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Maximum likelihood inference on cointegration and seasonal cointegration
Journal
of Econometrics
54 (1992) l-47.
North-Holland
Maximum likelihood inference on cointegration and seasonal cointegration Hahn
Shik Lee*
Tulane lJnil;ersiiy, New Orleans, LA 70118, USA Received
April
1990, final version
received
May 1991
Testing procedures for cointegration and seasonal cointegration are developed for nonstationary time series which have unit roots at seasonal frequencies as well as at the zero frequency. Using maximum likelihood inference, we estimate and test the hypothesis for the existence of cointegrating vectors at each frequency in the presence of unit roots at other seasonal frequencies. The basic finding is that several null hypotheses can be tested separately for each case of interest without any prior knowledge about the existence of cointegration relations at other frequencies. The asymptotic distribution theory is derived, and the critical values are generated by Monte Carlo simulations. The problems on the use of seasonally adjusted data are discussed, and an empirical example is illustrated by using Canadian data on unemployment and immigration rates.
1. Introduction
A vector of time series, all of which have unit roots, may have linear combinations which are stationary. In such a case, those variables are said to be cointegrated [see Granger (1981,1986)]. This idea of cointegration has been a major issue in econometric theory and practice over the last few years so that estimation and testing procedures have been widely developed [see Engle and Granger (19871, Phillips and Ouliaris (1988), Johansen (1988,19891, and Stock and Watson (198811. While there have been some theoretical extensions to more general settings since the original structure of cointegrated systems was first proposed, *I wish to thank and acknowledge my debt to Clive W.J. Granger and Robert F. Engle for many valuable suggestions and ideas they have given me throughout the preparation of paper. I am also grateful to Eric Chysels, Svend Hylleberg, Soren Johansen, Pierre Siklos, associate editor, and two anonymous referees for helpful comments and/or discussions earlier drafts.
0304-4076/92/$05.00
0 1992-Elsevier
Science
Publishers
B.V. All rights reserved
the this an on
2
H.S. Lee, Cointegration
and seasonal cointegration
tests
the idea of cointegration has mainly been devoted to the case where the series in the model are integrated, so that they become stationary after differencing. This case assumes that the roots of interest are precisely one, and there are no other unit roots in the system. Such a root corresponds to a zero-frequency peak in the spectrum and describes the long memory properties of the series. Many economic time series, however, exhibit strong seasonality which can be characterized by seasonal unit roots corresponding to peaks at seasonal frequencies in the spectrum. In this case, the two-step procedure in Engle and Granger (1987) turns out to be inappropriate for a cointegration test at the zero frequency. The lack of consistency due to the presence of seasonal unit roots is shown in Engle, Granger, and Hallman [henceforth EGH] (1989). The problem of the standard cointegration technique in this context motivates an alternative approach to testing for cointegration in the presence of unit roots at other seasonal frequencies. A few attempts have been made by Hylleberg, Engle, Granger, and Yoo [henceforth HEGY] (1990) and Engle, Granger, Hylleberg, and Lee [henceforth EGHL] (1990) to extend the usual cointegration technique to the case where the data have unit roots at both the zero and seasonal frequencies. In these papers, they suggest that the Engle-Granger type two-step procedure be applied to appropriately filtered series. Notice however that we need prior information on which unit roots are present in order to filter out seasonal unit root components and to test for cointegration with the filtered series; hence we need to pretest for seasonal unit roots, the implications of which have not yet been investigated. One may, in practice, circumvent these problems by simply using seasonally adjusted series, as the usual seasonal adjustment process such as the X-11 method should eliminate seasonal unit roots, if any, in the original time series data [see Bell and Wilcox (1990) and Lee and Siklos (1991a)]. In fact, usage of seasonally adjusted data has become a well-established practice in much empirical work. As is pointed out in the literature, however, seasonal adjustment might lead to mistaken inference on economic relationships (1984) and Ghysels among time series data [see, e.g., Bell and Hillmer (1988)], and it causes a significant loss of valuable information on important seasonal behavior in economic time series if seasonal fluctuations are an important source of variation in the system. Problems of this kind in seasonal adjustment processes do suggest that time series analysts should generally work with seasonally unadjusted data. In the case of seasonally unadjusted data, strong evidence for the presence of unit roots at some seasonal frequencies as well as at the zero frequency has been found in a large number of macroeconomic time series [see Ghysels, Lee, and Siklos (1992) for a survey]. If seasonal fluctuations of this kind are an important feature in economic time series, it is natural to extend the concept of cointegration to consider the possibility that common roots exist
H.S. Lee, Cointegration
and seasonal cointegration
3
tests
at seasonal frequencies other than zero, which results in the idea of seasonal cointegration discussed by HEGY (19901 and EGHL (1990).
The aim of this paper is two-fold. First, we extend the standard cointegration technique to a testing procedure for cointegration that does not require any prior knowledge about the presence of seasonal unit roots. Second, we also develop estimation and testing procedures for seasonal cointegration at any set of the zero and seasonal frequencies. Throughout this paper, we consider a nonstationary VAR process which has unit roots at seasonal frequencies as well as at the zero frequency. Using maximum likelihood inference developed in Johansen (1988), we derive tests for cointegration and seasonal cointegration in time series data at each frequency in the presence of unit roots at other frequencies. The definitions of an integrated seasonal process and seasonal cointegration are presented in section 2. Section 3 develops a general framework of the testing strategy for seasonal cointegration that looks at unit roots at all seasonal frequencies as well as the zero frequency by deriving the ML estimation of cointegrating vectors and test statistics for hypotheses about cointegrating vectors. This testing procedure is extended to examine an interesting case where all of the cointegrating vectors coincide. In section 4, finite sample properties of the testing procedure are investigated by Monte Carlo simulations, and an empirical example is also presented. The conclusion is in section 5. 2. Seasonal integration and cointegration It has been widely observed that many economic time series exhibit substantial seasonal components and a variety of possible models for seasonality have been attempted to describe seasonal patterns. The definition of integration in Engle and Granger (1987) can be generalized to include seasonal integration. Definition 2.1.
Let S(B) have a root with modulus 1 at frequency 0 [i.e., S(B) = (1 - eie B)] for ,9 E (- 7, ~1, and also let D(B) have all the unit roots at seasonal frequencies as well as the zero frequency, if any, which are different from 19.A series X, with no deterministic component is said to be seasonally integrated of order d at frequency 8, denoted as X, - I,(d), if d is the smallest integer for which the representation S( B)dD( B)x, = C( B)E,
(2.1)
has the following properties: (i) the spectrum of C(B)E~ is bounded away from zero and infinity at all frequencies; (ii) {Ed}is a sequence of serially
4
H.S. Lee, Cointegration
and seasonal cointegration
tests
uncorrelated random vectors with finite and constant unconditional (iii) the series is taken to be initialized by Ed, X, = 0 for t I 0.
variance;
Note that the presence of D(B) in (2.1) allows a series x, being Z,(d) as well as I,(&) for 19~# 0, and d # d’. This definition generalizes the ordinary Z(d) process which is a special case of Z&d) where 0 = 0 and D(B) = 1. For d = 0 with D(B) = 1, X, will simply be stationary as an ordinary Z(O) process. From this definition, we see that if x, N Z,(l) with D(B) = 1, then (i) variance of x, goes to infinity as t + 03;(ii) X, has ‘long memory’ so that an innovation has a permanent effect on the seasonal pattern of xt; (iii) the (asymptotic) spectrum of X, takes the form f(w) =A( w - 0)-2 for o near f3 so that it has infinite peak at frequency 8.’ Moreover, it is well known that (iv> X, is asymptotically uncorrelated with processes which have unit roots at other frequencies2 This property of seasonally integrated time series plays an important role in analyzing the asymptotic distribution of estimates and test statistics. For simplicity, only the value d = 0 and d = 1 will be considered in this paper, but many of the results can be generalized to the cases of higher order. Some examples of integrated seasonal processes are presented in HEGY (1990). A very interesting and familiar example for a quarterly data might be (1 -ZP)X,=&,, which can be rewritten
(2.2) as3
(1-B4)X,=(1-Z3)(1+B+B2+B3)X,m(1-B)yl,
(2.3)
=(l+B)(l-B+B2-B’)x,=(1+B)y2,
(2.4)
= (1 +ZP)(1
(2.5)
-P)x,
= (1 +ZP)y3,,+,.
Using the above definition, the process x, in (2.2) can be denoted as x, - Z,(l) for any 0 = 0, r, and &r/2. That is, it has four roots with modulus ‘Strictly speaking the spectrum of x, is not well defined here, since x, is nonstationary so that it does not have a finite variance. Apart from a possible problem at frequency 8, however, the usual definition of spectrum can still be employed in this case. See Granger and Newbold (1986, ch. 2). ‘Two series, say, y,! and y2, (t = 1,2,. , T), are said to be asymptotically uncorrelated if the correlation coefficient between them vanishes as T+ ~0. See Ghan and Wei (1988) and HEGY (1990). “The expression note 4.
y,, , + , (rather
than
y,,)
is used here for notational
convenience.
See foot-
H.S. Lee, Cointegration
and seasonal cointegration
tests
5
one: one at the zero frequency (w = O), one at two cycles per year [i.e., a half cycle per quarter (w = +>I, and a pair of complex roots at one cycle per year [i.e., a quarter cycle per quarter (w = +>I.” We can now generalize the idea of cointegration in Granger (1986) and Engle and Granger (1987) to define seasonal cointegration. Definition 2.2. Let all components of x, be seasonally integrated of order 1 of the vector x, are said to be at frequency 0, i.e., x, - 1,(l). The components seasonally cointegrated at frequency 8, denoted x, - U&l, l), if there exists a vector (Y(# 01 so that z, = LY’X~ - Z,(O). For a vector of nonstationary series which has unit roots at some seasonal frequencies as well as at the zero frequency, it is possible that a single cointegrating vector could eliminate all the unit roots in the series. To formalize this idea, the following definition adapted from EGH (1989) is also introduced. Dejinition 2.3. Let each component of X, be seasonally integrated of order 1 at some frequencies, not necessarily at the same frequencies for all components. The components of the vector x, are said to be fully cointegrated, denoted x, - CZ(1, l), if there exists a vector (Y (Z 0) so that z, = (Y’x, is stationary. These definitions can be reduced to the ordinary cointegration when 0 = 0 and D(B) = 1. The implications of seasonal cointegration are not immediately obvious but are quite similar to those of the ordinary cointegration as established by Engle and Granger (1987). In particular, seasonal cointegration would mean that an innovation has only a temporary effect on the seasonal behavior of z, = &x1, while it may have a permanent effect on the seasonal pattern of x,. This would occur if each of the series contains the same factor which is Z,(l). The close relationship between cointegration and error correction models also allows us to develop error correcting mechanisms when the variables in the system are seasonally cointegrated, which will be employed in the next section to discuss a general framework of the testing strategy for cointegration and seasonal cointegration. 3. Testing procedures
for seasonal cointegration
This section proposes a testing procedure for cointegration in the presence of unit roots at other seasonal frequencies, and develops a general framework of seasonal cointegration test for nonstationary VAR time series with 4Following HEGY (1990), the expressions w = $ and w = f are used in this paper the seasonal frequencies 0 = T and 0 = *r/2, respectively.
to signify
6
H.S. Lee, Cointegration and seasonal cointegration tests
unit roots at all seasonal frequencies as well as at the zero frequency. The testing strategy follows the framework in Johansen (1988) so that we can adapt some results from his paper.
