikkonen. R. LuukkonenlJournal o f Econometrics 81 (1997) 93-126

Y't=[Y~t nl

nl + n z = n ,

Y2t],

95

(1)

/12

and generated by the cointegrated system Ylt - - A y 2 t + ult,

(2a)

zl y2t = u2t,

(2b)

where A is the usual difference operator and ut = [u~t u2t] ' ' is a stationary process with zero mean and continuous spectral density matrix which is positive definite at the zero frequency. Hence Y2t is an integrated process o f order one which is not cointegrated while Ylt and Y2t are cointegrated. The initial value y0 m a y be any r a n d o m vector with a fixed probability distribution. T a k i n g first differences and rearranging yields the triangular error correction representation

(3)

A yt = J O ' y t - t + vt,

where J ' = [ - I n , 0], O'--[InL - A ] , and v t - - [ v ~ t v'2t]' is a nonsingular linear transformation o f ut defined by v l t - - u l t + A u 2 t and v2t = u 2 t (cf. Philli0s, 1991; Saikkonen, 1992). The process vt (and hence ut) is a s s u m e d to have an infiniteorder autorcgressive representation oo

~_, G j v t - j -- e,t,

(4)

Go=In,

j=O

where et is a sequence o f continuous i.i.d. (0,2~) r a n d o m vectors with Z7 positive definite and et have finite fourth moments. A further assumption is that the coefficient matrices Gj are l - s u m m a b l e so that oo

EJlIGjl[
(5)

j=l

This condition imposes a slight restriction on the temporal dependence o f the process yr. It is, for example, satisfied by all stationary and invertible A R M A processes and it implies that the process vt and, consequently, yt can be approximated by a finite-order autoregression (see Saikkonen, 1992). Specifically, from ( 3 ) and (4) one can obtain the approximate model K

A y t = qJYt-i + ~

j=!

l l j A y t _ j + et,

where oo

e,=~.-

E

j=K+ i

Gjv,-i.

t = K + 2, K + 3 , . . . ,

(6)

96

e. Saikkonen, I{ LuukkonenlJournal o f Econometrics 81 (1997) 03-126

Here K is supposed to be so large that Gi ~ O, j > K, so that et ~ e.t. Furthermore, due to cointegration, the coefficient matrix q~ has reduced rank and the structure K ~, = ~ o ' =

-

~

GjJO',

j=O

(7)

w h e r e the second equality defines the n x nl matrix 4~ which is o f full colu m n rank (at least for K large enough). Details o f the derivation o f ( 6 ) can be found in S a i k k o n e n ( 1 9 9 2 ) and S a i k k o n e n and Liitkepohl ( 1 9 9 6 ) and are not repeated here. i W e note, however, that the coefficient matrices I l j ( j = 1. . . . . K ) are functions o f O and Gy ( j = 1 , 2 , . . . ) , and, likewise ~ , they also depel,d on K. Furthermore, the sequence H / ( j = 1. . . . . K ) is absolutely s u m m a b l e as K ~ ¢x~. In practice, it is usual that the cointegrating rank, that is, the integer n~ is a priori not known. Therefore, it is o f interest to consider testing the null h y p o t h e s i s H(nl ): C o i n t e g r a t i n g rank equal to hi. Since the n u m b e r o f free elements in the p a r a m e t e r matrix ~ increases with nl the natural alternative h y p o t h e s i s in this setting is l~(nl ): C o i n t e g r a t i n g rank larger than n,. N o t e that the t b r m u l a t i o n implies that 0 ~
t The present formulation of the model is similar to that used in Saikkonen and Ltitkepohl (1996) and also in the appendix of Saikkoncn (1992). it is slightly simpler than the one given on p. 4 of Saikkonen (1992) where the definition of the matrix 4~ does not involve K and consequently the error term et is slightly more complicated than here. Condition (5) implies that the difference between these two formulations is asymptotically negligible in the sense that thc two definitions of the matrix 4~ differ by a quantity which is of order o(K - t ) (see Saikkonen, 1992, Eq. (A.17)).

P. Saikkonen, R. Luukko,~en l Journal o f Econometrics 81 (1997) 93-126

97

d a t a g e n e r a t i n g p r o c e s s satisfies ( 2 ) a n d ( 3 ) . T h i s is the m a i n r e a s o n w h y t h e n o r m a l i z i n g restrictions are used in this paper. T h e a b o v e m o d e l a s s u m e s that t h e r e is no intercept in t h e c o i n t e g r a t i n g relation ( 2 a ) . Since this is r a t h e r s e l d o m a realistic a s s u m p t i o n it is r e a s o n a b l e to c o n s i d e r a m o d e l w h i c h allows for this feature. S u p p o s e w e n o w o b s e r v e an n - d i m e n s i o n a l t i m e series zt, t - - 1, . . . . T , w h i c h is g e n e r a t e d b y (8)

zt - - m q- Yt,

w h e r e m (n x 1 ) is an u n k n o w n p a r a m e t e r v e c t o r a n d Yt is as in ( 2 ) or, e q u i v a lently, ( 3 ) . M u l t i p l y i n g ( 8 ) f r o m the left b y 19' a n d [0 In2]A y i e l d s z l , = It + Az2, + u l , ,

(9a)

Az2, = u2,,

(9b)

w h e r e / ~ = O ' m w h i l e z l t a n d z2t are o b v i o u s s u b v e c t o r s o f zt. T h i s is the e x t e n sion o f ( 2 ) c o n s i d e r e d b y S a i k k o n e n ( 1 9 9 2 ) , w h o p o i n t e d o u t that, a n a l o g o u s l y to ( 6 ) , o n e c a n obtain f r o m ( 9 ) the a p p r o x i m a t e m o d e l K

Az~ = v + qJzt-~ + ~

l l y A z t _ j + el,

t = K + 2,K + 3....

(6')

j----!

w h e r e t/, satisfies ( 7 ) , v = - - O / ~ , a n d et is as d e £ n e d b e l o w (6). 2 Since v = - - O O ' m = - tPm it f o l l o w s f r o m ( 6 ' ) that K

Azt = t P ( z t _ t -- m ) + ~ H j A z t - j + et, j=l

t =K

+ 2,K + 3,....

(6")

T h e null h y p o t h e s i s H ( n l ) c a n also be c o n s i d e r e d w i t h i n m o d e l s ( 6 ' ) a n d ( 6 " ) . A difficulty with the latter m o d e l is that it is n o n l i n e a r with r e s p e c t to the level p a r a m e t e r m and, unlike ( 6 ' ) , c a n n o t b e e s t i m a t e d b y o r d i n a r y least squares. It turns out, h o w e v e r , that .~t is w o r t h w h i l e to p a y for this c o m p l i c a t i o n . T h e s i m u l a t i o n results o f S e c t i o n 4 indicate that test p r o c e d u r e s b a s e d on m o d e l ( 6 " ) c a n be s u b s t a n t i a l l y m o r e p o w e r f u l than t h o s e b a s e d on ( 6 ' ) .

3. Test procedures 3.1.

Models

w i t h o u t a n i;2tercept

T h e e s t i m a t i o n a n d testing p r o c e d u r e s in S a i k k o n e n ( 1 9 9 2 ) are b a s e d o n the u n r e s t r i c t e d l e a s t - s q u a r e s e s t i m a t i o n o f a m i n o r m o d i f i c a t i o n o f ( 6 ) . T h e

2 Unlike in Saikkonen (1992) the same error term et appears both in (6) and (6t). As Eq. (9") of Saikkonen (!~92) r,mkes clear, the reason is that here the matrix 4b is allowed to depend on K. (Note also the printing error in the definition of the error term e~' below Eq. (9') of the previous paper. The correct definition should contain J/~ in place of/~.)

P. Saikkonen, R. LuukkonenlJournal o f Econometrics 81 (1997) 93-126

98

modification takes explicitly into account the fact that the first n~ c o l u m n s o f the coefficient matrix I l K are zero vectors. T h e purpose o f the modification was only to s i m p l i f y m a t h e m a t i c a l derivations and, since it has no effect on asymptotic results, it will not be used in this paper. Thus, we define xt = [ A y ' t _ 1 . . . A Y ' - K ] ' and H----[Hi . . . ILK], and write the estimated version o f ( 6 ) as A y t = ~ Y t - t + f l x t + ~:t,

(lO)

t = K + 2 , . . . , T,

w h e r e ~' and / 7 - - [ / 7 1 . . . /TK] are the o r d i n a r y least-squares estimators o f the coefficient matrices tp and H , respectively. T h e error covariance matrix ,~ is estimated by T

£

N -I

--'

~t ~t • t =K+2

w h e r e N = T -- K -- 1. Let 2t ~< - " - ~<2n be the solutions o f the generalized eigenvalue p r o b l e m -

z£t

(11)

= 0,

where C'~"

r

t

~ Y[- ]Y!--| t~.K+2

_

r

E

t=K+2

J. |.-. -. l X',

~ - - ~ + ) / r 2 x tt x \ - i t

r E

X

¢

/y/__ | •

t----K+2

In order to test the h y p o t h e s i s H(ni ) we n o w introduce the test st-..sii.~ /12

w=Z

G. j=l

It is clear that this test statistic is invariant to the " ~qmalization o f the cointeg r a t i n g vectors and its large values are critical. Test statistic W is a slight modification G• test statistic R discussed in S a i k k o n e n ( 1 9 9 2 ) . This latter test statistic ¢~: :.e defined in the same w a y as test statistic W e x c e p t that th~ i . . . . . :. C it. ( i i I is replaced b y

M'-

T

~, y,_,y~_,.

(12)

t=K+2

Thus, in the present version o f the test statistic the matrix M is corrected for the short-run d y n a m i c s o f the process. This correction has s o m e appeal because it takes the intertemporal d e p e n d e n c e o f the data into account. It also m a k e s the test v e r y close to the likelihood ratio test d e v e l o p e d b y Johansen ( 1 9 8 8 ) for finite-order G a u s s i a n V A R processes. The likelihood ratio test statistic is obtained b y techniques similar to those e m p l o y e d in reduced rank regression. In fact, it

P. Saikkonen. 1~ Luukkonenl Journal of Econometrics 81 (1997) 93-126

99

is not difficult to check that in our context an analog o f the likelihood ratio test statistic is n2

L R ----N~--~ log( l + f~jlN) j=l

(cf. Anderson, 1951, Eq. (3.2)). This expression o f the likelihood ratio test statistic differs from the one given by Johansen (1988). However, the two expressions are identical since in our notation Johansen (1988) uses the transformed eigenvalues ~ j / ( N + ~ j ) ( j - - I . . . . . n2). To be able to obtain the limiting distributions o f the above test statistics some assumptions are needed. Since we only use a finite-order autoregression as an approximation we have to assume that its order tends to infinity with the sample size T at a suitable rate. Therefore, we shall m a k e the following assumption which has previously been used by several authors (see e.g. Berk, 1974; Lewis and Reinsel, 1985; Said and Dickey, 1984; Saikkonen, 1991, 1992; and Saikkonen and Lfitkepohl, 1996).

A s s u m p t i o n 1. K is chosen as a function 6 f T such that f - - - , oo and K 3 / T - - , 0 as T ---, c~. Thus, we impose an upper bound for the rate at which the autoregressive order K is allowed to tend to infinity with the sample size. In most previous studies in the area, like those referred to above, it has also been assumed that the autoregressive order satisfies a lower bound condition, na.~nely, oo

T '/2

j---:K+l

IIGjlI

0

as T - - , o o .

(13)

However, recently Ng and Perron (1995) pointed out that this condition is actually not needed to obtain the limiting distribution o f the univariate unit root test o f Said and Dickey (1984). Since our test procedures can be thought o f as extensions o f this univariate unit root test it is no wonder that condition (13) can also be removed in this context. Being able to remove the lower bound condition (13) has also some practical implications concerning the choice o f the autoregressive order K. For instance, suppose one uses some c o m m o n model selection criterion, like AIC or BIC, to choose an appropriate value o f K. In the important special case where the data generation process has an A R I M A structure the selected order will then tend to infinity at the rate Op(Iog T), as shown by N g and Perron (1995) and Saikkonen (1995). This means that the selected order is generally not consistent with the lower bound condition (13) although it is consistent with Assumption I. It is also shown in Saikkonen (1995) that, under conditions essentially the same as we have assumed here, the application o f all conventional model selection criteria

I00

P. Saikkonen. R. L u u k k o n e n l J o u r n a l o f Econometrics 81 ( 1 9 9 7 ) 9 3 - 1 2 6

will yield an order which tends to infinity in probability. This is again consistent with A s s u m p t i o n 1. A detailed discussion o f the use and properties o f data-based order selection m e t h o d s is outside the scope o f this study, however. Unless otherwise stated A s s u m p t i o n 1 is supposed to hold throughout the paper. It will be convenient to denote

W ( r ) d W ( r ) ' ) t (lfo W ( r ) W ( r ) ' d r ~

~(n2)=tr(~

where W ( r ) is an n2-dimensional standard B r o w n i a n motion. N o w w e can state the following t h e o r e m where the hypothesis H(n~ ) is a s s u m e d and the s y m b o l ' - - - - > ' is used to signify w e a k convergence o f the associated probability m e a sures.

Theorem 3.1. Suppose that Yt (t = 1. . . . . T) is generated by ( 3 ) and ( 4 ) with the rows o f J and 6) possibly permuted. Suppose further that Assumption 1 holds. Then, W ~.. ~(n2) and L R .~ ~(n2), as T ~ oo. The p r o o f o f T h e o r e m 3. i is given in the appendix where it is shown that test statistic W and its previous version R and related by W = R + op( 1 ). W h e n the order o f the considered V A R process is fixed and finite this result is straightforward to obtain but s o m e complications occur w h e n the order is a l l o w e d to tend to infinity. F r o m the definitions it is also easy to deduce the L R = W + o p ( 1 ). Thus, u n d e r the hypothesis H(ni ), the three test statistics are a s y m p t o t i c a l l y equivalent and have the s a m e lirniting distribution as the likelihood ratio test statistic in the case o f a finite-order Gaussian V A R process.

3.2. Models with an intercept The test procedure o f S a i k k o n e n (1992), which allows for an intercept, is based on an estimated version ( 6 ' ) , that is, on AZt=V+gvZt._l+J~vXt"~vt,

t=K+2

. . . . . T,

(14)

where "~, W~. and / l v : = [/l~.i . . . /I,.K] are ordinary least squares estimators o f the parameters v, tp and H , respectively. Notice that, due to the relation Azt = Ayt, t > 1, the vector xt also appears here and is defined as x t - [Az~_ l ... AZt_K]' '. In this case the error covariance matrix Z" is estimated b y T

:.~ =

N -l

W. t =K

+2

~"v t ~", t .

P. Saikkonen. R. LuukkonenlJournal of Econometrics 81 (1997) 93-126

101

A n analog o f test statistic W can n o w be obtained in an obvious way. Let ~.vl ~< "'" ~<~.v,, be the solutions o f the generalized eigenvalue p r o b l e m (15)

v,~cv,~ - : . ~ 1 = o where

~=

T

~

~,-,~,-,-

T

Y~ t=K+2

t=K+2

/

-'~

T

t=K+2

--'/

--\--I

/

T

--

t=K+2

with ~'t-! and ~t d e m e a n e d versions o f the series zt-i and xt, respectively. In other words, ~'t- i = z t - m-- £'- i and £t = xt - £ where ~_ ~ and .~ are the sample m e a n s o f z t - ~ and xt (t = K + 2,..., T), respectively. In place o f test statistic W we n o w have n2

,,.

j=l

The analogous previous test statist';e discussed in Saikkonen ( 1 9 9 2 ) and denoted by Re here is defined in the same way as test statistic W~ except that the matrix C" in ( 1 5 ) is replaced by T

t=K+2

A n obvious analog o f the likelihood ratio test statistic is n o w tt2

LRv = N ~

log(I +

j=l

~v,/N).

The following t~¢orem shows that, under the hypothesis H(ni ), test statistics W~ and LRv have the same limiting distribution as test statistic Rv and tim corresponding likelihood ratio test statistic obtained by Johansen ( 1 9 9 1 ) for finite order Gaussian V A R processes. The limiting distribution can be represented by the r a n d o m variable ~ ( n 2 ) - - t r ( 0~ l~'(r)dW(r)')' ( ~ 14,'(r)W(r)' d r ~ I

x(~,Y(r)dW(r)') where l ~ ' ( r ) = W ( r ) -

f~ W(s)ds (0 <~r<~ l ) is a d e m e a n e d Brownian motion.

Theorem 3.2. Suppose that zt (t--- 1. . . . . T) is oenerated by ( 8 ) and chat the assumptions stated in Theorem 3.1 hold. Then, W~ ~. ~(n2) and LRv --~- ~(n2), as T---> cx~.

!02

P. Saikkonen. IZ LuukkonenlJournal o f Econometrics 81 (1997) 93-126

Next, we shall consider tests based on model ( 6 " ) . The idea is to replace the level p a r a m e t e r m b y an appropriate e s t i m a t o r and proceed in the same w a y as in Section 3.1 or, equivalently, w h e n the value o f m is k n o w n . A m a j o r p r o b l e m with this approach is the estimation o f m. Since v = - ~ m , a natural estimator is - - ~ ' f ~ ¢ but this is not a g o o d choice since, due to the a s y m p t o t i c singularity o f the matrix ~,, it d i v e r g e s as the sample size tends to infinity. Specifically, using results in S a i k k o n e n (1992, 1995) it is not difficult to s h o w that in s o m e directions o f the p a r a m e t e r space the rate o f d i v e r g e n c e is as fast as O p ( T ) . To find a better estimator, first define

d, ==,

t--! -

&:,-j,

t -- 1. . . . . K + 1,

}=i

and K+I

t--K+2

. . . . . T,

j=l

w h e r e /}v, = In + ~v + lZlvl, B~j = ffl,j - f f l , . j _ t , j = 2 . . . . . K , a n d / } v . g + i = - - / l , g . F r o m these definitions and ( 1 4 ) it can be seen that /}~/ ( j = 1, . . . . K + I ) are least-squares estimators obtained b y fitting an autoregression o f order K + ! to the levels series so that z t = ~ + B ~ t z t - l + " "" + B~.K + l z t - g - ! + ~:~t ( t = K + 2 . . . . . T ) and In - - / ~ , . . . . . /~v.K+l = -- ~U, (Cf. S a i k k o n e n a n d Liitkepohl, ! 996 b e l o w ( 2 . 8 ) ) . N e x t define C ' l - - I , , t--I

Ct=ln--

Y~.Bvj,

t=2 ..... K+

1,

j=!

and Ct = -tp,. for t = K + 2 . . . . , T. Note that we can also write C't = - ~ . - / J r , for t = 2 . . . . . K + 1. With these definitions we n o w introduce the estimator

t-!

1

To see the m o t i v a t i o n o f this estimator, consider ( 6 " ) a u g m e n t e d for t = 1 . . . . , K + 1 with the m i s s i n g pre-sample observations a s s u m e d to be z t = m , t = - - K , . . . , 0. Then, i f B t . . . . . B x + t denote the autoregressive coefficient matrices o f the levels formulation o f the model, w e h a v e t--!

z, = m + ~

B j ( z j -- m ) + et,

t-- I,...,K

+ I

j---I

(16)

K+i

zt=m

+ Y]~ B j ( z j -- m ) + et,

j=i

t--K

+ 2,K + 3 .....

P. Saikkonen, 1~ LuukkonenlJournal o f Econometrics 81 (1997) 93-12o

103

O f course, the relation between the coefficient matrices By (j---- I, . .. , K 4 - 1 ) , and / / j ( j - - 1 . . . . . K ) is exactly the same as the relation between the corresponding estimators described above. Suppose that zt in ( 1 6 ) is a Gaussian V A R process o f a fixed and finite order K 4- 1. Then, et : Et ~-, n.i.d.(0, E ) and, i f /~j ( j -- 1. . . . . K ÷ 1 ) and ~ were m a x i m u m likelihood estimators based on (16), the estimator ~fi would clearly he the m a x i m u m likelihood estimator o f the parameter m. O f course, we do not have m a x i m u m likelihood estimators but this observation shows that the estimator ~fi can be interpreted as a first step in an iterative m a x i m u m likelihood estimation algorithm. T h e next step w o u l d be to replace m in ( 1 6 ) by the estimator ~ and obtain obvious n e w estimators for the parameters By ( j - - 1. . . . . K 4- 1 ) and 2:. Iterating this procedure until c o n v e r g e n c e yields m a x i m u m likelihood estimators. The estimator t~ can thus be motivated as an approximate m a x i m u m likelihood estimator. T h e approximation results hecause the procedure is not iterated and because our assumptions arc m o r e general than those a s s u m e d above. This interpretation o f the estimator n~ was suggested by a referee. Since the error term e, in ( 1 6 ) is approximately white noise with covariance matrix Z the estimator t~ can alternatively be interpreted as a feasible version o f an approximately optimal generalized least-squares estimator. Asymptotic properties o f the estimator ~ are given in the following l e m m a where J~ -- [0 In2].