3.1. The seasonal cointegration
model
We will consider the case where the observed data is a sequence of random vectors X, with components (xlr, xZt,. . . , xnt)’ drawn sequentially from n-dimensional Gaussian random vectors {E[} with mean zero and variance matrix a. Thus the process can be written as x, =
@,x,_,+ dj2xrp2
+ . . . +@ p x I--p t=1,2
+et,
,...,
(3.1)
T,
where E,, E*, . . . , Em are i.i.d. N,(O, 0) and X, are hxed for t s 0. This is a VAR model of order p, and the n x n matrices (@,,@,, . . . ,@,, 0) are parameters to be estimated on the basis of T observations. Since we allow the process X, to have unit roots at seasonal frequencies as well as at the zero frequency, the determinant I@(z)l of the matrix polynomial Q(z)
=I-@,z-@,z2-
... -Qpz”
will have roots on the unit circle; one at the zero frequency and at least one at some seasonal frequencies, with all roots of l@(z>l = 0 satisfying Iz I 2 1. We shall mainly consider a simple case in which Asx, is stationary, where A, = 1 -B” and s is the number of observations taken per year, so that the determinant I@(z)/ has unit roots at the zero frequency (w = 0) and all seasonal frequencies w = j/s (j = 1,2,. . . , s/2). Thus the general structure of such a process can also be discussed by rewriting the model (3.1) as Asx,=17,S,(B)x,_,
+ ...
+~,~S,~(B)x,_,
+FI~A.~x,_~ + . . . +Ap_,AF~,_p+r
+AlAsX,-l + E,,
(3.2)
where S,(B) is a seasonal filter that will eliminate unit roots at all frequenties other than the one in the filtered series Yk,(_] = S#?)x,_, (k = \ 1,2,. . . ) s), and Il,‘s (k = 1,. . . , s> and Ai’s (i = 1,. . ,p - sl are approprrately redefined. Note that among the unit roots of l@(z)1 some of them are complex pairs, hence that the model (3.2) can be rewritten to include, if needed, lagged as terms Y~,,-~ of Y~,~+~. Notice also that this expression can be interpreted
7
H.S. Lee, Cointegration and seasonal cointegration tests
an error correction model for the process with unit roots both at the zero and some seasonal frequencies. This representation is quite general in that the coefficient matrices II, might be different at different lags as well as at different frequencies. Hence the cointegrating vectors and the coefficients of the error correction terms may also be different at different frequencies and, in some cases, at different lags. For ease of exposition, only the value s = 4 will be discussed here, but the results can naturally be extended, for example, to monthly data (i.e., s = 12) with minor modifications [see Beaulieu and Miron (1990)]. Now the model (3.2) will be reduced to
+A,dz,~~-~
+ . . . +Ap-4A4~t_p+4
+ E,,
(3.3)
where ylt=S,(B)x,=(1+B+B2+B3)XI,
and n, = - @(1)/4, II2 = @( - H/4, II3 = Real[@(i)1/2, Lid = - Im[@(i)1/2, A. = - c[!P-i)‘41@ (for i = 1,2,. . . , p - 4), and [(p - i)/4] denotes the IaZgest in&ger in [;‘,“- i)/4]. Notice that S,(B) is a seasonal filter that removes unit roots at all seasonal frequencies (i.e., w = i and a>, while S,(B) and S,(B) eliminate unit roots at seasonal frequency w = f and o = i, respectively, as well as at the zero frequency. This implies that y,,, y,,, and y,, have unit roots only at frequency w = 0, o = 4, and w = a, respectively. Since the coefficient matrices II,, . . . , II4 may convey the information concerning long-run behavior of the series, we need to investigate the properties of these matrices in order to determine whether or not the components of x, are seasonally cointegrated in the presence of unit roots at other frequencies. Note that if the matrix IIk has full rank, then the series does not contain unit roots at the corresponding frequency. If the rank of II, is zero, seasonal cointegration relationship at that frequency does not exist. In the intermediate case where 0 < rank(l7,) = r
the
term
Y~,~, and
will be employed
8
H.S. Lee, Cointegration and seasonal cointegration tests
Hence the natural hypothesis of seasonal cointegration can be formulated as the condition that rank(ll,) < r for corresponding frequency against the alternative that rank(ll,) > r. The possibility of seasonal cointegration at each frequency is discussed in the following subsections. Owing to the asymptotic uncorrelatedness between any two series with unit roots at different frequencies, it turns out that several null hypotheses can be tested for each case of interest without any prior information on the existence of cointegration relations at other frequencies.
3.2. Cointegration
at frequency
o = 0
When we want to test for cointegration at the zero frequency in the presence of unit roots at some seasonal frequencies, the hypothesis of interest can be given by H,: rank(U,) I r for some r < 12 against an alternative that rank(Z7,) > r. We can alternatively express this as H,:
n, = ?,a;,
where y1 and (Yeare n X r matrices. In order to test this hypothesis, we need to estimate the parameters y, and a,. They cannot be estimated, however, since they form an overparametrization of the model [see Johansen (19SS)]. But we can estimate the space spanned by y, and by cyI, and the estimates of sp(y,) and sp(cu,) can be used to test the hypothesis H,. We shall first show that testing H, for the model (3.3) can be done in a similar fashion to the procedure in Johansen (1988). This follows from the fact that the same distributions are obtained when it is not known a priori that they are seasonally cointegrated at some seasonal frequencies other than zero. We can consider two cases. First, if they are seasonally integrated of order 1 at frequency w = 4, for example, but not seasonally cointegrated at that frequency, then the term n, yZ I_, would be nonstationary and the true 17, would be equal to null matrix. Since the variables y,,,_, and y,,,_ , have unit roots at different frequencies, we know that they are asymptotically uncorrelated. Note also that both will be asymptotically uncorrelated with lags of A4x, which are stationary. Hence the distribution of the test statistic for H, will not be affected by the inclusion of the term y,,,- ,. Second, if they are seasonally cointegrated, then the term U, y,,,_ 1 will be stationary. In this case, the regression analysis is similar to the procedure in Johansen (1988) except for the augmented variable y2,+ 1. We shall now consider the maximum likelihood estimation of the parameters in the unrestricted model (3.3). Here we need to maximize the likelihood function with respect to the parameters (A,, . . . , FI~_~, fl> and (n,, . . . , Ii’,). is equivalent to an For fixed values of IZk’s (k = 1,. . . , 4) the ML estimation OLS regression of (A,x, - C~,117kyk,,_ ,) on the lagged differences.
H.S. Lee, Cointegration and seasonal cointegration tests
9
Let the product moment matrices of the variables in the model (3.3) be such that
Mmi,
= T-’
5
A‘,x,_~A~x;_~
for
i, j=O ,...,
~,,,_,A,x:_~
for
k = 1,. . . ,4,
p-4,
(3.4)
r=l T MIX,,
=
T-’
c
(3.5)
t=1
j=O ,..*, P - 4, T
-,I
=
T-’
c
for
Yk,r-&-I
l,...,
k,l=
4.
(3.6)
I=1
Then, from the normal equations provided by the OLS regression of the form T
T
p-4
C A,x,A,x~_~=
4
C Ai C A4xt-iA4xi-j+
t=1
i=l
T
k~lnkt~,yk,t-,d4x:-j,
r=1
(3.7) we
have 4
p-4
Mm,,=
c
A,MXXij+
c
for
IIkMYXkj
j= I,...,
p-4.
k=l
i=l
(3.8) This can be solved for Ai to give p-4 &II,,...,
fl,)
=
c j=l
MxXojMxXii
-
;
nkmkjMxii,
(3.9)
k=l
where MXX; ‘, = [MxX’j] is the inverse of MXX, .+, the matrix with elements M_XXij (i, j = 1,. . . , p - 4). By substituting these into eq. (3.31, we can get the residuals R, of the OLS regression. Alternatively, we can obtain the residuals by first regressing A4x, on the lagged differences [i.e., A4x,_j’s for j=l , . . . , p - 41 giving the residuals R,,, and then regressing each yk, t _ 1 on the lagged differences giving the residuals R,, (for k = 1,. . . ,4) and finally forming R, = R,, - c”,,,Il,R,,.
10
H.S. Lee, Cointegration
Now the concentrated
= Ifp/2
likelihood
and seasonal cointegration
function
becomes
proportional
R,,
exp
tests
to
R,,- t&R,, k=l
(3.10) Hence the MLE of the parameters IIk’s can be estimated by a usual OLS regression of R,,on Rkr's. That is, the MLE of IZ,‘s are equivalent to the estimates of the following regression equation:
4, = fl,R,t + HzR,t + fl,R,* + 4&; For notational the residuals,
convenience, i.e.,
let Sij denote
i,j=O
+ 5,.
the product
I’.‘, 4,
(3.11) moment
matrices
of
(3.12)
t=1
be the matrix with elements Sij (i, j = 1,. . . ,4) and let S,, .+ = and let S,, D = (Z7,. . . I&,) and R,, = CR',, ...R&j',the MLE (S a,. . . S,,). By denoting of Z7 is given by
(3.13)
We can also write (3.14)
~=~,,,,-~~,*~,j,~;,*,
2,‘s (i = 1,. . . , p - 4) can be derived by substituting the 1,. . . ,4) into eq. (3.9). model under H 1: IZ, = ylcJ,. For fixed value of II,, the (ZI, I7s II,) is equivalent to an OLS regression of CR,,, -IT,R,,) on (R2(R,,Rdr). The normal equations provided by the OLS regression are and the estimates of estimated fik’s (k = Now consider the ML estimation for
j=l k=l
1=1
7. . * , 4.
(3.15)
H.S. Lee, Cointegration
Using the notation
s”j
=
5 k=l
and seasonal cointegration
(3.12), eq. (3.15) can be rewritten
11
tests
as
(3.16)
nkskjY
or equivalently,
s,j
-n&j
=
i
17,skj,
k=2
which can be solved for Ilk’s (k = 2,3,4)
li,( Iz,) = ;
to give
(3.17)
( S”js’k - n,S,jS’k),
j=2
where S;; = [Sik] is the inverse of S,,, the matrix with elements Sjk (j, k = 2,3,4X By substituting these into (3.111, we get the residuals Q, from the OLS regression. Alternatively, we can also get the residuals by first regressing R,, on (R,, Rxt Rdt) to give the residuals Q,,, and then regressing R,, on (R,, R,, Rdt) to give the residuals Q,,, and finally forming Q, = Q,, - n,Q,,. Hence the concentrated likelihood function becomes proportional to
L(II,,R)
= I0l-““exp
[
-~,~,(Q,,-,,,Q,~).~-~(Q,,-,,,Q,,)
1
(3.18) so that the MLE of the parameter IJI, can be estimated by a usual OLS regression of Q,, on Q,,. By denoting Dij = T- ‘CT_ lQ,,Qi, (i, j = 0,. . . ,4), we can write -1 = D,,D,‘,
(3.19)
I h = Do0 - D,,D,‘D,,,
(3.20)
and the estimates of fik’s (k = 2,3,4) and ai’s (i = 1,. . . , p - 4) can be derived by substituting the estimated fi, into (3.17) and (3.9) sequentially. For fixed (or, we can maximize (3.18) over yr and R by an OLS regression of
7
12
Q,t
H.S. Lee, Cointegration
and seasonal cointegration
tests
on CX’,Q,, to obtain
.
fl(a1> = D”cl- 4u’y*(~;DII~I)
-I
(3.22)
4D,“.