L e m m a 3.1. Suppose that zt (t = 1, .... T) is generated by ( 8 ) where yt satisfies (3) and (4). Suppose further that Assumption 1 holds. Then, O ' ( t ~ m ) - - O p ( T -1/2) and J~(sfi -- m)=op(Kl/2). The first result o f L e m m a 3.1 shows that linear combinations o f ~ determined by coi1~tegrating vectors are consistent estimators o f corresponding linear c o m binations o f m. The order o f consistency is the. usual one, that is, O p ( T - l / 2 ) . Since O'm = p and O'----[In, --A] an estimator o f tt can be constructed from n~ and an estimator o f A. Suppose an estimator A has the usual super consistency property A = A 4 - O p ( T - ! ). Then, if ~ ' = [In, --,4], L e m m a 3.1 implies that ~ ' ~ - - O ' t ~ - - o p ( T -~/2) and e ' t f i is an estimator o f p with the s a m e asymptotic properties as O't~. Using the results in the p r o o f o f L e m m a 3. ! and those in Saikkonen (1992, 1995) it is possible to derive the limiting distribution o f O ' ~ . This result is omitted, however, because it is not needed in this paper. The second result o f L e m m a 3.1 implies that linear combinations o f n~ which are not determined by cointegrating vectors are not consistent estimators o f corresponding linear combinations o f m. In fact, in the direction o f J± the estimator n~ can diverge but only at the slow rate Op(K-m/2). This slow rate o f divergence suffices for our p u r p o ~ s although better results can be obtained with stronger assumptions. For instance, if the lower b o u n d condition ( 1 3 ) is a s s u m e d one can readily show that J~0fi -- m ) - - O p ( l ) (see the p r o o f o f L e m m a 3.1). As a comparison, recall that the estimator _ ~ - m ~ always diverges at the rate O p ( T ) .

104

P. Saikkonen, t~ LuukkonenlJournal o f Econometrics 81 (1997) 93-126

Analogous results have previously appeared in some univariate unit root tests in which the intercept is not identified and therefore its estimator is not consistent (see Schmidt and Phillips, 1992; Elliot et al., 1996). In the present context the inconsistency o f J [ ~ can similarly be thought o f as a consequence o f an identification problem. Now, we can introduce the final test statistics o f this section. Replace the unknown level parameter m in ( 6 " ) by the estimator r~ and consider the leastsquares regression Azt---@m(Zt-1--1~)-~-lTmxt-~mt,

t=K+2,...,T,

(17)

where ~m and /lm----[/Tml --. /lmK] are ordinary least-squares estimators o f the parameters ~P and / / , respectively. Although -~v or any other consistent estimator o f the error covariance matrix could be employed we shall use the estimator based on (17), i.e., T t:-K+2

Let ~ml ~< "'" ~< ~,-n be the solutions o f the generalized eigenvalue problem [@me@" -- AZml--0,

(18)

where ¢~ is defined by replacing Y t - ! in the definition o f C by Yt-i : - z t - i - - t f i and defining x t again as x t = [Az~_ ! ... Az~_K]'. With these definitions we can now define test statistic //2

W m - - Y~ ~mj j=!

which is an obvious analog o f test statistic W defined in the previous section. An analog o f the previous likelihood ratio type test statistic can correspondingly be defined as n2

LRm = N

log( 1 + j=l

It is worth noting, however, that even if the data are generated by a Gaussian V A R process o f a fixed and finite order K + 1 test statistic LRm is not a proper likelihood ratio test statistic. In this respect it differs from test statistics LR and LR~. The obvious reason is that the estimators in (18) are not m a x i m u m likelihood estimators although they can be interpreted as approximate m a x i m u m likelihood estimators and therefore test statistic LRm can be motivated as an approximate likelihood ratio test statistic. The limiting distributions o f test statistics W,, and LR,n are given in the following theorem.

P. Saikkonen. 1~ L u u k k o n e n l J o u r n a l o f Econometrics 81 ( 1 9 9 7 ) 9 3 - 1 2 6

T h e o r e m 3.3. U n d e r t h e c o n d i t i o n s o f T h e o r e m ~(n2) a s T - - , o o .

3.2, W,n

;.

105

~ ( n 2 ) a n d LRm

Thus, under the null h y ) o t h e s i s test statistics W,,, and LR., have the same limiting distribution as their analogs W and L R obtained w h e n the value o f the level parameter m is known. The limiting distribution is therefore different from that o f test statistics Wv and LRv and the associated asymptotic critical values are considerably smaller than f~ose o f test statistics Wv and LRv (see e.g. Reinsel and Ahn, 1992). This suggests that tests based on test statistics W., and LR., should be m.qre powerful than tests based on test statistics W~ and LR,,. T h e simulation results o f the next section show that in general this is indeed the case. We close this section with a discussion o f some related test procedures. Referees have c o m m e n t e d on the relation between test statistic LRm and the likelihood ratio test statistic obtained in T h e o r e m 2.2 o f Johansen ( 1 9 9 1 ) in the context o f a Gaussian finite-order V A R process (see also Johansen and .luselius, 1990). In our case this test is related to testing the joint hypothesis qJ = O O ' and v - - - O / t in (6'). It is not difficult to see that this test can be based on the generalized eigenvalue problem

(19)

I q ' * c * q'*' - ,,t$',,I = o , where ~'* --[~v ~] and T c*

,i

=

=;_,z,_, t=K+2

T -

E t=K+2

T

t(

\

t

/ -I

T t=K+2

Xt~t_ I

with z t = [ z ~ !]'. If A ~ < - - - ~ < ' ~ n are the solutions o f ( 1 9 ) then this approach yields the test statistic n2 LR* = N ~ log( 1 + ~ / N ) . j=l

It w a s pointed out by a referee that, using results from partitioned regression, one readily f n d s that solving the generalized eigenvalue problem ( 1 9 ) is equiva:~nt to performing a reduced rank regression on Azt = Lt'(zt_m - ffzv) + l l x t + ~t,

t = K + 2 , . . . , I",

(20)

~--i where ~v =--~Pv ~ Thus, the difference between test statistic LR* and our test statistic LR,. is that different estimators are used for the parameter vector m. As we already noted, the estimator ~,, has the undesirable property that it diverges in the direction o f J ± at the fast rate Op(T). This implies that the limiting distribution o f test statistic LR* is different from that o f test statistic LR,. or from what is obtained w h e n the value o f m is known. In the case o f a Gaussian finite

106

P. Saikkonen. P~ LuukkonenlJournal o f Econometrics 81 (1997) 93-126

order V A R process the limiting distribution o f test statistic LR* is obtained in Theorem 2.2 o f Johansen (1991) and we would expect that the same result also applies in our more general context, although no formal p r o o f will be provided. The above derivation o f test statistic LR* is interesting, however, because it motivates the search for alternative estimators o f m whose properties in the direction o f J_L are better than those o f ~v. The estimator ~ used in this paper is one possibility in this respect although other estimators, like iterated versions o f n~, might also be considered. It can be seen from the appendix that the results o f Theorem 3.3 do not change if n~ is replaced by any other estimator ~* with the properties O ' ( ~ * -- m ) - - Op(T -1/2) and J_L(m* -- m ) - - o p ( K l / 2 ) . T h i s implies, for instance, that any estimator n~* satisfying t~* - - i ~ - - O p ( T - I / 2 ) can be used in place o f t~. Other related test procedures have been obtained in the univariate special case n = n2 = 1. First note that in this case test statistic Wm provides a new test for testing a unit root in univariate infinite-order autoregressive processes. When stationary alternatives are considered one should use the obvious square root o f Wm, however, because in that case the testing problem is one sided. The test obtained in this way is similar to the two step version o f a test developed by Pantula et al. (1994) for the first-order Gaussian autoregressive process. These authors make the same initial value assumption as in (16) and base their test on the m a x i m u m likelihood estimator o f the autoregressive parameter. They also discuss extensions to higher order autoregressive processes. Another related univariate test is the D F - G L S ~' test o f Elliot et al. (1996) which is asymptotically equivalent to the test based on the above-mentioned special case o f test statistic Win. Elliot et al. (1996) show that their test has optimal power in large samples. There are two major differences between the approach o f these authors and the present one. First, when Elliot et ai. (1996) estimate the level parameter by generalized least squares they do not use an estimator o f the autoregressive parameter but employ a chosen value local to unity. This is feasible in the univariate case bat seems difficult to apply in the present context because the counterpart o f the autoregressive parameter is the matrix q& The second difference is that Elliot et al. (1996) do not take the short-run dynamics o f the process into account in the generalized least-soLuares estimation o f the level ~ e ~ r . In our case this would mean using d z t -- ~Pvzt-I and --~'v in place o f dt and Ct, t >12, respectively.

4. Simulation study In this section we will report results o f a limited simulation study which examines the finite sample sizes and powers o f the test procedures described in the previous section. The two models used in this simulation study are take~ from a recent paper o f Yap and Reinsel (1995). One o f them is a second order V A R

P. Saikkonen. R. Luukkonen/Journal o f Econometrics 81 (1997) 93-126

107

model and the other one a first-order A R M A model. 3 Since in practice intercept terms are almost a l w a y s included in cointegration relations, the test procedures o f Section 3.2, w h i c h are designed for this case, are only considered. In al! experiments w e used the sample size T = 2 0 0 and the nominal 5 % significance level. The n u m b e r o f replications w a s a l w a y s 10000. In the generated series the n o r m a l l y distributed innovations ~t w e r e obtained b y using a r a n d o m n u m b e r generator in the N A G subroutine library on a V A X 8800 c o m p u t e r at the University o f Helsinki. 4.1. A s e c o n d - o r d e r

JAR

process

In our first experiments a trivariate second order autoregressive process w a s considered and the series w e r e generated b y A z t -~ tllzt_! d - F l l A Z t _ l

d-e.t,

t = 1,...,200,

w h e r e zo = z - ! - - 0 and Et ,-~ n.i.d.(0,2~). The p a r a m e t e r matrices follows: [--0.080 0.224 --0.152"] [ 0 . 4 7 0.20 Ill= 0.177 0.046 -0.254 / , E= 0.2_'? 0.32 0.000 --0.102 0.129.1 0.18 0.27

(21) are defined as 0.18] 0.27 0.30

and ~u = C d i a g ( ~ l , ~ 2 , ~ b 3 ) C -1 -- 13 with C -~ =

--0.29 --0.01 -0.75

--0.47 --0.85 1.39

--0.57] 1.00 . --0.55

F o l l o w i n g Yap and Reinsel ( 1 9 9 5 ) , w e considered three different cases described in Table 1 where, in each case, the first r o w (a) corresponds to the situation u n d e r the null hypothesis whereas the second and third r o w s ( b ) and (c) correspond to situations under the corresponding alternative hypothesis. Thus, the value ~bi = 1 indicates a unit root in the process and the n u m b e r o f the ~i parameters less than unity indicates the n u m b e r o f cointegration relations. O n e w o u l d expect that cases w h e r e the value o f a ~b~ p a r a m e t e r is close to but b e l o w unity are difficult for the tests because then the process contains a l o n g - m e m o r y stationary c o m p o n e n t w h i c h is not easy to separate f r o m nonstationary int~;grated components. For each case and each replication, w e c o m p u t e d values o f test statistics Rv, W~, LRv, Wm and LRm for testing the null hypothesis H ( n l ) . Three different values o f the order p a r a m e t e r K w e r e considered. T h e y w e r e K = 1,3 anti 5.

3 These models were taken from a discussion paper version of Yap and Reinscl (1995). Unfomtnately, the models given in the discussion paper were slightly different from those in the published article although the reported simulation results were the same in both cases. This apparently explains the discrepancy which can be seen between some of our results and those reported by Yap and Reins¢l (1995).

~. Saikkonen, R. LuukkonenlJournal o f Econometrics 81 (1997) 93-126

108

Table I Three cases of model (21) 01

02

03

H(n! )

H(nl )

1.0 0.9 0.7

0.8 0.8 0.8

0.7 0.7 0.7

nl = 2 nl ~ 2

nl = 3 n! -~ 3

nj = 2

n~ = 3

Case 2 (a) 2 (b) 2 (c)

1.0 1.0 1.0

1.0 0.9 0.8

0.7 0.7 0.7

n! = I nl = I nl = i

nl i> 2 n! > / 2 nl i> 2

Case 3 (a) 3 (b) 3 (c)

!.0 I.D 1.0

1.0 1.0 i.0

1.0 0.9 0.8

nl = 0 nl = 0 nl ----0

nz i> 1 nl /> 1 nl 1> I

Case ! (a) I (b) I (c)

The value o f K or the order o f the autoregression fitted to the levels series is indicated by writing Rv(K + 1 ) and similarly for the other test statistics. The value K = 1 corresponds to the true order o f the process whereas the two other values correspond to overspecified orders. The results are presented in Table 2 where the cases indicated by (a) display the empirical sizes o f the various tests based on critical values given in Table 1 o f Reinsel and Ahn (1992). The cases indicated by (b) and (c) display the corresponding powers which are size adjusted, based on simulated critical values obtained from the cases indicated by (a). The empirical sizes o f the likelihood ratio type test statistics LR~ and LRm are reasonable except in Case 3 where the rejection frequencies are somewhat too high. This is particularly the case when the autoregressive order is overspecified. The size properties o f test statistic W~ and W.; are generally inferior to those o f test statistics LRv and LR,~ and the size distortions o f test statistic Rv are so large that this test statistic cannot be rec3mmended for general use. The empirical powers o f Table 2 show that the new test statistics Wm and LRm are more powerful than their previous counterparts W~ and LRv. In several cases the superiority o f test statistics Wm and LRm is also very clear. The empirical powers o f test statistic Wm and LRm are always very close to each other and the same is true for the empirical powers o f test statistics ~K. and LRv. There are cases where test statistics Rv is the most powerful one but, due to the serious size problems o f this test statistic, results o f its power can be considered as rather insignificant. 4 The choice o f the autoregressive order has an expected effect on

4 O f c o u r s e , the a p p r o a c h u s e d for test statistics R and Rv c o u l d also b e a p p l i e d in the c o n t e x t o f m o d e l ( 6 H) to obtain a test statistic a s y m t o t i c a l l y e q u i v a l e n t to Wm a n d LRm. W e w o u l d e x p e c t the finite s a m p l e p o w e r o f this test statistic to b e a l w a y s c o m p a r a b l e to that o f test statistic Rv a n d o R e n c o n s i d e r a b l y grea,~,r. H o w e v e r , s i n c e its size properties w o u l d p r e s u m a b l y not b e m u c h better than t h o s e o f test statistic Rv w e d e c i d e d n o t to c o n s i d e r this possibility.

P. Saikkonen, R. Luukkonen/Journal o f Econometrics 81 (1997) 93-126

109

Table 2 Empirical sizes and size adjusted powers of various test statistics for testing cointegration in model (21) at the nominal 5% significance level Case Test statistic

l(a)

2(a)

3(a)

l(b)

2(b)

3(b)

I(c)

2(c)

3(c)

Rv(2) W,(2) LRv(2) Win(2) LRm(2)

0.063 0.056 0.052 0.058 0.057

9.094 0.073 0.063 0.07 ! 0.064

0.132 0.084 0.064 0.095 0.077

0.804 0.784 0.784 0.982 0.982

0.701 0.700 0.693 0.850 0.849

0.341 0.303 0.302 0.372 0.367

0.999 0.999 0.999 !.000 1.000

0.991 0.981 0.980 0.980 0.980

0.866 0.796 0.784 0.823 0.817

Rv (4)

0.070

O. ! 38

0.260

0.737

0.615

0.3 ! 5

0.991

0.952

0.789

Wv(4) LR,,(4) LRm(4)

0.053 0.050 0.057 0.056

0.074 0.063 0.070 0.063

0.104 0.078 0.I I I 0.091

0.664 0.664 0.942 0.942

0.518 0.515 0.696 0.696

0.242 0.242 0.283 0.281

0.976 0.976 0.994 0.994

0.834 0.828 0.911 0.91 !

0.593 0.585 0.635 0.626

Rv(6) Wv(6) LR,(6) Win(6) LR,n(6)

0.079 0.050 0.045 0.054 0.053

0.195 0.078 0.065 0.074 0.065

0.408 0.126 0.096 0. ! 33 0.110

0.671 0.551 0.551 0.876 0.876

0.539 0.368 0.369 0.531 0.530

0.282 0.182 0.182 0.224 0.225

0.965 0.888 0.888 0.972 0.972

0.875 0.600 0.595 0.767 0.765

0.702 0AI7 0.409 0.45 ! 0.449

Win(4)

t h e p o w e r p r o p e r t i e s o f t h e tests. A n o v e r s p e c i f i c a t i o n o f t h e o r d e r g e n e r a l l y w e a k e n s t h e p o w e r s o f all t h e tests. 4.2.

A f i r s t - o r d e r ARA'IA p r o c e s s

In this section we c o n s i d e r a trivariate A R M A process. T h e series are g e n e r a t e d

by Azt = q J z t - i - I - E t - F i e f - l ,

t

=

1,...,200,

(22)

w h e r e a g a i n z o = z - i = 0 a n d ~t--~n.i.d.(0,2~). T h e p a r a m e t e r m a t r i c e s ~P a n d Z a i c a s d e s c r i b e d i n t h e p r e v i o u s s e c t i o n f o r m o d e l ( 2 1 ) w h i l e /~ = C~ d i a g ( 0 . 2 9 7 , - - 0 . 2 0 2 , 0 ~ ) C ~ -I w i t h 0 r = - - 0 . 5 , 0 , 0 . 5 , and

Cy--

--0.816 --0.624 --0.488

--0.657 --0.785 0.475

--0.822 1 0.566[. 0.174.1

F o r e a c h r e p l i c a t i o n , t h e t e s t s t a t i s t i c s c o n s i d e r e d i n S e c t i o n 4.1 w e r e c o m p u t e d w i t h the s a m e c h o i c e s o f the order p a r a m e t e r K. In this case no 'true" v a lu e of K exists and an appropriate choice apparently depends on the strength of the moving average part of the process. The elements of the diagonal matrix in the

! 10

P. Saikkonen, 17, LuukkonenlJournal o f Econometrics 81 (1997) 93-126

definition o f F! measure this aspect of the process. The closer they are to unity in absolute value the stronger the moving average part is and the larger the order of the approximating autoregression has to be to provide a good approximation. The moving average part is moderately strong when Ov = :i:0.5 and rather weak when ~b~ ----0. The results are displayed in Tables 3-5. Again the cases indicated by (a) display the empirical sizes of the various test statistics based on critical values given in Table 1 of Reinsel and Ahn (1992) whereas the cases indicated by (b) and (c) display the corresponding size adjusted powers. The empirical sizes vary depending on the parameter configurations and the value of the autoregressive order. A general conclusion is that the size properties of test statistic Rv are again very poor. Except for Case 3 the empirical sizes of the likelihood ratio type test statistics LRv and LRm look reasonable provided the autoregressive order is chosen properly. The size properties of test statistics Wv and Wm are again somewhat inferior to those of test statistics LRv and LRm. Choosing two lags to the autoregressive approximation is always too little. In Cases 1 and 2 four and six lags work reasonably well except for test statistic Rv. In Case 3 the results get worse when the number of lags is increased fi'om four to six. This indicates that these size distortions cannot be removed by a proper choice of the autoregressive order. Table 3 Empirical sizes and size adjusted powers o f various test statistics for testing cointegration in Case 1 o f model (22) at the nominal 5% significance level (O! = !, 0.9, or 0.8, ~2 = 0.8, qb3 = 0.7) (a) Ol = 1

Test statistic

( b ) 01 = 0 . 9

( e ) O! = 0.8

--0.5

0.0

0.5

--0.5

0.0

0.5

--0.5

0.0

0.5

0.095 0.084 0.078 0.087 0.085

0.098 0.084 0.079 0.090 0.088

0.099 0.085 0.080 0.096 0.094

0.889 0.870 0.870 0.987 0.987

0.889 0.869 0.869 0.985 0.985

0.892 0.872 0.872 0.983 0.983

0.996 0.996 0.996 0.997 0.997

0.997 0.996 0.996 0.997 0.997

0.996 0.996 0.996 0.998 0.998

Wv(4) LR~.(4) Win(4) LRm(4)