The likelihood profile now becomes proportionai to ].f?(~~,>j -“’ SO that, as shown in Johansen (1988), it remains to solve the minimization problem
min
I”;D,,a, -
‘Y;D,oD,olDo,a, 1
I”p,,a,
[a,1
(3.23)
>
I
where the minimization is over all n x r matrices LY,.We can now estimate CX,by choosing the first r eigenvectors of D,, D,‘DoI with respect to D, ,. Based upon Johansen (1988, theorem l), the test statistic for the hypothesis J-I, shall be formulated in the following theorem, which provides inference about the number of important cointegration relations at the zero frequency. The proofs of our theorems are given in the appendix. Theorem 3. I. The likelihood ratio test statistic for the hypothesis that H,: l7, = y,o!, where y, and ayI are n x r matrices, i.e., that there are at most r cointegrating vectors at zero frequency, is -2ln(Q)
= -T
2 ln(l i=r+ I
(3.24)
-i,,i),
where /i l,r+l,..., i I,n are the (n - r) smallest squared partial canonical correlations of R, with respect to Ro, given R,, R,, and R,, i.e., the first (n -r) eigenvalues of D,,D,‘DO, with respect to D,,. The test statistic of this hypothesis is then asymptotically distributed as (3.25) where W( r> is an (n - r&dimensional standard Brownian motion on the unit interval [0,11. Remark 1. From the result in Johansen (19881, it is clear that the test for n, = -y,c.~‘,will have the asymptotic distribution in (3.25) if 112 = L’s = fl, = 0 (null matrix), since the model then can be written in the form A4x, =D,Y,,~_,
+A,A,x,_,
+ . . . +&-4Az,~t-~+a
+ “t.
H.S. Lee, Cointegration
and seasonal cointegration
tests
13
Asymptotic uncorrelatedness of y,,(_, with Y~,~_,, y3,[_, and Y~,,-~ will lead us to the same distribution whether or not it is known a priori that some of the IIk’s are zero. Notice that the MLE of the space spanned by (Y, is the space spanned by the r canonical variates corresponding to the r largest squared partial canonical correlations between y, f_, and A4x, given lagged A4x,‘s 1 and other yk f_ ,‘s (k = 2,3,4X (A,x,-,,...,A,+,+, 3.3. Seasonal cointegration at seasonal frequency w = + This subsection proposes a testing procedure to determine whether or not the components of X, are seasonally cointegrated at seasonal frequency w = 4 in the presence of unit roots at the zero frequency and/or at other seasonal frequencies. The hypothesis of interest can be formulated as
where y2 and (Ye are n x r matrices. The testing procedure is very similar to the one developed in the previous subsection except that the role of Rr, and R,, is reversed. That is, the MLE of the parameter ZI, with no restriction on II,, II,, and 114 can be estimated by an OLS regression of Q,, on Qzt, where Qzt is the residual from regression of RZt on CR,, R,, R4t). We can write
(3.26)
d = D,,
- D”,
DT*‘D,,
(3.27)
)
and the estimates of fi,‘s (k = 1,3,4) and 2;‘s (i = 1,2,. . . , p - 4) can now be derived by using the estimated 17,. By arguments similar to Theorem 3.1, we can formulate the results for the test statistic of the hypothesis H, in the following theorem. Theorem 3.2. The LR test statistic for the hypothesis that there are at most r seasonal cointegrating vectors at frequency w = t is -2ln(Q)
= -T
k i=r+
ln(l
-i,,i),
(3.28)
I
where i 2,r+lY..J2,n are the (n - r) smallest squared partial canonical correlations between R, and R,, given R,, R,, and R,, which are just the first (n - r) eigenvalues of D20D&1D02 with respect to D,,. Furthermore, this test statistic has the same asymptotic distribution as (3.25).
14
H.S. Lee, Cointegration and seasonal cointegration tests
Remark 2. Using the notation in Lemma A.1 [see eqs. (ASa) and (A.5b) in the appendix], it is easy to see that /d dW, W; 2 /d dWz W;, which was first observed by Fuller (1976, p. 370) for the univariate case. The same asymptotic distribution for testing seasonal cointegration at seasonal frequency o = i as (3.25) follows from this observation.
3.4. Seasonal cointegration at seasonal frequency w = f To test the hypothesis of seasonal cointegration at frequency w = f, we need to look at the two parameter matrices, 17, and II4, simultaneously since the cointegrating vectors and the coefficients of the error correction term may be different at different lags. As is pointed out in HEGY (1990), this might imply that we need to consider the polynomial cointegrating vector (PCIV) since one vector is sought to eliminate two unit roots C&i) [see Engle (1987)], hence that a much more complicated testing procedure than the previous two cases may be required for testing the joint hypothesis on Ii’, and I14. For simplicity of exposition, it will be assumed throughout in this paper that cointegration, if any, is contemporaneous. Then the hypothesis of interest can be formulated as a joint test such that
where y3, yq, and LYEare n X r matrices. For fixed values of I73 and 114, the ML estimation for (IZ, II,> is equivalent to an OLS regression of CR,,, - n3R,, - f14R4f) on CR,, R2,). We can also show that the MLE of the parameter matrix [n, II,] can be estimated by an OLS regression of Q,, on Q3* and Qdt, where Q3, and Qdr are the residuals from regression of R,, and RJr on (R,, Rzt), respectively. Then we have
r=1
where
-
[Q,tQJQ,tQ,,l
r=1
D,,D,-,’>
(3.29)
D,,,D,;‘D,,,,
(3.30)
=
h = D,,
5
-I
H.S. Lee, Cointegration and seasonal cointegration tests
15
The estimates of fir, I?,, and a,‘s can then be derived by substituting the estimated [II, III41 into the normal equations of the form similar to (3.17) and (3.9) sequentially. For fixed (Ye, we can maximize the concentrated likelihood function
x (Qo,- n,Q,t - KQ4,) over y3, y4, and obtain
(Y is a 2n X 2r matrix
(Y=
By applying minimization
min
of
Q,, on abQ,, and ~;QA, to
=D,,,-D,qa(a~Dqqa)-la~Dq”,
ii where
0 by an OLS regression
1
(3.32)
such that
aq O 0 aq . i
I
the matrix problem
relation
used in (3.23), it now remains
b’Dqqcx - cx’DqOD,,‘D OqffI
[%,I
to solve the
(3.33) ’
Ia’Dqq~I
where the minimization is over all n x r matrices (Y We can then formulate the test statistic for the hypothesis H, in the follow&g theorem. The LR test statistic for the hypothesis that there are at most r Theorem 3.3. seasonal cointegrating L:ectors at frequency w = + is -2ln(Q)
= -T
i i=r-t
ln(l -i,,j-&i).
(3.34)
1
where /ik,r+ ,, . . .,/i, n are the (n - r) smallest squared partial canonical correlations of R, (k = j, 4) with respect to R,, gicen R, and R,, i.e., the first
16
H.S. Lee, Cointegration and seasonal cointegration tests
(n - r) eigenvalues of DkODi’DOk with respect statistic for H, is asymptotically distributed as
tr
‘(dW, W,l +dW,
W,)
to D,,
jji’(dWC W,-dW,
(k = 3,4).
W,,)
\ L’(W,W:+
W,W,l)dr
X
/‘( V$Wi - w,Wi) 0
dr
/‘(WSWi0
The test
-I
W,W,l)dr
/‘( W,Wi + W’,Wi) dr 0 I
(3.35)
where W,(r) and W,(r) in (n - r ) dimensions.
are mutually
independent
standard
Brownian
motions
Based on the structure of the error correction mechanism for a process with unit roots both at the zero and seasonal frequencies given by (3.3), we can further assume that y4 = 0 (i.e., 17, = 0) with little effect on the test for seasonal cointegration at frequency w = d when cointegration is contemporaneous. This observation is very useful in practice, since in this case the test can be performed by looking at the matrix IIJI, only [see section 41. That is, if we restrict our attention to the case where it is assumed that U4 = 0, the hypothesis H, will be reduced to the condition that
and the testing strategy will then be similar to that for seasonal cointegration at frequency w = 4 except that R3, takes the role of R,,. The MLE of the parameter U3 with no restriction on II, and f17, can be estimated by an OLS
H.S. Lee, Cointegration
and seasonal cointegration
tests
17
of Q,, on Q3t, to obtain
regression
(3.36) fi = D,, - D,, D,‘D,,
(3.37)
.
By arguments similar to those in previous subsections, we can formulate the test statistic for the hypothesis H, as a simple corollary to Theorem 3.3. Under the assumption that IT, = 0, the LR test statistic for the Corollary 3.4. hypothesis that there are at most r seasonal cointegrating vectors at frequency w=$is -2ln(Q)
= -T
i ln(1 -ij,i), i=r+ 1
(3.38)
,.
where i 3,r+, , , . , , A 3,n are the (n - r ) smallest squared partial canonical correlations of R, with respect to R,, given R, and R,. The test statistic for H, is asymptotically distributed as
x
‘(W,dW,‘+W,dW,‘)
(3.39)
.
/0
I
Remark 3. Notice that for r = n - 1, i.e., for testing (n - 1) seasonal cointegrating relations at frequency w = i, we obtain the asymptotic distribution with one-dimensional Brownian motions. In this case, (3.35) will be reduced to /l(w,dW,+
1[
W$dW,) *+
0
j’(w,dW,-W.dWJ 0
/‘(W,*
+ WS”)dr
0
which is the limiting distribution of, where t,? and t-statistics on rTT3and r4 for seasonal unit root test at frequency EGHL (199O)l. Similarly, (3.39) will be reduced to ‘(WdW,+
WdW,)
tSTl are the w = $ [see
H.S. Lee, Cointegrution and seasonal cointegration tests
18
which is tantamount to the square of the Dickey-Hasza-Fuller for biannual seasonality [see also Chan and Wei (1988, corollary
distribution 3.3.8)1.
3.5. Full cointegration The testing procedures for seasonal cointegration have so far considered the general error correction model of the form (3.3) so that the cointegrating vectors may be different at different frequencies. It is however possible that some of the cointegrating vectors coincide but some do not. If (Ye= a4, one cointegrating vector, say CY~,will remove all seasonal unit roots. This might be particularly interesting when it is expected that the seasonality in the time series data is due to the same source. This is the case examined in EGH (1989) for electricity demand. Similarly, the cointegrating vectors GJ,, (Y*, and cyy may coincide, and therefore one ccintegrating vector, say (Y~, could eliminate unit roots of the series at all frequencies. In this case, we have full cointegration as defined in Definition 2.3. Since an efficiency gain in the estimates of the cointegrating vectors is naturally expected by imposing the restrictions between the cointegrating vectors, it may be worth checking the existence of such relationships between the cointegrating vectors once seasonal cointegration relations at different frequencies are identified. The general framework of seasonal cointegration tests proposed in previous subsections is extended here to discuss full cointegration. Suppose that the cointegrating vectors LY,, crz, and a4 coincide (i.e., say, Ly, = (Y, = ay = a,). Then the error correction model (3.3) will be reduced to A4x, =n,(B)x,_,
+A,A4xt-,
+ .