0.075 0.053 0.048 0.061 0.059

0.079 0.053 0.050 0.063 0.061

0.081 0.055 0.050 0.063 0.062

0.790 0.702 0.702 0.948 0.948

0.796 0.706 0.706 0.946 0.946

0.802 0.717 0.7 ! 7 0.947 0.947

0.984 0.965 0.965 0.988 0.988

0.985 0.966 0.966 0.989 0.989

0.984 0.968 0.968 0.989 0.989

Rv(6) Wv(6) LRv(6) W,,,(6) LRm(6)

0.085 0.046 0.041 0.053 0.052

0.084 0.046 0.043 0.055 0.054

0.085 0.046 0.043 0.055 0.054

0.712 0.565 0.565 0.890 0.890

0.716 0.581 0.581 0.881 0.881

0.738 0.597 0.597 0.875 0.875

0.953 0.870 0.870 0.965 0.965

0.955 0.882 0.882 0.964 0.964

0.962 0.893 0.893 0.960 0.960

Rv(2) Wt,(2) LRv(2)

Win(2) LRm(2)

Rv(4)

P. Saikkonen. i~ Luukkoneni Journal o f Econometrics 81 (1997) 93-126

111

Table 4 Empirical sizes and size adjusted powers o f various test statistics for testing cointegration in Case 2 o f model (22) at the nominal 5% significance level (41 = I, 42 = !, 0.9, or 0.8, 43 = 0.7) (a) 42 = I

( c ) 42----0.8

( b ) 4 2 = 0.9

Test statistic

--0.5

0.0

0.5

--0.5

0.0

0.5

--0.5

0.0

0.5

Rv(2) W,(2) LRv(2) W'm(2) LR,,,(2)

0.115 0.085 0.073 0.095 0.083

0.123 0.092 0.080 0.095 0.084

0.124 0.091 0.077 0. i 07 0.093

0.31 ! 0.321 0.319 0.438 0.439

0.339 0.362 0.361 0.503 0.506

0282 0.317 0.321 0.447 0.446

0.872 0.871 0.867 0.925 0.925

0.882 0.884 0.881 0.935 0.935

0.870 0.882 0.880 0.935 0.934

Rv(4) Wv(4) LRv(4) W,,,(4) LRm(4)

0.138 0.071 0.060 0.073 0.064

0.141 0.072 0.062 0.072 0.064

0.151 0.076 0.063 0.075 0.068

0.312 0.272 0.270 0.406 0.403

0.316 0.284 0.283 0.432 0.431

0.294 0.276 0.273 0.420 0.421

0.796 0.696 0.688 0.822 0.821

0.805 0.703 0.696 0.828 0.826

0.808 0.728 0.720 0.832 0.832

Rv(6) ;~,,(6) LRv(6) Win(6) LRm(6)

0.195 0.069 0.057 0.073 0.064

0. ! 96 0.073 0.061 0.071 0.062

0.209 0.081 0.069 0.073 0.065

0.284 0.225 0.225 0.327 0.327

0.288 0.235 0.234 0.358 0.357

0.282 0.223 0.224 0.355 0.354

0.697 0.507 0.501 0.648 0.647

0.707 0.518 0.513 0.674 0.672

0.729 0.534 0.532 0.699 0.697

A s to the empirical powers in Tables 3-5, the general picture is very similar to that obtained from Table 2. In most cases the n e w test statistics W and L R . are more powerful than their competitors and often by a fairly wide margin. Test statistic Rv is most powerful in some cases but its poor size properties make this result rather meaningless, at least from a practical point o f view. In Case 3(c) test statistics Wv and LRv are sometimes more powerful than the n e w test statistics Wm and LR,,, but their superiority is not great.

5. C o n c l u s i o n This paper has studied various test procedures w h i c h can be used for testing the cointegrating rank in infinite order V A R processes. The associated asymptotic distribution

theory

was obtained

by weakening

previous

assumptions

of the rela-

between the sample size and selected order o f the V A R process. Some o f the considered tests were straightforward extensions o f previous likelihood ratio tests or their close analogs obtained for finite order Gaussian V A R processes. In addition to these extensions s o m e entirely n e w tests were also d e v e l o p e d for tion

112

P. S a i k k o n e n . 17,. LuukkonenlJournal o f Econometrics 81 ( 1 9 9 7 ) 9 3 - 1 2 6

Table 5 Empirical sizes and size adjusted powers o f various test statistics for testing cointegration in Case 3 o f model (22) at the nominal 5% significance level (01 = 02 = I, 03 = 1, 0.9, or 0.8) ( a ) 03 = i

( b ) 03 = 0.9

( c ) 03 = 0 . 8

0"~

0~

0~

Test statistic

--0.5

0.0

0.5

--0.5

0.0

0.5

--0.5

0.0

0.5

Rv(2) W~.(2) LR,(2) Win(2) LR,,(2)

0.190 0.130 0. 100 0.135 0. 110

0.160 0.104 0.080 0.I 14 0.090

0.374 0.235 0.195 0.258 0.222

0.355 0.374 0.372 0.468 0.464

0.273 0.270 0.270 0.324 0.322

0.582 0.579 0.570 0.590 0.584

0.887 0.884 0.875 0.926 0.922

0.831 0.80 i 0.790 0.837 0.832

0.998 0.996 0.994 0.950 0.948

Rr(4) W~(4) LRv(4) W,n(4) LRm(4)

0.274 0.106 0.081 0. ! 10 0.086

0.265 0.100 0.077 0. ! 07 0.089

0.341 0.109 0.082 0.122 0. 100

0.297 0.262 0.260 0.332 0.330

0.273 0.232 0.232 0.275 0.273

0.600 0.470 0.466 0.496 0.490

0.760 0.635 0.624 0.724 0.715

0.766 0.613 0.603 0.659 0.651

0.993 0.933 0.927 0.864 0.859

Rv(6) W,(6) LRv(6) Win(6) LRm(6)

0.420 0.125 0.094 0.127 0.105

0.409 0. I24 0.091 0.129 0.105

0.465 0.119 0.090 0.130 0.107

0.294 0.187 0.189 0.250 0.248

0.246 0.182 0.183 0.218 0.217

0.535 0.343 0.338 0.372 0.368

0.647 0.425 0.421 0.515 0.506

0.675 0.427 0.422 0.472 0.465

0.978 0.748 0.733 0.674 0.666

m o d e l s with intercept terms included in the cointegrating relations. A limited simulation study was performed to investigate the finite sample properties o f the considered tests. The main findings o f this simulation study were the following. (i) The likelihood ratio type tests had better size properties than their asymptotically equivalent alternatives. However, our results also indicate that one can always find cases where even these tests have problems with their size. In particular, it is possible that the null hypothesis is rejected quite too frequently. (ii) The test based on the approach used in Saikkonen ( 1 9 9 2 ) had severe size problems so that tests o f this type are not r e c o m m e n d e d for general use. (iii) The n e w tests developed in the paper were generally more powerful than their competitors. Quite often their superiority w a s considerable although there were cases where the power o f the n e w tests and their previous alternatives was about the same.

M a t h e m a t i c a l appendix W e shall m a k e use o f results already e m p l o y e d in Saild
P. Saikkonen, 1~ LuukkonenlJournal of Econometrics 81 (1997) 93-126

113

the lower bound condition (13). When these results are applied it will be understood that they are valid without the lower bound condition even if this is not always explicitly mentioned. Since some o f the subsequent derivations are rather long and tedious they will not be given in full detail. A more detailed treatment is available upon request. Before proving Theorem 3.1 some general remarks, also relevant for other proofs, will be made. First recall that we can assume throughout that the triangular error correction form is as in (3). Unless otherwise stated, it will be assumed that all partitions o f vectors and matrices are conformable to that o f yt in (1). The notation Bi.y = B ~ i - BoBj~tByi is used for any positive-definite matrix B = [Bij]i,j=l,2. The symbol H" [[ is used for the Euclidean norm so that, for an arbitrary matrix B, IIBII = [tr(B'B)] !/2. By IIBII, we mean the operator norm o f the matrix B, that is, the square root o f the largest eigenvalue o f B'B. These norms are connected by the well-known inequality

lIB,B211 ~ liB, II lIB211,

(A.1)

which holds for any conformable matrices Bl and B2. The transformation matrix

,.] will be employed in several places. For instance, we have

= ÷ a -~ = [÷,

~2]

(A.2)

where ---@2= q'IA_A_+ @2. Furthermme, under the conditions o f Theorem 3.1, tJ~'l -- ¢i~4- o p ( K - I/2 )

(A.3)

-~--~2-- Op( T - ! )

(A.4)

,~ -- ~' -4-op(K -1 )

(A.5)

and

(see Lemma A3 and the proof o f "~hc,Jrem 3.1 o f Saikkonen, 1995). If the lower bound condition (13) is assumed ~ahe order terms in (A.3) and (A.5) can be improved to O p ( ( K / T ) 1/2) and Op,[T-l/2), respectively [see Theorem 3.2 and result (A.14) in Saikkonen (1992):and Theorem 3 o f Saikkonen and Lfitkepohl (1996)]. Analogs of ( A . 3 ) - ( A . 5 ) ere also valid when obtained fi'om the least squares regression (14) which incl~les an intercept (see Saikkonen, 1995).

P r o o f o f Theorem 3.1. Denote 9 = [E~]~j=,,2 = ÷ ' ~ - ~ ~'

P. Saikkonen. IL L u u k k o n e n l J o u r n a i o f Econometrics 81 (1997) 93-126

114

and note that ,~1 ~< "'" ~< 2",, are the eigenvalues o f the matrix C v. The matrices C and 17 are transtbrmed as (7 = [__C,~]i.j=t.2 = A C A '

and

I2 = [__##]i.j=l.2 = ___~,~-1__~

where ~ is as in (A.2). This transformation does not change eigenvalues so that the matrices C V and C # have identical eigenvalues. W e shall first consider asymptotic properties o f the matrix (7. F r o m the definitions it follows that C = M -- G where G = [Gij]i.j=l,2 = t=K+2 ~

ul,t-! Y2,,-,

x,,

~,

t=K+2

x, xtt

~

t=K+2

t xttu~.t_, Y2.t-t]

and M = [Mij]i./=L2 = A M A ' is the matrix o f s u m s o f squares and cross products o f the c o m p o n e n t s o f [ U i , t _ Y2,t--lt y ( t = K + 2, . . . , T). B y ( 2 ) w e have

A y t = A - "I

- [ Au2t ult]

Therefore,

xt = Hqt U t-K ¢ U tl.t_K_t] ~ and H ( n K x ( n K + n l ) ) where qt = [ u ~ _ 1 . . . o b s e r v a b l e ) transformation matrix o f full r o w rank. Denote

T

Sqq = N - '

is a suitable (un-

T

E

q,q~

and

S,q = N - '

t=K+2

~E ",.,-,qL t=K+2

Then, i f ~rqq =Eqtq~ and 2~lq - - E u l . t - l q ~ , w e have

[ISqq -- Z~,qq ][I - - OP ( u / r l / 2 )

(A.6)

llS|q -- ~,lq [[ = O p ( ( g / T ) l / 2 ) .