+&~A~~,-,,+~
+&r,
(3.40)
three lags. This indicates where II,(B) = yA(B)& and yA(B) has potentially that a single linear combination may eliminate all the unit roots in the time series data. The hypothesis of interest can now be formulated as a joint test such that
x r matrices. By arguments similar to those in the previous subsection, we will restrict our attention to the case where it is assumed that yA, = yA, = yA, = 0 (i.e., flA, = nA, = flAX = 0) for ease of exposition. However, the results can be extended in a straightforward way as in Theorem 3.3.6 Under the assumption that yA(B) = yAB’, the model
where YA,, ?'A:, YA,' YA,? and LY~are n
‘Alternatively, estimate II,(E)
the index model in (3.40).
estimation
procedure
by Reinsel
(1983) can be employed
to
H.S. Lee, Cointegration
(3.40) can be written
procedure
H,:
tests
19
as
A‘+, =IT/v_~ The testing
and seasonal cointegration
+A,A,x,_,
+ . . . +Ap-J~qp+z,
+&t.
for the hypothesis
rr, = r,4aa
can now be formulated
in the following
theorem.
The LR test statistic for the hypothesis that there are at most r Theorem 3.5. full cointegrating oectors is -2ln(Q)
= -T
5
ln(l --iA,;),
(3.41)
i=r+l
are the (n - r) smallest squared canonical correlations where i A,r+l,...&4,n of R, with respect to R,, where RAl is the residual from regression of x,_~ on A4x,_j’s (j = 1,. . . , p - 4). The test statistic for HA is asymptotically distributed as
where Wj(r)‘s are mutually independent standard Brownian motions in (n - r) dimensions. Remark 4. Note that for r = n - 1, i.e., for testing (n - 1) full cointegrating relations, we get the limit distribution with one-dimensional Brownian motions. In this case, (3.42) will be reduced to
which is the square and Fuller (1984).
of the ‘seasonal
4. Finite sample behavior:
unit root’ distribution
Some simulation
in Dickey,
Hasza,
results
Since inference on cointegration and seasonal cointegration developed in the previous section has largely been based upon the asymptotic distribution theory, it is of great importance to investigate the finite sample properties of
20
H.S. Lee, Cointegration
and seasonal cointegration tests
Table Quantiles tn -r)
in the distribution
1o/c
Frequency w=o,; u=$ Full CI
1
2
3
0.00 0.00 0.00
1
of the seasonal
5%
10%
50%
0.01 0.01 0.00
0.02 0.02 0.02
0.58 0.52 0.49
0 = 0, I
1.27
1.99
2.54
5.42
m=; Full CI
0.37 0.32
1.08 0.86
1.51 1.30
4.68 4.01
w=o,: &I=_:Full Cl
6.03 4.61 2.79
7.79 6.38 4.09
x.93 7.49 5.13
14.3 12.1 10.6
cointegration 904 2.89 2.82 2.79 10.5 9.69 X.82 21.6 19.3 17.0
test.”
95% 3.94 3.81 3.80
97.5% 5.08 5.05 5.02
99% 6.73 6.70 6.69
12.5
14.4
16.3
11.7
10.6
13.1 12.7
15.3 14.6
24.3 21.7 20. I
26.4 24.1 21.9
29.8 27.3 24.9
“The stochastic integral is approximated by sum of 400 i.i.d. Gaussian variables, i.e., 400-step Gaussian random walks are used for the standard Brownian motions in the asymptotic distributions of the LR statistics: (3.25), (3.39), and (3.42). 30,000 replications.
the testing procedure. In this section, the finite sample distributions of the test statistics are examined by Monte Carlo simulations, and an empirical application to Canadian data on unemployment and immigration rates is presented. The asymptotic quantile distributions are first constructed by replacing the stochastic integral by the sum of T = 400 random Gaussian variables, using the GAUSS programming language. Table 1 reports the limiting distributions of the LR statistics [(3.25), (3.391, and (3.42)], which can be used as the critical values for cointegration and seasonal cointegration at each frequency. The figures are obtained through 30,000 replications. Notice that the 95% acceptance regions of the critical values are not very different from each other, except for the n = 3 case. To examine the finite sample behavior of the LR statistics in (3.24), (3.28), (3.38), and (3.41), simple simulation experiments are designed such that a sequence of random vectors X, with elements (x,~ xzl.. . x,,~)’ are generated by
x,
=x,-4
+ E,
)
t=l
,...,
T,
(4.1)
=xg = 0. Now the where &i, Ed,. . . , Ed are i.i.d. N(0, I,,) and x_~ =xP2 =x-, seasonal cointegration model (3.3) will take a simple form
H.S. Lee, Cointegration
and seasonal cointegration
tests
21
By construction, there are no seasonal cointegration relations among the elements of the vector X, at any frequencies (w = 0, i, or a>. It is then clear that the true lIk’s are null matrices [i.e., rank(L’,) = 01 for k = 1,. . . ,4. Hence it follows that the distribution of the test statistics for the null of II, = ykak = 0 provides information on the size of the test. The figures are computed through 10,000 replications in each case for n = 1,2,3 and for T = 50,100,300. The results are presented in table 2, from which we can obtain the actual rejection frequency of the null hypothesis for the cointegration test when the variables are generated by (4.11.’ Notice first that we tend to reject the true null hypothesis of no (seasonal) cointegration [i.e., r = 01 more often than is presumed if we use the asymptotic distribution in table 1. This result suggests that the finite sample quantile distribution in table 2, rather than the asymptotic distribution in table 1, should be used in practice as the critical values for the cointegration and seasonal cointegration tests. It is, however, interesting to see whether such a discrepancy between table 1 and table 2 becomes negligible as T grows. We can see that the actual rejection incidence is close to the nominal size of the test except for small T CT= 501. Notice also that the larger the number of observations T, the closer the empirical size of the test to the significance level. This indicates that the actual rejection frequency approaches the asymptotic rejection probability as T grows. Since we know in this case that the true Dk’s are zeros, it may be worth checking the effects of inclusion of the terms that are asymptotically uncorrelated. This result is shown in table 3. The first row of each case [denoted by
of Econometrics
54 (1992) l-47.
North-Holland
Maximum likelihood inference on cointegration and seasonal cointegration Hahn
Shik Lee*
Tulane lJnil;ersiiy, New Orleans, LA 70118, USA Received
April
1990, final version
received
May 1991
Testing procedures for cointegration and seasonal cointegration are developed for nonstationary time series which have unit roots at seasonal frequencies as well as at the zero frequency. Using maximum likelihood inference, we estimate and test the hypothesis for the existence of cointegrating vectors at each frequency in the presence of unit roots at other seasonal frequencies. The basic finding is that several null hypotheses can be tested separately for each case of interest without any prior knowledge about the existence of cointegration relations at other frequencies. The asymptotic distribution theory is derived, and the critical values are generated by Monte Carlo simulations. The problems on the use of seasonally adjusted data are discussed, and an empirical example is illustrated by using Canadian data on unemployment and immigration rates.
1. Introduction
A vector of time series, all of which have unit roots, may have linear combinations which are stationary. In such a case, those variables are said to be cointegrated [see Granger (1981,1986)]. This idea of cointegration has been a major issue in econometric theory and practice over the last few years so that estimation and testing procedures have been widely developed [see Engle and Granger (19871, Phillips and Ouliaris (1988), Johansen (1988,19891, and Stock and Watson (198811. While there have been some theoretical extensions to more general settings since the original structure of cointegrated systems was first proposed, *I wish to thank and acknowledge my debt to Clive W.J. Granger and Robert F. Engle for many valuable suggestions and ideas they have given me throughout the preparation of paper. I am also grateful to Eric Chysels, Svend Hylleberg, Soren Johansen, Pierre Siklos, associate editor, and two anonymous referees for helpful comments and/or discussions earlier drafts.
0304-4076/92/$05.00
0 1992-Elsevier
Science
Publishers
B.V. All rights reserved
the this an on
2
H.S. Lee, Cointegration
and seasonal cointegration
tests
the idea of cointegration has mainly been devoted to the case where the series in the model are integrated, so that they become stationary after differencing. This case assumes that the roots of interest are precisely one, and there are no other unit roots in the system. Such a root corresponds to a zero-frequency peak in the spectrum and describes the long memory properties of the series. Many economic time series, however, exhibit strong seasonality which can be characterized by seasonal unit roots corresponding to peaks at seasonal frequencies in the spectrum. In this case, the two-step procedure in Engle and Granger (1987) turns out to be inappropriate for a cointegration test at the zero frequency. The lack of consistency due to the presence of seasonal unit roots is shown in Engle, Granger, and Hallman [henceforth EGH] (1989). The problem of the standard cointegration technique in this context motivates an alternative approach to testing for cointegration in the presence of unit roots at other seasonal frequencies. A few attempts have been made by Hylleberg, Engle, Granger, and Yoo [henceforth HEGY] (1990) and Engle, Granger, Hylleberg, and Lee [henceforth EGHL] (1990) to extend the usual cointegration technique to the case where the data have unit roots at both the zero and seasonal frequencies. In these papers, they suggest that the Engle-Granger type two-step procedure be applied to appropriately filtered series. Notice however that we need prior information on which unit roots are present in order to filter out seasonal unit root components and to test for cointegration with the filtered series; hence we need to pretest for seasonal unit roots, the implications of which have not yet been investigated. One may, in practice, circumvent these problems by simply using seasonally adjusted series, as the usual seasonal adjustment process such as the X-11 method should eliminate seasonal unit roots, if any, in the original time series data [see Bell and Wilcox (1990) and Lee and Siklos (1991a)]. In fact, usage of seasonally adjusted data has become a well-established practice in much empirical work. As is pointed out in the literature, however, seasonal adjustment might lead to mistaken inference on economic relationships (1984) and Ghysels among time series data [see, e.g., Bell and Hillmer (1988)], and it causes a significant loss of valuable information on important seasonal behavior in economic time series if seasonal fluctuations are an important source of variation in the system. Problems of this kind in seasonal adjustment processes do suggest that time series analysts should generally work with seasonally unadjusted data. In the case of seasonally unadjusted data, strong evidence for the presence of unit roots at some seasonal frequencies as well as at the zero frequency has been found in a large number of macroeconomic time series [see Ghysels, Lee, and Siklos (1992) for a survey]. If seasonal fluctuations of this kind are an important feature in economic time series, it is natural to extend the concept of cointegration to consider the possibility that common roots exist
H.S. Lee, Cointegration
and seasonal cointegration
3
tests
at seasonal frequencies other than zero, which results in the idea of seasonal cointegration discussed by HEGY (19901 and EGHL (1990).
The aim of this paper is two-fold. First, we extend the standard cointegration technique to a testing procedure for cointegration that does not require any prior knowledge about the presence of seasonal unit roots. Second, we also develop estimation and testing procedures for seasonal cointegration at any set of the zero and seasonal frequencies. Throughout this paper, we consider a nonstationary VAR process which has unit roots at seasonal frequencies as well as at the zero frequency. Using maximum likelihood inference developed in Johansen (1988), we derive tests for cointegration and seasonal cointegration in time series data at each frequency in the presence of unit roots at other frequencies. The definitions of an integrated seasonal process and seasonal cointegration are presented in section 2. Section 3 develops a general framework of the testing strategy for seasonal cointegration that looks at unit roots at all seasonal frequencies as well as the zero frequency by deriving the ML estimation of cointegrating vectors and test statistics for hypotheses about cointegrating vectors. This testing procedure is extended to examine an interesting case where all of the cointegrating vectors coincide. In section 4, finite sample properties of the testing procedure are investigated by Monte Carlo simulations, and an empirical example is also presented. The conclusion is in section 5. 2. Seasonal integration and cointegration It has been widely observed that many economic time series exhibit substantial seasonal components and a variety of possible models for seasonality have been attempted to describe seasonal patterns. The definition of integration in Engle and Granger (1987) can be generalized to include seasonal integration. Definition 2.1.
Let S(B) have a root with modulus 1 at frequency 0 [i.e., S(B) = (1 - eie B)] for ,9 E (- 7, ~1, and also let D(B) have all the unit roots at seasonal frequencies as well as the zero frequency, if any, which are different from 19.A series X, with no deterministic component is said to be seasonally integrated of order d at frequency 8, denoted as X, - I,(d), if d is the smallest integer for which the representation S( B)dD( B)x, = C( B)E,
(2.1)
has the following properties: (i) the spectrum of C(B)E~ is bounded away from zero and infinity at all frequencies; (ii) {Ed}is a sequence of serially
4
H.S. Lee, Cointegration
and seasonal cointegration
tests
uncorrelated random vectors with finite and constant unconditional (iii) the series is taken to be initialized by Ed, X, = 0 for t I 0.
variance;
Note that the presence of D(B) in (2.1) allows a series x, being Z,(d) as well as I,(&) for 19~# 0, and d # d’. This definition generalizes the ordinary Z(d) process which is a special case of Z&d) where 0 = 0 and D(B) = 1. For d = 0 with D(B) = 1, X, will simply be stationary as an ordinary Z(O) process. From this definition, we see that if x, N Z,(l) with D(B) = 1, then (i) variance of x, goes to infinity as t + 03;(ii) X, has ‘long memory’ so that an innovation has a permanent effect on the seasonal pattern of xt; (iii) the (asymptotic) spectrum of X, takes the form f(w) =A( w - 0)-2 for o near f3 so that it has infinite peak at frequency 8.’ Moreover, it is well known that (iv> X, is asymptotically uncorrelated with processes which have unit roots at other frequencies2 This property of seasonally integrated time series plays an important role in analyzing the asymptotic distribution of estimates and test statistics. For simplicity, only the value d = 0 and d = 1 will be considered in this paper, but many of the results can be generalized to the cases of higher order. Some examples of integrated seasonal processes are presented in HEGY (1990). A very interesting and familiar example for a quarterly data might be (1 -ZP)X,=&,, which can be rewritten
(2.2) as3
(1-B4)X,=(1-Z3)(1+B+B2+B3)X,m(1-B)yl,
(2.3)
=(l+B)(l-B+B2-B’)x,=(1+B)y2,
(2.4)
= (1 +ZP)(1
(2.5)
-P)x,
= (1 +ZP)y3,,+,.
Using the above definition, the process x, in (2.2) can be denoted as x, - Z,(l) for any 0 = 0, r, and &r/2. That is, it has four roots with modulus ‘Strictly speaking the spectrum of x, is not well defined here, since x, is nonstationary so that it does not have a finite variance. Apart from a possible problem at frequency 8, however, the usual definition of spectrum can still be employed in this case. See Granger and Newbold (1986, ch. 2). ‘Two series, say, y,! and y2, (t = 1,2,. , T), are said to be asymptotically uncorrelated if the correlation coefficient between them vanishes as T+ ~0. See Ghan and Wei (1988) and HEGY (1990). “The expression note 4.
y,, , + , (rather
than
y,,)
is used here for notational
convenience.
See foot-
H.S. Lee, Cointegration
and seasonal cointegration
tests
5
one: one at the zero frequency (w = O), one at two cycles per year [i.e., a half cycle per quarter (w = +>I, and a pair of complex roots at one cycle per year [i.e., a quarter cycle per quarter (w = +>I.” We can now generalize the idea of cointegration in Granger (1986) and Engle and Granger (1987) to define seasonal cointegration. Definition 2.2. Let all components of x, be seasonally integrated of order 1 of the vector x, are said to be at frequency 0, i.e., x, - 1,(l). The components seasonally cointegrated at frequency 8, denoted x, - U&l, l), if there exists a vector (Y(# 01 so that z, = LY’X~ - Z,(O). For a vector of nonstationary series which has unit roots at some seasonal frequencies as well as at the zero frequency, it is possible that a single cointegrating vector could eliminate all the unit roots in the series. To formalize this idea, the following definition adapted from EGH (1989) is also introduced. Dejinition 2.3. Let each component of X, be seasonally integrated of order 1 at some frequencies, not necessarily at the same frequencies for all components. The components of the vector x, are said to be fully cointegrated, denoted x, - CZ(1, l), if there exists a vector (Y (Z 0) so that z, = (Y’x, is stationary. These definitions can be reduced to the ordinary cointegration when 0 = 0 and D(B) = 1. The implications of seasonal cointegration are not immediately obvious but are quite similar to those of the ordinary cointegration as established by Engle and Granger (1987). In particular, seasonal cointegration would mean that an innovation has only a temporary effect on the seasonal behavior of z, = &x1, while it may have a permanent effect on the seasonal pattern of x,. This would occur if each of the series contains the same factor which is Z,(l). The close relationship between cointegration and error correction models also allows us to develop error correcting mechanisms when the variables in the system are seasonally cointegrated, which will be employed in the next section to discuss a general framework of the testing strategy for cointegration and seasonal cointegration. 3. Testing procedures
for seasonal cointegration
This section proposes a testing procedure for cointegration in the presence of unit roots at other seasonal frequencies, and develops a general framework of seasonal cointegration test for nonstationary VAR time series with 4Following HEGY (1990), the expressions w = $ and w = f are used in this paper the seasonal frequencies 0 = T and 0 = *r/2, respectively.
to signify
6
H.S. Lee, Cointegration and seasonal cointegration tests
unit roots at all seasonal frequencies as well as at the zero frequency. The testing strategy follows the framework in Johansen (1988) so that we can adapt some results from his paper.
3.1. The seasonal cointegration
model
We will consider the case where the observed data is a sequence of random vectors X, with components (xlr, xZt,. . . , xnt)’ drawn sequentially from n-dimensional Gaussian random vectors {E[} with mean zero and variance matrix a. Thus the process can be written as x, =
@,x,_,+ dj2xrp2
+ . . . +@ p x I--p t=1,2
+et,
,...,
(3.1)
T,
where E,, E*, . . . , Em are i.i.d. N,(O, 0) and X, are hxed for t s 0. This is a VAR model of order p, and the n x n matrices (@,,@,, . . . ,@,, 0) are parameters to be estimated on the basis of T observations. Since we allow the process X, to have unit roots at seasonal frequencies as well as at the zero frequency, the determinant I@(z)l of the matrix polynomial Q(z)
=I-@,z-@,z2-
... -Qpz”
will have roots on the unit circle; one at the zero frequency and at least one at some seasonal frequencies, with all roots of l@(z>l = 0 satisfying Iz I 2 1. We shall mainly consider a simple case in which Asx, is stationary, where A, = 1 -B” and s is the number of observations taken per year, so that the determinant I@(z)/ has unit roots at the zero frequency (w = 0) and all seasonal frequencies w = j/s (j = 1,2,. . . , s/2). Thus the general structure of such a process can also be discussed by rewriting the model (3.1) as Asx,=17,S,(B)x,_,
+ ...
+~,~S,~(B)x,_,
+FI~A.~x,_~ + . . . +Ap_,AF~,_p+r
+AlAsX,-l + E,,
(3.2)
where S,(B) is a seasonal filter that will eliminate unit roots at all frequenties other than the one in the filtered series Yk,(_] = S#?)x,_, (k = \ 1,2,. . . ) s), and Il,‘s (k = 1,. . . , s> and Ai’s (i = 1,. . ,p - sl are approprrately redefined. Note that among the unit roots of l@(z)1 some of them are complex pairs, hence that the model (3.2) can be rewritten to include, if needed, lagged as terms Y~,,-~ of Y~,~+~. Notice also that this expression can be interpreted
7
H.S. Lee, Cointegration and seasonal cointegration tests
an error correction model for the process with unit roots both at the zero and some seasonal frequencies. This representation is quite general in that the coefficient matrices II, might be different at different lags as well as at different frequencies. Hence the cointegrating vectors and the coefficients of the error correction terms may also be different at different frequencies and, in some cases, at different lags. For ease of exposition, only the value s = 4 will be discussed here, but the results can naturally be extended, for example, to monthly data (i.e., s = 12) with minor modifications [see Beaulieu and Miron (1990)]. Now the model (3.2) will be reduced to
+A,dz,~~-~
+ . . . +Ap-4A4~t_p+4
+ E,,
(3.3)
where ylt=S,(B)x,=(1+B+B2+B3)XI,
and n, = - @(1)/4, II2 = @( - H/4, II3 = Real[@(i)1/2, Lid = - Im[@(i)1/2, A. = - c[!P-i)‘41@ (for i = 1,2,. . . , p - 4), and [(p - i)/4] denotes the IaZgest in&ger in [;‘,“- i)/4]. Notice that S,(B) is a seasonal filter that removes unit roots at all seasonal frequencies (i.e., w = i and a>, while S,(B) and S,(B) eliminate unit roots at seasonal frequency w = f and o = i, respectively, as well as at the zero frequency. This implies that y,,, y,,, and y,, have unit roots only at frequency w = 0, o = 4, and w = a, respectively. Since the coefficient matrices II,, . . . , II4 may convey the information concerning long-run behavior of the series, we need to investigate the properties of these matrices in order to determine whether or not the components of x, are seasonally cointegrated in the presence of unit roots at other frequencies. Note that if the matrix IIk has full rank, then the series does not contain unit roots at the corresponding frequency. If the rank of II, is zero, seasonal cointegration relationship at that frequency does not exist. In the intermediate case where 0 < rank(l7,) = r
the
term
Y~,~, and
will be employed
8
H.S. Lee, Cointegration and seasonal cointegration tests
Hence the natural hypothesis of seasonal cointegration can be formulated as the condition that rank(ll,) < r for corresponding frequency against the alternative that rank(ll,) > r. The possibility of seasonal cointegration at each frequency is discussed in the following subsections. Owing to the asymptotic uncorrelatedness between any two series with unit roots at different frequencies, it turns out that several null hypotheses can be tested for each case of interest without any prior information on the existence of cointegration relations at other frequencies.
3.2. Cointegration
at frequency
o = 0
When we want to test for cointegration at the zero frequency in the presence of unit roots at some seasonal frequencies, the hypothesis of interest can be given by H,: rank(U,) I r for some r < 12 against an alternative that rank(Z7,) > r. We can alternatively express this as H,:
n, = ?,a;,
where y1 and (Yeare n X r matrices. In order to test this hypothesis, we need to estimate the parameters y, and a,. They cannot be estimated, however, since they form an overparametrization of the model [see Johansen (19SS)]. But we can estimate the space spanned by y, and by cyI, and the estimates of sp(y,) and sp(cu,) can be used to test the hypothesis H,. We shall first show that testing H, for the model (3.3) can be done in a similar fashion to the procedure in Johansen (1988). This follows from the fact that the same distributions are obtained when it is not known a priori that they are seasonally cointegrated at some seasonal frequencies other than zero. We can consider two cases. First, if they are seasonally integrated of order 1 at frequency w = 4, for example, but not seasonally cointegrated at that frequency, then the term n, yZ I_, would be nonstationary and the true 17, would be equal to null matrix. Since the variables y,,,_, and y,,,_ , have unit roots at different frequencies, we know that they are asymptotically uncorrelated. Note also that both will be asymptotically uncorrelated with lags of A4x, which are stationary. Hence the distribution of the test statistic for H, will not be affected by the inclusion of the term y,,,- ,. Second, if they are seasonally cointegrated, then the term U, y,,,_ 1 will be stationary. In this case, the regression analysis is similar to the procedure in Johansen (1988) except for the augmented variable y2,+ 1. We shall now consider the maximum likelihood estimation of the parameters in the unrestricted model (3.3). Here we need to maximize the likelihood function with respect to the parameters (A,, . . . , FI~_~, fl> and (n,, . . . , Ii’,). is equivalent to an For fixed values of IZk’s (k = 1,. . . , 4) the ML estimation OLS regression of (A,x, - C~,117kyk,,_ ,) on the lagged differences.