(A.7)

and

These results can be obtained from the p r o o f o f L e m m a A.2 o f Saikkonen (1991 ). In the s a m e w a y one also obtains

Ii

IIS2qtl ~ f N - ' ~ t=K+2

Y2,t--lq~

II

= Op((l/2)

-

(A.8)

N o w , w e can study asymptotic properties o f the matrix (7. W e start with mull and observe that

N - ! Gt i -- S t q H ' ( H ~ , q q H ' ) - I HS~q = - - S l q H ' ( H ~ q q H ' ) - 1H[Sqq -- ~qq]Ht(HSqqH ' ) - i HS;q.

(A.9)

P. Saikkonen, R. LuukkonenlJournal of Econometrics 81 (1997) 93-126

115

Our assumptions imply that the eigenvalues o f the matrix ~E~ are bounded and bounded away from zero. Hence, from L e m m a A.2 o f Saikkonen and Lfitkepohl (1996) it follows that an analog o f (A.6) also holds for the corresponding inverses and, from L e m m a A . l o f the same paper, one therefore obtains

IIH'(HZqqH')-~HIIE = O(I)

and

IIH'(HSqqH')-IHIll = Op(1). (A.|0)

Furthermore, since the covariance function of u, is absolutely summable, [[Z'lq[[ is bounded uniformly in K and it follows from (A.7) that

llSlqII -

op( l ).

(A. l X)

Now, using (A.10) and (A.II) in conjunction with (A.6) and (A.I) one can see that the Euclidean norm of the fight-hand side of (A.9) is of order Op(K/Tl/2)--Op(1). From (A.7) it similarly follows that the error of replacing S~q by Z1q in the second term on the leR-hand side of (A.9) is of order Op(1). Finally, by the law of large numbers, N-IMIj = E t l l , t _ l u tl,t-! + Op(l) SO that, since C 1! -- M_M_! ! -- Gi i, we can conclude that

N-IC--ll : EUl,t-! Uti,t-! -- ~ l q H ' ( H ~ , q q H t ) - l H ~ q ":t-op(l).

(A.12)

The two first terms on the right-hand side o f (A.12) define the covariance matrix o f the prediction error obtained by linearly predicting ul,t--! by Hqt or, equivalently, by xt. The same prediction error is obtained by linearly predicting u~,r-i by { A u l , t - j , u2,t-j, j "- 1. . . . . K}. As K--~ oo, the smallest eigenvalue o f the covariance matrix o f this prediction error must be bounded away from zero since this is the case for Eqq, the covariance matrix o f the random vector qt = [u,_' l . . . utt_K utl,t_r_l ]'. Hence, we have shown that, with a probability approaching one,

N-ICll

>~Int,

~ > 0.

(A.13)

Since N - I G l 2 - - S l q H ' ( H S q q H ' ) - I H S ~ q arguments similar to those used to obtain (A.13) can also be used to show that N - I G 1 2 : O p ( K l / 2 ) . Hence, since N-IM___12 - - O p ( l ) [see Saikkonen (1992, p. 26)] and C i e -----Ml2 + Gl2, we have N - I C I 2 : Op(gl/2).

(A.14)

From N -2 G22 = N - 1SzqH'(HSqqH')- !H S ~ it similarly follows that [ I N - 2 G ~ [[ -Op(K/T). Thus, if we partition M--[Mij]i,j=l,2, we have M2z = M g z and N-2CC22 -- N-2M22 -{- O p ( ( g / T ) !/2 ).

(A. 15)

Here the error term could be O p ( K / T ) but the above result will be convenient in the proof o f Theorem 3.3. The limiting distribution o f test statistic W can now be derived by using the above results and arguments similar to those in the p r o o f o f Theorem 4.1 o f

116

P. Saikkonen, R. LuukkonenlJournal o f Econometrics 81 (1997) 93-126

Saikkonen (1992). First notice that ~l ~< "'" ~<~n are the eigenvalues o f the matrix C ~" in the metric X and

From (A.3), (A.4), (A.14) and (A.15) one obtains N - I ~ C W I = N - - l ~ i C l l ~ + Op(1). Hence, using (A.3), ( A . 1 2 ) and ( A . 1 3 ) w e can show that, with a probability approaching one, the n i largest eigenvalues o f N -m V'C ~ ' are b o u n d e d and b o u n d e d a w a y from zero. Noticing ( A . 5 ) w e can thus conclude that ~.y, j = n2 + 1. . . . . n, tend to infinity at the rate Op(N). Let O_L(n X n2) be a matrix such that ~ _ O ± = I,,~ and ~'L W! ---0. Then, since N - 2 M 2 2 converges weakly to a m a trix which is positive definite (a.s.) [see ( A . 1 0 ) in S a i k k o n e n (1992)] we can use ( A . 4 ) and ( A . 1 5 ) to show that q~ W C ~"O_L = O p ( l ) . This result and the Poincare s e p a r a t i o . theorem (see Rao, 1973, p. 64) imply that ) V - O p ( l ), j - - 1 . . . . . n2. In the san2e w a y as in the p r o o f o f T h e o r e m 4.1 o f Saikkonen ( 1 9 9 2 ) we n o w consider ~.1 ~< "-- ~<~.~ as solutions o f the generalized eigenvalue problem [ P - - ) , C - I [ = 0 . Denote E_*(A)= [ff__~-j(g)]= ~ - A C - t and note that, w h e n IETI(A)I 0, the equality 1F_*(A)I- 0 can be expressed as

IFT~(A)I IF~.~(),)I-

0.

C o n s i d e r the nl solutions o f tFT,(A)I = 0 and notice that

F___*I I(3,) -----@I,~-I @l -- :.CT.t2 -- /?l, -- :tCT-t~, where /?ll--¢/2~--lqb + °p(I) by (A.3) and (A.5). F r o m (A.14) and (A.15) it also follows that C -+- O p ( K / T ) . N - 1 C t - 2 -- N - ' --11

(A.16)

Using this result, (A.12), and ( A . 1 3 ) one can show that the nl solutions o f I F ~ I ( A ) I = 0 tend to infinity at the rate Op(N). 5 N o w recall that we showed above that the n2 smallest solutions o f IF*(A) I - - 0 are o f order Op(1 ). Therefore, asymptotically they cannot solve IF_h(~.)l = 0 and w e can consider ~.j, j = 1. . . . . n2, as solutions o f IF_~.,(~.)I = 0 or, equivalently, [N2F~.I(A)[ = 0 . Furthermore, in the analysis below we can assume that A = O ( I ). By L e m m a A.2 o f Saikkonen and Liitkepohl ( 1 9 9 6 ) the error terms in ( A . 1 5 ) and ( A . 1 6 ) do not change w h e n the involved matrices are replaced by the corresponding inverses. This fact in conjunction with ( A . 1 3 ) and ( A . 1 4 ) shows that

5 N o t i c e that, unlike stated in S a i k k o n e n ( 1 9 9 9 , p. 26), o n e c a n n o t directly c o n c l u d e from this that Aj = O p ( N ) , j : n2 + ! . . . . . n. A similar r e m a r k c o n c e r n s the related p r e v i o u s statement a b o u t t h e smallest e i g e n v a l u e s Aj, j ---- 1. . . . . n2. T h e r e f o r e the p r e s e n t p r o o f differs f r o m the p r e v i o u s o n e at t h e s e points.

P. Saikkonen. R. Luukkonen/Journal o f Econometrics 81 (1997) 93-126

117

C_.-~C21C-~.~=Op(KI/2/T2). Thus, as A = O ( I ) can be assumed, we can use the definitions and the inversion formul~ of a partitioned matrix to obtain =

-

=

÷

Op(f'/2/r2).

By similar arguments we also have F~l (A) = ~11--),C~.J = TY~ + O p ( T - l ). Hence, Lemma A.2 of Saikkonen and Lfitkepohl (1996) gives =

+ Op(Y

).

As for F~2(A ), it follows from ( A . 1 3 ) - ( A . 1 5 ) that N-2C2.t = N - 2 M 2 2 - [ Op((K/T) 1/2) so that, using Lemma A.2 of Saikkonen and Lfitkepohl (1996) again, F~2('~) = ---.V22 -- ~C~.] = ----V22-- ~ I r ~ i - I - o p ( T - 2 ) .

Thus, since ~ 2 t -- . ~ - 1 ~ , --Op(T - I ) and __VHI---(~J~-I~I)-I = O p ( l ) , we can conclude from the above discussion t h a t N2F~.I(~,)=N2(__V2.1 - - , ~ [ 2 2 1 ) -[op(l ). Hence, by the continuity of eigenvalues, W --- t r ( M 2 2 ~ 2. I ) -~ Op( 1 ).

It was shown in Saikkonen (1992, pp. 2 6 - 2 7 ) that R = t r ( M 2 2 / 7 2 . 1 ) + op(l) =~ ~(n2). Equation (A.30) of that paper implies that the same result is also obtained here because the previous definition of the matrix • is asymptotically equivalent to the present one and because the limiting dmtribution of the estimator ---~2 does not require the lower bound condition (13), as shown in the proof of Theorem 3.1 o f Saikkonen (1995). Thus, test statistic W has the stated limiting distribution. The asymptotic equivalence of test statistics W and LR discussed below Theorem 3.1 follows from a standard mean value expansion of the logarithmic function. This completes the proof. []

Proof of Theorem 3.2. In the same way as in the proof of Theorem 3.1 we first transform the matrices ~v and C by using the transformation matrix A_ and obtain analogs of ( A . 3 ) - ( A . 5 ) . Next, using arguments in S~:.k~onen (1992, p. 22) it can be shown that analogs of ( A . 6 ) - ( A . 8 ) and furthermore ( A . 1 3 ) - ( A . 1 5 ) hold even in the present context. The proof can then be completed by using arguments entirely similar to those in the proof o f Theorem 3.1 [cf. Saikkonen, 1992, p. 27). Details are omitted. []

Proof of Lemma 3.1. For simplicity, we shall assume the asymptotically negligible initial value conditions Y0 ----0 and vt -- 0, t ~ 0. For notational convenience the subscript v will be suppressed from the estimators B,j, ~,,/1~ and Z, and similarly from quantities derived from them. This should cause no confusion since we shall only use such properties of these estimators which are not affected by the inclusion of the intercept in the model.