H.S. Lee, Cointegration and seasonal cointegration tests
9
Let the product moment matrices of the variables in the model (3.3) be such that
Mmi,
= T-’
5
A‘,x,_~A~x;_~
for
i, j=O ,...,
~,,,_,A,x:_~
for
k = 1,. . . ,4,
p-4,
(3.4)
r=l T MIX,,
=
T-’
c
(3.5)
t=1
j=O ,..*, P - 4, T
-,I
=
T-’
c
for
Yk,r-&-I
l,...,
k,l=
4.
(3.6)
I=1
Then, from the normal equations provided by the OLS regression of the form T
T
p-4
C A,x,A,x~_~=
4
C Ai C A4xt-iA4xi-j+
t=1
i=l
T
k~lnkt~,yk,t-,d4x:-j,
r=1
(3.7) we
have 4
p-4
Mm,,=
c
A,MXXij+
c
for
IIkMYXkj
j= I,...,
p-4.
k=l
i=l
(3.8) This can be solved for Ai to give p-4 &II,,...,
fl,)
=
c j=l
MxXojMxXii
-
;
nkmkjMxii,
(3.9)
k=l
where MXX; ‘, = [MxX’j] is the inverse of MXX, .+, the matrix with elements M_XXij (i, j = 1,. . . , p - 4). By substituting these into eq. (3.31, we can get the residuals R, of the OLS regression. Alternatively, we can obtain the residuals by first regressing A4x, on the lagged differences [i.e., A4x,_j’s for j=l , . . . , p - 41 giving the residuals R,,, and then regressing each yk, t _ 1 on the lagged differences giving the residuals R,, (for k = 1,. . . ,4) and finally forming R, = R,, - c”,,,Il,R,,.
10
H.S. Lee, Cointegration
Now the concentrated
= Ifp/2
likelihood
and seasonal cointegration
function
becomes
proportional
R,,
exp
tests
to
R,,- t&R,, k=l
(3.10) Hence the MLE of the parameters IIk’s can be estimated by a usual OLS regression of R,,on Rkr's. That is, the MLE of IZ,‘s are equivalent to the estimates of the following regression equation:
4, = fl,R,t + HzR,t + fl,R,* + 4&; For notational the residuals,
convenience, i.e.,
let Sij denote
i,j=O
+ 5,.
the product
I’.‘, 4,
(3.11) moment
matrices
of
(3.12)
t=1
be the matrix with elements Sij (i, j = 1,. . . ,4) and let S,, .+ = and let S,, D = (Z7,. . . I&,) and R,, = CR',, ...R&j',the MLE (S a,. . . S,,). By denoting of Z7 is given by
(3.13)
We can also write (3.14)
~=~,,,,-~~,*~,j,~;,*,
2,‘s (i = 1,. . . , p - 4) can be derived by substituting the 1,. . . ,4) into eq. (3.9). model under H 1: IZ, = ylcJ,. For fixed value of II,, the (ZI, I7s II,) is equivalent to an OLS regression of CR,,, -IT,R,,) on (R2(R,,Rdr). The normal equations provided by the OLS regression are and the estimates of estimated fik’s (k = Now consider the ML estimation for
j=l k=l
1=1
7. . * , 4.
(3.15)
H.S. Lee, Cointegration
Using the notation
s”j
=
5 k=l
and seasonal cointegration
(3.12), eq. (3.15) can be rewritten
11
tests
as
(3.16)
nkskjY
or equivalently,
s,j
-n&j
=
i
17,skj,
k=2
which can be solved for Ilk’s (k = 2,3,4)
li,( Iz,) = ;
to give
(3.17)
( S”js’k - n,S,jS’k),
j=2
where S;; = [Sik] is the inverse of S,,, the matrix with elements Sjk (j, k = 2,3,4X By substituting these into (3.111, we get the residuals Q, from the OLS regression. Alternatively, we can also get the residuals by first regressing R,, on (R,, Rxt Rdt) to give the residuals Q,,, and then regressing R,, on (R,, R,, Rdt) to give the residuals Q,,, and finally forming Q, = Q,, - n,Q,,. Hence the concentrated likelihood function becomes proportional to
L(II,,R)
= I0l-““exp
[
-~,~,(Q,,-,,,Q,~).~-~(Q,,-,,,Q,,)
1
(3.18) so that the MLE of the parameter IJI, can be estimated by a usual OLS regression of Q,, on Q,,. By denoting Dij = T- ‘CT_ lQ,,Qi, (i, j = 0,. . . ,4), we can write -1 = D,,D,‘,
(3.19)
I h = Do0 - D,,D,‘D,,,
(3.20)
and the estimates of fik’s (k = 2,3,4) and ai’s (i = 1,. . . , p - 4) can be derived by substituting the estimated fi, into (3.17) and (3.9) sequentially. For fixed (or, we can maximize (3.18) over yr and R by an OLS regression of
7
12
Q,t
H.S. Lee, Cointegration
and seasonal cointegration
tests
on CX’,Q,, to obtain
.
fl(a1> = D”cl- 4u’y*(~;DII~I)
-I
(3.22)
4D,“.
The likelihood profile now becomes proportionai to ].f?(~~,>j -“’ SO that, as shown in Johansen (1988), it remains to solve the minimization problem
min
I”;D,,a, -
‘Y;D,oD,olDo,a, 1
I”p,,a,
[a,1
(3.23)
>
I
where the minimization is over all n x r matrices LY,.We can now estimate CX,by choosing the first r eigenvectors of D,, D,‘DoI with respect to D, ,. Based upon Johansen (1988, theorem l), the test statistic for the hypothesis J-I, shall be formulated in the following theorem, which provides inference about the number of important cointegration relations at the zero frequency. The proofs of our theorems are given in the appendix. Theorem 3. I. The likelihood ratio test statistic for the hypothesis that H,: l7, = y,o!, where y, and ayI are n x r matrices, i.e., that there are at most r cointegrating vectors at zero frequency, is -2ln(Q)
= -T
2 ln(l i=r+ I
(3.24)
-i,,i),
where /i l,r+l,..., i I,n are the (n - r) smallest squared partial canonical correlations of R, with respect to Ro, given R,, R,, and R,, i.e., the first (n -r) eigenvalues of D,,D,‘DO, with respect to D,,. The test statistic of this hypothesis is then asymptotically distributed as (3.25) where W( r> is an (n - r&dimensional standard Brownian motion on the unit interval [0,11. Remark 1. From the result in Johansen (19881, it is clear that the test for n, = -y,c.~‘,will have the asymptotic distribution in (3.25) if 112 = L’s = fl, = 0 (null matrix), since the model then can be written in the form A4x, =D,Y,,~_,
+A,A,x,_,
+ . . . +&-4Az,~t-~+a
+ “t.
H.S. Lee, Cointegration
and seasonal cointegration
tests
13
Asymptotic uncorrelatedness of y,,(_, with Y~,~_,, y3,[_, and Y~,,-~ will lead us to the same distribution whether or not it is known a priori that some of the IIk’s are zero. Notice that the MLE of the space spanned by (Y, is the space spanned by the r canonical variates corresponding to the r largest squared partial canonical correlations between y, f_, and A4x, given lagged A4x,‘s 1 and other yk f_ ,‘s (k = 2,3,4X (A,x,-,,...,A,+,+, 3.3. Seasonal cointegration at seasonal frequency w = + This subsection proposes a testing procedure to determine whether or not the components of X, are seasonally cointegrated at seasonal frequency w = 4 in the presence of unit roots at the zero frequency and/or at other seasonal frequencies. The hypothesis of interest can be formulated as
where y2 and (Ye are n x r matrices. The testing procedure is very similar to the one developed in the previous subsection except that the role of Rr, and R,, is reversed. That is, the MLE of the parameter ZI, with no restriction on II,, II,, and 114 can be estimated by an OLS regression of Q,, on Qzt, where Qzt is the residual from regression of RZt on CR,, R,, R4t). We can write
(3.26)
d = D,,
- D”,
DT*‘D,,
(3.27)
)
and the estimates of fi,‘s (k = 1,3,4) and 2;‘s (i = 1,2,. . . , p - 4) can now be derived by using the estimated 17,. By arguments similar to Theorem 3.1, we can formulate the results for the test statistic of the hypothesis H, in the following theorem. Theorem 3.2. The LR test statistic for the hypothesis that there are at most r seasonal cointegrating vectors at frequency w = t is -2ln(Q)
= -T
k i=r+
ln(l
-i,,i),
(3.28)
I
where i 2,r+lY..J2,n are the (n - r) smallest squared partial canonical correlations between R, and R,, given R,, R,, and R,, which are just the first (n - r) eigenvalues of D20D&1D02 with respect to D,,. Furthermore, this test statistic has the same asymptotic distribution as (3.25).
14
H.S. Lee, Cointegration and seasonal cointegration tests
Remark 2. Using the notation in Lemma A.1 [see eqs. (ASa) and (A.5b) in the appendix], it is easy to see that /d dW, W; 2 /d dWz W;, which was first observed by Fuller (1976, p. 370) for the univariate case. The same asymptotic distribution for testing seasonal cointegration at seasonal frequency o = i as (3.25) follows from this observation.
3.4. Seasonal cointegration at seasonal frequency w = f To test the hypothesis of seasonal cointegration at frequency w = f, we need to look at the two parameter matrices, 17, and II4, simultaneously since the cointegrating vectors and the coefficients of the error correction term may be different at different lags. As is pointed out in HEGY (1990), this might imply that we need to consider the polynomial cointegrating vector (PCIV) since one vector is sought to eliminate two unit roots C&i) [see Engle (1987)], hence that a much more complicated testing procedure than the previous two cases may be required for testing the joint hypothesis on Ii’, and I14. For simplicity of exposition, it will be assumed throughout in this paper that cointegration, if any, is contemporaneous. Then the hypothesis of interest can be formulated as a joint test such that
where y3, yq, and LYEare n X r matrices. For fixed values of I73 and 114, the ML estimation for (IZ, II,> is equivalent to an OLS regression of CR,,, - n3R,, - f14R4f) on CR,, R2,). We can also show that the MLE of the parameter matrix [n, II,] can be estimated by an OLS regression of Q,, on Q3* and Qdt, where Q3, and Qdr are the residuals from regression of R,, and RJr on (R,, Rzt), respectively. Then we have
r=1
where
-
[Q,tQJQ,tQ,,l
r=1
D,,D,-,’>
(3.29)
D,,,D,;‘D,,,,
(3.30)
=
h = D,,
5
-I
H.S. Lee, Cointegration and seasonal cointegration tests
15
The estimates of fir, I?,, and a,‘s can then be derived by substituting the estimated [II, III41 into the normal equations of the form similar to (3.17) and (3.9) sequentially. For fixed (Ye, we can maximize the concentrated likelihood function
x (Qo,- n,Q,t - KQ4,) over y3, y4, and obtain
(Y is a 2n X 2r matrix
(Y=
By applying minimization
min
of
Q,, on abQ,, and ~;QA, to
=D,,,-D,qa(a~Dqqa)-la~Dq”,
ii where
0 by an OLS regression
1
(3.32)
such that
aq O 0 aq . i
I
the matrix problem
relation
used in (3.23), it now remains
b’Dqqcx - cx’DqOD,,‘D OqffI
[%,I
to solve the
(3.33) ’
Ia’Dqq~I
where the minimization is over all n x r matrices (Y We can then formulate the test statistic for the hypothesis H, in the follow&g theorem. The LR test statistic for the hypothesis that there are at most r Theorem 3.3. seasonal cointegrating L:ectors at frequency w = + is -2ln(Q)
= -T
i i=r-t
ln(l -i,,j-&i).