P. S a i k k o n e n ,

118

R. L u u k k o n e n l J o u r n a l

o f E c o n o m e t r i c s 81 ( 1 9 9 7 ) 9 3 - 1 2 6

T h e a s s u m e d initial value conditions i m p l y that in ( 1 6 ) w e h a v e e t - - e t for t - - 1 . . . . , K + 1 while for t > ~ K + 2, et is as defined in (6). Thus, replacing B j in ( 1 6 ) b y / } j ( j -- I, . . . , K + 1 ) gives the regression equation d , = C t m + et, t = 1 . . . . .

(A.17)

T,

where t--I

t---- 1 , . . . , K + 1

e.t -- ~ , ( B j -- B j ) Y t _ j ,

j=l

K

et - - (t~.l _

tp)yt_

l _

~

([lj

-- rlj)Ay,_j,

t =K+2

. . . . . T.

j=l

Set ~ , =[_~_,, ~ , 2 ] = C t A _ - ' and, as before, ~ = [~', ---~2] = @~-=- By the dot'initions and the above discussion we can then write A_(th -- m ) = ___.u-i~

(A.I8)

where K+!

and

[]

T

E

Next, w e shall analyze the matrix __U and start by c o n s i d e r i n g the quantities __Cj, j = 2 . . . . . K + 1. Since ¢ ~ j = - ~ ' - / l j - ! w e h a v e C j I = -- ~ul - / l j - ~ , l a n d

~j2 = -----~2 --/~ry-,.z where H j 2 = Hj,A + Hi2. We also define / I , , = @= + HI, and l=ljt--flit- l=lj_,.,, j--2 ..... K. Then H__"- [_~_, ... /IK], where H j -[ ~ j l Rj2], is exactly as in S a i k k o n e n (1992, p. 19) except tbat the a s y m p totically irrelevant restriction /~rK! = 0 has not b e e n used. ( N o t e also the printing error in the p r e v i o u s definition o f / l j l - ) T h e theoretical counterparts o f H j a n d H_0_"are defined in an o b v i o u s m a n n e r a n d d e n o t e d by H j = [_H_jl Hj2] a n d F / = [_H_l . . . / / r ] , respectively. B y L e m m a A . 3 ( i ) o f S a i k k o n e n ( 1 9 9 5 ) w e h a v e

II~- ~II =

op(K-I/2)

o

Since l l j 2 = / / y t A +/-/j2 and Cj2 = -- ---~2 - - / I j - l , 2 f o r w a r d to d e d u c e f r o m this that

(A.19) ( J = 2 , . . . , K + 1 ) it is straight-

K+I

:E II~j211 = Op(l). j=l

(A.20)

P. Saikkonen, 1~ L u u k k o n e n l J o u r n a l o f E c o n o m e t r i c s 81 ( 1 9 9 7 ) 9 3 - 1 2 6

119

As for C j l , we first observe that j--!

~jl = --(~l +/lj-i,t)

-- - - ~

H__-~i,

j -- 2 , . . . , K + 1,

(A.21)

i=l

where the latter equation follows from the definition o f the sequence l l j t . A r g u ments similar to those in the case o f (A.20) show that max

l <_j<~K+!

II--CjlII = Op( i ).

(A.22)

Using the above results in conjunction with ( A . 3 ) - ( A . 5 ) it can now be shown that N-~___011 = ~ , ~ . - l ~ + o p ( l ) , __012= O p ( l ) and ___022= O p ( l ) . Furthermore, i f __0- l = 10iJ]/j=l.2, these results and the inversion formula for a partitioned matrix yield N~_UII -- (~t.E'-140 - ! 4 - % ( 1 )

(A.23)

0___22 = Op( i )

(A.24)

__012 -- O p ( N - 1).

(A.25)

and

To analyze the asymptotic behavior o f the vector ~ an order o f consistency is needed for the estimator B - [ / ~ I ... Blc]. As is ( A . 2 ) o f Saikkonen and Lfitkepohl (1996), define ~,'-L_H_ - //KI] and ="~- L ~ _ - H K,,], and note that I1~- Sll =Op(K'--1/2), by L e m m a A.3(i) o f Saikkonen (1995). From the proofs o f Theorems 1 and 2 of Saikkonen and Lfitkepohl (1996), one also obtains B -- B - - ( ~ -- S ) P I 4- Z where B - - [ B I . . . BK], IlZll---~Op(N-I), and Pi is a nonstochastic matrix ~ach that IlPaill =O(1). Thus, making use o f ( A . I ) , w e find that

l i b - BII =

(A.26)

Using this result and well-known properties of integrated processes it is straightforward to show that

1 <<.t~K+!

j=!

This result can be used to analyze the residual ~t for t---- I , . . . , K 4- 1. For other values o f t we shall use the representation et'-et--~__2Y2.t-l--(~--~-~)qt,

K + 2 . . . . ,T,

(A.28)

which can be obtained from the earlier definition o f et with transformations similar to those above (A.19) [of. (A.2) of Saikkonen (1992)]. Now, from (A.28)

120

P. S a i k k o n e n , !~ L u u k k o n e n l J o u r n a l o f E c o n o m e t r i c s 81 ( 1 9 9 7 ) 9 3 - 1 2 6

and the definitions of ~1 and ~, it follows that =

_

t= 1

t= 1

+

C t t ~,-I y~.

j= 1

-- B j ) Y t - j

+

(A.29) where e, fi2 and ti are the sample means of et, y 2 . t - i and qt ( t = K + 2 , . . . , T ) , respectively. Using of (A.5), (A.22) and (A.27) it can be shown that the first and second terms on the right-hand side of (A.29) are of order Op(I ). From the definition of et below (6) and well-known properties of stationary processes it follows that ~ : O p ( T - l / 2 ) . This result in conjunction with (A.3) and (A.5) implies that the third term on the right-hand side of (A.29) is of order Op(l). Next note that -v2 =Op(TI/2) and I I ~ ] [ = O p ( ( K / T ) !/2) (see Saikkonen 1992, pp. 21-22). Hence, since II~-_=ll--op(K-l/2), it follows from ( A . 3 ) - ( A . 5 ) t h a t the fourth and fifth terms on the right-hand side of (A.29) are of orders O p ( l ) and op(l ), respectively. As a whole we have thus shown that T-I/2~ ! -- Op(l).

(A.30)

Next, con~ider K+l

K+I

t--I

t= 1

t= I

j= 1

T t=K+2

(A.31) The analysis given for the last three terms on the right-hand side of (A.29) readily shows that the third term on the right-hand side o f (A.31) is of order o p ( l ) while the second one is of order op(K !/2) by (A.5), (A.20) and (A.27). Finally, since ~t2 = -----~2- /It-l,2 for t = 2 , . . . , K + 1, we can use (A.4) and (A.19) to show that the first term on the right-hand side of (A.31) is o f order Op(l). Thus, we have h_-2= op(K !/2).

(A.32)

From (A.18) one now obtains O ' ( ) h - m)=O__llh_ I + _U_Ul2~2--Op(r-l/2 ) and J _ ~ ( m - m)---__U21h_'!-~-U____22~2=Op(Kl/2 ) where the order terms are justified by ( A . 2 3 ) - ( A . 2 5 ) , (A.30) and (A.32). This completes the proof. It may be noted, however, that if the lower bound condition (13) is assumed we have H ~ , - 3,[] - - O p ( ( K / r ) 1/2) (see Saikkonen (1992, pp. 20-21) and the right-hand side of (A.26) can be changed to O p ( ( K / T ) l / 2 ) . This implies that the right-hand side of

P. S a i k k o n e n , IL L u u k k o n e n l J o u r n a i o f E c o n o m e t r i c s 81 ( 1 9 9 7 ) 9 3 - 1 2 6

121

(A.27) can be improved to O p ( ( K 3 / T ) l / 2 ) = o p ( l ) and we have h"2 = O p ( l ) . T h e latter result o f the lemma can thus be improved to J_~(n~ - - m ) = O p ( l ) . [] Before proving Theorem 3.3 an auxiliary result will be given. First note that ~ s and/Tm are ordinary least-squares estimators obtained from the model Azt -" ~ f ' t - ! + l l x t + at,

where Yt-~ = z t - t

.

- t~ and a t : e t

[.,,] Y2t

LJr-( ~

(A.33)

t = K + 2 , . . . , T, -t- t~Ot(t~t -- m ) .

-- m )

=

fii2t

Set

"

In the same way as in (A.28) w e can transform (A.33) to Azl = ---~2.P2,t-i + ~q, + at,

t = K + 2 . . . . . T,

(A.34)

where 9' 2 = OA + q~z = O, #, = [~?~t q~,]' with ~]tt = [ u "l , , _ ! - .. u"l , , _ r _ l ] , ' a n d q2t----[u2,t_l

U2,t_K]. The

.-.

coefficient m a t r i x

-- is defined in the same w a y

as in t h e p r o o f o f L e m m a 3.1 except for an irrelevant permutation o f columns. Let ~m and ---~,,,2 be the (infeasible) least squares estimators of = and _~2,respectively, obtained from (A.34). As in Saikkonen (1992, p. 18) we then have 9",,2 = tP,,,O± where O L = [.4' In.]L Next, consider an analog o f (A.34) defined by t=K+2

Azt=qJ2y2,t-l+~.qt+et,

. . . . . T,

(A.35)

! ... 1~tI , t - - K - - l J11 where now qt =[q~t q~t]' and q l t =[Ul,t--I " Thus, (A.35) is derived from (6 #) in the same way as (A.34) is derived from (A.33). Let ~ and ---~2 be the least-squares estimators o f S and qJ2, respectively, obtained from (A.35). Then 9"2 = tPO-L where ~' is now the least-squares estimator o f qJ obtained from ( 6 " ) with m treated as known. N o w we can prove following.

Lemma

A. 1. U n d e r t h e c o n d i t i o n s o f T h e o r e m 3.3, T

N~_.m2 ~- N~___.2 + ~ 0 ' ( ~

- m)N-'

~_. Y ~ , t - , t=K+2

x

Y2,t-lY2,t-t 2

~',nl = • + op(K -1/2) and £,m = Z + o p ( K - t ).

d- Op(l)

122

P. Saikkonen, R. LuukkonenlJournal o f Econometrics 81 (1997) 93-126

/lm--'[~m

P r o o f . Denote fit =[t]~ ,P[.t-l]' and (/Zm -- A ) D r ! =

T ~

t=K+2

atfftDr (Dr ~

T ~

--~--~m2]and note ,hat

)--1

t=K+2

ptfftDr

,

(A.36)

where A = [ ~ ~ ' 2 ] = [ S 0] and D r = d i a g [ N - n / 2 1 n X + n , N - ! In2]. W e shall consider replacing /5t in ( A . 3 6 ) by Pt = [q~ Y2.t' ! ] ' - We can write qlt = ~ " -- (z ® O ' ) ( ~ -- m )

and

f'et = Y2t -- J~_(,'n -- m ) ,

where z ----[1 --- I ] ' ( ( K + 1) x 1). Recall that the sample m e a n s o f qt and Y2,t-t (t----K + 2 , . . . , T ) are o f orders O p ( ( K / T ) 1/2) and Op(Tl/2), respectively, while [ ] ( t ® O ' ) ( , ' ~ - m ) l l = O p ( ( K / T ) t/2) and ( l f i - m ) d _ t _ = o p ( K 1/2) by L e m m a 3.1. Using these results it is straightforward, though s o m e w h a t tedious, to show that

IDrt=~r I T +2 Pt f f D r

T

-- D r _~+2 Pt p [ D r

II

= O p ( ( K / T ) !/2).

(A.37)

Next note that

T

T

t=K+2

t=f +2

--Nl/2~(t~ -- m ) ' ( d ® O ) -- N I / 2 ~ O ' ( r ~ -- m)(t~ -- m ) ' ( d ® O )

(A.38)

where ~ is the sample m e a n o f qlt (t = K + 2 , . . . , T ) . Using a r g u m e n t s similar to those referred to above one can here show that the n o r m o f the left-hand side is o f order O p ( ( K / T ) 1/2) and that the same result holds with t]! t and qlt replaced by q2t. Hence, it follows that

r

N -~/2 ~

at#~ -- N -

,,= ,(-,

.ll =

OP((K/T)'/2)"

(A.39)

From the definition o f et given below (6), ( A . 7 ) o f S a i k k o n e n (1992), and L e m m a A.2(ii) o f Saikkonen ( 1 9 9 5 ) we can conclude that the Euclidean n o r m o f the latter matrix on the left-hand side o f ( A . 3 9 ) is o f order Op((T/K)l/2). Thus, ( A . 3 9 ) yields

ii_, a,q,ll , : 2 ,

(A.40)

Furthermore, by L e m m a 3.1 and the result # = O p ( T - l / 2 ) ,

T

T

N -l ~ "t N-I , a t Y 2 , t - - i -Y ~ etY2,t- 1 -- O 0 ' O f i -- m ) f ~ t=K+2 t----K+2

= --~(r~ -- m ) ' J ± -- ~ 0 ' ( ~ -- Op((K/T)l/2).

-- m)(tfi -- m ) ' J ± (A.41)

P. Saikkonen, R. Luukkonen/ Journal of Econometrics 81 (1997) 93-126

123

Denote T

Dr ~

pip,Dr

= R = [Rij],j=t,2

R - ' = [Rqli,j=l,2,

and

t=K+2

where the partitions are conformable to that o f P t - [q[ y2.t_l] ' ~. L e m m a A.I of Saikkonen (1995) gives IIR'211, ----Op((K/r) '/2) and R 22 - - O p ( l ). Thus, using ( A . 3 6 ) - ( A . 3 8 ) it readily follows that T N~_~_.m2:- N -I/2 ~

T atqttf~12 -[-N -I Y~ atf,,t2,t_l R22 -+-op(l).

t=K+2

(A.42)

t=K+2

The first term on the right-hand side is of order op(l ) by (A.40) znd the abovementioned result lira2111 = Op((g/T)l/2). Hence, N~_.m2 "-" N - I

~-~ etY2,t--! ' t~K-t-2

-t-~O'(t~ -- m)f,~

N-

~

' Y2,t--lY2,t-i

t~-K+2

N-

2

T

)--1

~_, Y2,t-,Y~,,-I

+ Op(l )

t=K+2

where we have made use (A.41) and Lemma A.l(iii) o f Saikkonen (1995) which justifies the replacement o f k 22 by the indicated inverse. Lemma A.2(i) of the same paper implies that in the last expression it is further possible to replace et by et. The first assertion of the lemma follows from this and the discussion in Saikkonen ( 1992, pp. 20-21 ). To prove the second assertion, first note that (A.21) and the definition of the matrix ~, (see below (A.25)) give K+I

j=l

where the matrix JK = [ - J ' ... - - J ' In,]' ((Kn + nl ) × n l ) has the property iIJxlll = ( K + l y/2. Similarly, O--- =~EJKand therefore @l - O = ( ~ m - ~=)Jx- From (A.36) and arguments similar to those used to analyze (A.42) it now follows that T @l -- ¢~ : N - I

~

atqttelljK 4" O p ( ( K / T ) l / 2 ) .

t=K+2

Recall that £,qq = Eqtq't and note that [[/~ll _Z~ql Ill = O p ( K / T I / 2 ) by Lemma A . l ( i ) of Saikkonen (1995). From this result, (A.39) and (A.40), one obtains T -

=

E t=K+2

e,

Z q'J c +

op(g/rl/2).

124

P. Saikkonen, R. LuukkonenlJournal o f Econometrics 81 (1997) 93-126

The arguments given in the proof of L e m m a A.3 o f Saikkonen (1995) show that the last expression is o f order o p ( K - l / 2 ) , as required. As to the third assertion, its proof is very similar to that o f L e m m a A.3(iii) o f Saikkonen (1995), so only a brief outline will be given. First note that [ [ A m - All = op(K - I / 2 ) by the derivations in the proof o f the first assertion o f the lemma. Then observe that eT,,,t-- at -- (fi-m -- A).bt so that arguments similar to those in the above-mentioned previous proof show that in the definition o f Zm the replacement o f ~mt by at causes an error which is of order op(K - ! ). Next, since at = et + O O ' ( t h - m), one can use L e m m a 3.1 to show that a t can further be replaced by e t . After this, it suffices to use an argument already given in the p r o o f of L e m m a A.3(iii) o f Saikkonen (1995). []

P r o o f o f T h e o r e m 3.3. Following the p r o o f o f Theorem 3.1 we first define = [~ij]i,j=l.2 = A C A '

and

--~-Vm-- [V___m.ij]i.j=l,2 = l~//m'~ml_~_~m,

where ---~m= [~,,,' ---~m2]= tPmA---t" Then ~-ml~ < " " ~< 2m, are the eigenvalues o f the matrix ~ #m" The mairix C can tm written as C ' = M - - ( ~ where M___--[M___,•]ij=l.2 and ( ~ = [Gij]i,j=l,2 are defined by replacing u l . , - t and Y2.t-I in the definitions o f M and G by ~l,t-1 and fi2,t-t, respectively (see the beginning o f the proof o f Theorem 3.1). Using 171.t--I and .P2,t-t we similarly define analogs of the matrices Sqq, Stq and S2q and denote them by Sqq, Slq and S2a, respectively. Then, using (A.38), (A.40) and L c m m a 3.1 it can tm shown that ( A . 6 ) - ( A . 8 ) also hold with Sq¢, Slq and S2q replaced by Sqq, Siq and S2q, respectively. By similar arguments it can further tm seen that the matrices M and M are related as N-IM----li = N-tM---! l + op(l), N-tm___~t2 = N - I m l 2 -- O'(rh--m)fit2 + Op(1) = O p ( l ) and N-2M____"22 = N-2M_M_22+ Oo((K/T)I/2). Using the above results w e can n o w repeat the derivations used to prove ( A . 1 2 ) - ( A . 1 5 ) and show that these results also hold with C~j replaced by ---Co" Now, the p r o o f proceeds in the same way as that of Theorem 3.1. The first step is to consider ~.mi ~ " - " <.~Amn as eigenvalues o f the matrix ---~mC___~ m "~' in the metric ,~,~ and show that 2mj, j = n2 + l , . . . , n , tend to infinity at the rate O p ( N ) whereas ) . m j = O o ( 1 ) for . ] = 1 , . . . , n 2 . Here analogs o f ( A . 3 ) - ( A . 5 ) obtained from L e m m a A.I are used in addition to the above-mentioned results. Next, d e n o t e E ( ~ L ) = ~ _ i j ( A ) ] i . j = l , 2 =-~--Vm- , ~ - , - t and use L e m m a A.I to show that ---#m,I I = ~,Z--! ~ + Op(l ). In the same way as in the p r o o f o f Theorem 3.1 one can then show that the ni solutions o f IF__"lk(A)l = 0 tend to infinity at the rate Op(N). Thus, the n2 smallest eigenvalues o f C V m solve [N2~2.1(2)l = 0 where N 2 ~ 2 . 1 ( 2 ) = N 2 ( f ' m , 2 . i --,;al4~ l ) + Op(1) by arguments similar to those in the p r o o f o f T h e o r e m 3.1. Note that the only property o f the (infeasible) estimator ---~m2 w e have so far needed is ---,~m2--Op( N - I ). To complete the proof w e also

P. Saikkonen, R. Luukkoiwn/ Journal o f Econometrics 81 (1997)93-126

125

need the representation provided in Lemma A.I. First, in the same way as in (A.30) o f Saikkonen (1992) we can use the definition o f #_~ to obtain N2V___~,2., - - N2_~_.~2Q._~m2,

where Q - - ~ ' - E~' qJ~,(-'~ m ' ~ t ~/'~t ~' )--'. ~-- ~ , t . Let Q be the analog o f Q defined by using E and • in place o f Zm and ~'ml, respectively. From Lemma A.1 w e can then conclude that Q = Q + Op(l ) a n d , since Q O = 0, also that ! - ~ 2 = N 2 _~Q,.~_~2 -~- Op( l ), H e n c e N 2 "tllm2Q~___.m ~m = tr(J~f22_~Vm,2.1 ) ~t- Op( 1 ) = tr(~f22 ~.....t2Q___.~2) -~- Op( 1 ),

As discussed in the proof o f Theorem 3.1, the limiting distribution o f the estimator ---~2 is here the same as in Theorem 4.1 o f Saikkonen (1992) and the previous definition o f the matrix • is asymptotically equivalent to the present one. The arguments given in Saikkonen (1992, pp. 2 6 - 2 7 ) thus show that the first term in the last expression converges weakly to ~(n2). This completes the proof. []

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