(3.34)
1
where /ik,r+ ,, . . .,/i, n are the (n - r) smallest squared partial canonical correlations of R, (k = j, 4) with respect to R,, gicen R, and R,, i.e., the first
16
H.S. Lee, Cointegration and seasonal cointegration tests
(n - r) eigenvalues of DkODi’DOk with respect statistic for H, is asymptotically distributed as
tr
‘(dW, W,l +dW,
W,)
to D,,
jji’(dWC W,-dW,
(k = 3,4).
W,,)
\ L’(W,W:+
W,W,l)dr
X
/‘( V$Wi - w,Wi) 0
dr
/‘(WSWi0
The test
-I
W,W,l)dr
/‘( W,Wi + W’,Wi) dr 0 I
(3.35)
where W,(r) and W,(r) in (n - r ) dimensions.
are mutually
independent
standard
Brownian
motions
Based on the structure of the error correction mechanism for a process with unit roots both at the zero and seasonal frequencies given by (3.3), we can further assume that y4 = 0 (i.e., 17, = 0) with little effect on the test for seasonal cointegration at frequency w = d when cointegration is contemporaneous. This observation is very useful in practice, since in this case the test can be performed by looking at the matrix IIJI, only [see section 41. That is, if we restrict our attention to the case where it is assumed that U4 = 0, the hypothesis H, will be reduced to the condition that
and the testing strategy will then be similar to that for seasonal cointegration at frequency w = 4 except that R3, takes the role of R,,. The MLE of the parameter U3 with no restriction on II, and f17, can be estimated by an OLS
H.S. Lee, Cointegration
and seasonal cointegration
tests
17
of Q,, on Q3t, to obtain
regression
(3.36) fi = D,, - D,, D,‘D,,
(3.37)
.
By arguments similar to those in previous subsections, we can formulate the test statistic for the hypothesis H, as a simple corollary to Theorem 3.3. Under the assumption that IT, = 0, the LR test statistic for the Corollary 3.4. hypothesis that there are at most r seasonal cointegrating vectors at frequency w=$is -2ln(Q)
= -T
i ln(1 -ij,i), i=r+ 1
(3.38)
,.
where i 3,r+, , , . , , A 3,n are the (n - r ) smallest squared partial canonical correlations of R, with respect to R,, given R, and R,. The test statistic for H, is asymptotically distributed as
x
‘(W,dW,‘+W,dW,‘)
(3.39)
.
/0
I
Remark 3. Notice that for r = n - 1, i.e., for testing (n - 1) seasonal cointegrating relations at frequency w = i, we obtain the asymptotic distribution with one-dimensional Brownian motions. In this case, (3.35) will be reduced to /l(w,dW,+
1[
W$dW,) *+
0
j’(w,dW,-W.dWJ 0
/‘(W,*
+ WS”)dr
0
which is the limiting distribution of
WdW,)
tSTl are the w = $ [see
H.S. Lee, Cointegrution and seasonal cointegration tests
18
which is tantamount to the square of the Dickey-Hasza-Fuller for biannual seasonality [see also Chan and Wei (1988, corollary
distribution 3.3.8)1.
3.5. Full cointegration The testing procedures for seasonal cointegration have so far considered the general error correction model of the form (3.3) so that the cointegrating vectors may be different at different frequencies. It is however possible that some of the cointegrating vectors coincide but some do not. If (Ye= a4, one cointegrating vector, say CY~,will remove all seasonal unit roots. This might be particularly interesting when it is expected that the seasonality in the time series data is due to the same source. This is the case examined in EGH (1989) for electricity demand. Similarly, the cointegrating vectors GJ,, (Y*, and cyy may coincide, and therefore one ccintegrating vector, say (Y~, could eliminate unit roots of the series at all frequencies. In this case, we have full cointegration as defined in Definition 2.3. Since an efficiency gain in the estimates of the cointegrating vectors is naturally expected by imposing the restrictions between the cointegrating vectors, it may be worth checking the existence of such relationships between the cointegrating vectors once seasonal cointegration relations at different frequencies are identified. The general framework of seasonal cointegration tests proposed in previous subsections is extended here to discuss full cointegration. Suppose that the cointegrating vectors LY,, crz, and a4 coincide (i.e., say, Ly, = (Y, = ay = a,). Then the error correction model (3.3) will be reduced to A4x, =n,(B)x,_,
+A,A4xt-,
+ .
+&~A~~,-,,+~
+&r,
(3.40)
three lags. This indicates where II,(B) = yA(B)& and yA(B) has potentially that a single linear combination may eliminate all the unit roots in the time series data. The hypothesis of interest can now be formulated as a joint test such that
x r matrices. By arguments similar to those in the previous subsection, we will restrict our attention to the case where it is assumed that yA, = yA, = yA, = 0 (i.e., flA, = nA, = flAX = 0) for ease of exposition. However, the results can be extended in a straightforward way as in Theorem 3.3.6 Under the assumption that yA(B) = yAB’, the model
where YA,, ?'A:, YA,' YA,? and LY~are n
‘Alternatively, estimate II,(E)
the index model in (3.40).
estimation
procedure
by Reinsel
(1983) can be employed
to
H.S. Lee, Cointegration
(3.40) can be written
procedure
H,:
tests
19
as
A‘+, =IT/v_~ The testing
and seasonal cointegration
+A,A,x,_,
+ . . . +Ap-J~qp+z,
+&t.
for the hypothesis
rr, = r,4aa
can now be formulated
in the following
theorem.
The LR test statistic for the hypothesis that there are at most r Theorem 3.5. full cointegrating oectors is -2ln(Q)
= -T
5
ln(l --iA,;),
(3.41)
i=r+l
are the (n - r) smallest squared canonical correlations where i A,r+l,...&4,n of R, with respect to R,, where RAl is the residual from regression of x,_~ on A4x,_j’s (j = 1,. . . , p - 4). The test statistic for HA is asymptotically distributed as
where Wj(r)‘s are mutually independent standard Brownian motions in (n - r) dimensions. Remark 4. Note that for r = n - 1, i.e., for testing (n - 1) full cointegrating relations, we get the limit distribution with one-dimensional Brownian motions. In this case, (3.42) will be reduced to
which is the square and Fuller (1984).
of the ‘seasonal
4. Finite sample behavior:
unit root’ distribution
Some simulation
in Dickey,
Hasza,
results
Since inference on cointegration and seasonal cointegration developed in the previous section has largely been based upon the asymptotic distribution theory, it is of great importance to investigate the finite sample properties of
20
H.S. Lee, Cointegration
and seasonal cointegration tests
Table Quantiles tn -r)
in the distribution
1o/c
Frequency w=o,; u=$ Full CI
1
2
3
0.00 0.00 0.00
1
of the seasonal
5%
10%
50%
0.01 0.01 0.00
0.02 0.02 0.02
0.58 0.52 0.49
0 = 0, I
1.27
1.99
2.54
5.42
m=; Full CI
0.37 0.32
1.08 0.86
1.51 1.30
4.68 4.01
w=o,: &I=_:Full Cl
6.03 4.61 2.79
7.79 6.38 4.09
x.93 7.49 5.13
14.3 12.1 10.6
cointegration 904 2.89 2.82 2.79 10.5 9.69 X.82 21.6 19.3 17.0
test.”
95% 3.94 3.81 3.80
97.5% 5.08 5.05 5.02
99% 6.73 6.70 6.69
12.5
14.4
16.3
11.7
10.6
13.1 12.7
15.3 14.6
24.3 21.7 20. I
26.4 24.1 21.9
29.8 27.3 24.9
“The stochastic integral is approximated by sum of 400 i.i.d. Gaussian variables, i.e., 400-step Gaussian random walks are used for the standard Brownian motions in the asymptotic distributions of the LR statistics: (3.25), (3.39), and (3.42). 30,000 replications.
the testing procedure. In this section, the finite sample distributions of the test statistics are examined by Monte Carlo simulations, and an empirical application to Canadian data on unemployment and immigration rates is presented. The asymptotic quantile distributions are first constructed by replacing the stochastic integral by the sum of T = 400 random Gaussian variables, using the GAUSS programming language. Table 1 reports the limiting distributions of the LR statistics [(3.25), (3.391, and (3.42)], which can be used as the critical values for cointegration and seasonal cointegration at each frequency. The figures are obtained through 30,000 replications. Notice that the 95% acceptance regions of the critical values are not very different from each other, except for the n = 3 case. To examine the finite sample behavior of the LR statistics in (3.24), (3.28), (3.38), and (3.41), simple simulation experiments are designed such that a sequence of random vectors X, with elements (x,~ xzl.. . x,,~)’ are generated by
x,
=x,-4
+ E,
)
t=l
,...,
T,
(4.1)
=xg = 0. Now the where &i, Ed,. . . , Ed are i.i.d. N(0, I,,) and x_~ =xP2 =x-, seasonal cointegration model (3.3) will take a simple form
H.S. Lee, Cointegration
and seasonal cointegration
tests
21
By construction, there are no seasonal cointegration relations among the elements of the vector X, at any frequencies (w = 0, i, or a>. It is then clear that the true lIk’s are null matrices [i.e., rank(L’,) = 01 for k = 1,. . . ,4. Hence it follows that the distribution of the test statistics for the null of II, = ykak = 0 provides information on the size of the test. The figures are computed through 10,000 replications in each case for n = 1,2,3 and for T = 50,100,300. The results are presented in table 2, from which we can obtain the actual rejection frequency of the null hypothesis for the cointegration test when the variables are generated by (4.11.’ Notice first that we tend to reject the true null hypothesis of no (seasonal) cointegration [i.e., r = 01 more often than is presumed if we use the asymptotic distribution in table 1. This result suggests that the finite sample quantile distribution in table 2, rather than the asymptotic distribution in table 1, should be used in practice as the critical values for the cointegration and seasonal cointegration tests. It is, however, interesting to see whether such a discrepancy between table 1 and table 2 becomes negligible as T grows. We can see that the actual rejection incidence is close to the nominal size of the test except for small T CT= 501. Notice also that the larger the number of observations T, the closer the empirical size of the test to the significance level. This indicates that the actual rejection frequency approaches the asymptotic rejection probability as T grows. Since we know in this case that the true Dk’s are zeros, it may be worth checking the effects of inclusion of the terms that are asymptotically uncorrelated. This result is shown in table 3. The first row of each case [denoted by
] shows the distribution from the partial canonical correlations between R, and R, corrected for the effects of R,'s(I# k),while the second row [denoted by CR)]shows the quantiles from the canonical correlations between R,,and R,.As expected, the distribution of the test statistics turns out to vary little by the inclusion of asymptotically uncorrelated terms. The power properties of the test have also been examined for (n = 2, r=l)and(n=3,r=1,2).Forthecasewheren-r=l(i.e.,n=2,r=lor n = 3, r = 21, the data are assumed to be generated by Yf =
where
2x, +
x, is drawn x,
‘X1t
BYl
3
2, = 3x, + EZI)
(4.3)
from (4.1). For n = 3 and r = 1, we assume
+ X2t + &,I 7
Y, ‘Xl<
-
2x2,
+Eytt
2, ‘X2t
that -
2x1,
+c,r,
(4.4) 7The results for o = f are suppressed in the tables below, since they are essentially similar to those for w = 0. Note that for n = 1 the result shows the behavior of the seasonal unit root tests at each frequency.
H.S. Lee, Cointegration and seasonal cointegration tests
22
Table 2 Quantiles
(n-r)
T
1%
in the distribution
5%
10%
(1) Seasonal
of the LR statistics.”
50%
cointegration
90%
95%
97.5%
4.52 4.29 4.25
5.90 5.67 5.36
99%
test at w = 0, +
1
50 100 300
0.00 0.00 0.00
0.01 0.01 0.01
0.03 0.03 0.02
0.67 0.62 0.60
2
50 100 300
1.40 1.32 1.31
2.29 2.08 2.06
2.91 2.65 2.60
6.17 5.82 5.60
3
50 100 300
7.26 6.48 6.33
9.30 8.48 8.27
10.7 9.74 9.43
1
50 100 300
0.00 0.00 0.00
0.01 0.01 0.01
0.02 0.02 0.02
0.60 0.58 0.55
3.32 3.15 3.02
2
50 100 300
0.49 0.48 0.48
1.20 1.15 1.08
1.81 1.76 1.64
5.14 4.88 4.67
10.8 10.2 9.82
13.0 12.1 11.9
15.1 14.1 13.4
17.9 16.7 16.0
3
50 100 300
5.23 5.03 4.80
7.35 6.71 6.29
8.62 7.97 7.52
23.2 21.5 20.1
25.8 24.0 22.5
28.6 26.5 24.3
32.5 29.6 27.5
(2) Seasonal
17.0 15.5 14.8
cointegration
14.7 13.5 12.7
(3) Full cointegration
3.25 3.09 3.04 12.0 11.3 10.7
14.1 13.1 12.7
16.1 15.1 14.5
1x.9 17.2 16.X
26.0 23.3 22.2
29.3 26.1 25.0
32.5 28.5 26.8
36.3 31.6 29.4
test at w = L 5.95 5.79 5.47
4.67 4.45 4.20
-
7.61 7.34 6.82
test
1
50 100 300
0.00 0.00 0.00
0.01 0.01 0.00
0.02 0.02 0.02
0.56 0.52 0.50
3.27 3.07 2.88
2
50 100 300
0.38 0.37 0.36
0.93 0.89 0.86
1.38 1.34 1.29
4.39 4.18 4.04
9.99 9.43 9.13
3
50 100 300
3.19 2.86 2.71
4.90 4.70 4.39
6.23 5.84 5.35
12.1 11.2 10.7
7.5 1 7.25 6.86
20.3 19.1 18.0
4.50 4.35 4.07
5.72 5.68 5.25
7.34 7.28 6.84
12.1 11.4 11.1
14.2 13.2 12.8
16.3 15.9 15.1
23.2 22.0 20.6
25.9 24.4 22.7
29.0 27.3 24.9
‘The figures are obtained through 10,000 replications using the regression equation of the form (4.2). The finite sample critical values for the regression models with and without deterministic terms (i.e., a constant, seasonal dummies, and/or a linear trend) are presented in Lee and Siklos (1991b).
by (4.1). The result, summarized in tables 4, where (x,~, xZt) are generated indicates that the power of the test for seasonal cointegration in rejecting the false null hypothesis increases with T. Tables 4 also show the size properties of the test when the variables in the model are seasonally cointegrated at each frequency with cointegrating rank r = 1 or r = 2. We might derive the same conclusion in this case as those on table 2. That is, the actual rejection
H.S. Lee, Cointegration
and seasonal cointegration
tests
23
Table 3 Comparison between (R, &R,)
(n -r)
T
1%
5%
10%
and (QO & Qk). 50%
90%
95%
99%
(1) Seasonal cointegration test at 0 = 0, $ 1
2
3
100(Q) 100(R) 300(Q) 300(R)
0.00 0.00 0.00 0.00
0.01 0.01 0.01 0.01
0.03 0.03 0.02 0.03
0.62 0.63 0.60 0.63
100(Q) 100(R) 300(Q) 300(R)
1.32 1.33 1.31 1.29
2.08 2.10 2.06 2.05
2.65 2.69 2.60 2.58
5.82 5.83 5.60 5.57
100(Q)
6.48 6.49 6.33 6.36
8.48 8.53 8.27 8.21
9.74 9.74 9.43 9.40
100(R) 300(Q) 300(R)
15.5 15.3 14.8 14.7
3.09 3.09 3.04 3.06
4.29 4.22 4.25 4.27
7.25 7.18 6.86 6.98
11.3 11.1 10.7 10.7
13.1 13.2 12.7 12.7
17.2 17.1 16.8 16.5
23.3 23.1 22.2 22.2
26.1 25.7 25.0 24.8
31.6 30.8 29.4 29.6
(2) Seasonal cointegration test at w = 4 1
2
3
4.45 4.40 4.20 4.19
7.34 7.31 6.82 6.91
100(Q) 100(R) 3ONQ) 300(R)
0.00 0.00 0.00 0.00
0.01 0.01 0.01 0.00
0.02 0.02 0.02 0.02
0.58 0.58 0.55 0.55
3.15 3.10 3.02 3.01
100(Q) 100(R) 300(Q) 300(R)
0.48 0.50 0.48 0.49
1.15 1.19 1.08 1.10
1.76 1.77 1.64 1.65
4.88 4.85 4.67 4.66
10.2 10.4 9.82 9.75
12.1 12.7 11.9 11.7
16.7 16.4 16.0 15.9
100(Q) 100(R) 300(Q) 300(R)
5.03 4.78 4.80 4.75
6.71 6.72 6.29 6.35
7.97 7.94 7.52 7.49
21.5 20.8 20.1 20.1
24.0 23.8 22.5 22.7
29.6 29.6 27.5 27.4
13.5 13.3 12.7 12.7
frequency approaches the theoretical rejection probability as the number of observations T becomes larger. The LR statistic in (3.38) for seasonal cointegration at frequency w = f is constructed under the assumption that U, = 0. In order to examine the influence of exclusion of IId on the test for seasonal cointegrating relationship at frequency o = a, the distribution of the LR statistics with y4,+, term is also presented in tables 4. It turns out that inclusion of the term y,,,_ 1 has little effect on the distribution of the test statistics, suggesting that the assumption might be reasonable from a practical point of view when cointegration is contemporaneous as in (4.3) and (4.4). Given that the finding of polynomial cointegrating vectors is rare, this observation is useful in practice since the test can then be performed by simply looking at the matrix f13 only. We shall now examine the existence of cointegration and/or seasonal cointegration relations at different frequencies using Canadian data on
24
H.S. Lee, Cointegration
and seasonal cointegration
tests
Table 4a Quantiles
in the distribution
True model:
T
1%
100 300
15.1 49.3
)J, = 2x, + E,,
where
5%
10%
(1) Seasonal
100 300
17.8 55.6
0.00 0.00
100 300
28.5 97.3 0.00 0.00 Cf. seasonal
100 300 100 300
28.8 98.2 0.00 0.00
cointegration
cointegration 35.2 110
0.01 0.01
27.4 71.1 0.68 0.65
100 300
49.1 175 0.00 0.00
55.6 182 0.00 0.00
90%
45.6 127 0.65 0.59
46.2 127
0.02 0.02
59.2 188 0.02 0.02
99%
40.7 89.4
46.7 95.4
37.1 84.9 3.39 3.31
58.6 147
4.61 4.61
0.70 0.58
62.8 153
3.37 3.23
test at w = 4 [no assumption 35.5 110
95%
8.27 7.08
test at w = f [with I74 = 01
(3) Full cointegration 100 300
by (4.1)
test at w = 0, i
0.02 0.02
cointegration
0.01 0.00
50%
0.03 0.03
32.6 101
31.7 105
x, is generated
19.3 59.2
0.01 0.01
(2) Seasonal 100 300
of the LR statistics.
(n - r) = (2 - 1) = 1
59.5 147 3.49 3.24
4.67 4.43
69.2 162 7.32 7.62
on IT,] 63.4 153 4.77 4.57
70.1 163 7.90 7.66
test
72.9 212 0.57 0.44
87.7 236 3.26 2.73
94.0 243 4.52 3.75
104 257 8.29 6.38
unemployment and immigration rates. The choice of the example comes from the previous empirical literature on the relationship between unemployment and immigration [see Marr and Siklos (1988) for a survey of the literature]. Quarterly data for the period of 1962.Ql-1985.Q4 are used with IMM, = immigration/labor force and lJNf = unemployment rate. As is shown in fig. 1, it is clear that both series have a strong seasonal pattern as well as a random walk character. From this preliminary finding, we can expect to find seasonal unit roots in both series, and hence we can possibly test and estimate seasonal cointegration relationships. For the choice of lag length p in the model (3.31, the usual model selection criteria such as the AIC and SC are employed and Box-Pierce Q-statistics are also examined to test for uncorrelatedness of the residuals. First, seasonal unit root tests for each series are carried out at each frequency and the
H.S. Lee, Cointegration and seasonal cointegration tests
25
Table 4b Quantiles
in the distribution
True model:
2, = 3x, + EZI.
Yr = 2x, + ey,,
of the LR statistics.
(n - r) = (3 - 2) = 1 where
x, is generated
r
T
0
100 300
35.4 113
41.7 120
44.0 126
56.4 144
1
100 300
11.7 47.0
14.3 51.2
15.9 53.4
22.4 63.8
2
100 300
1%
5% (1) Seasonal
0.00 0.00
0.01 0.00
(2) Seasonal
50%
10% cointegration
90%
95%
99%
70.1 165
74.6 172
83.6 180
29.3 74.6
31.9 77.7
37.9 86.5
test at w = 0, +
0.03 0.02 cointegration
by (4.1)
0.71 0.62
3.32 3.27
4.47 4.37
7.15 6.97
test at w = $ [with 114 = 01
0
100 300
64.4 211
73.7 222
77.4 230
92.7 254
110 279
116 289
127 307
1
100 300
24.6 91.2
28.2 98.3
30.2 102
38.2 116
47.9 130
50.9 134
57.0 142
2
100 300
0.00 0.00 Cf. seasonal
0.00 0.01
0.65 0.59
0.02 0.03
cointegration
3.14 3.06
test at 0 = + [no assumption
4.06 4.09 on
7.21 7.15
&I
0
100 300
67.4 213
75.9 224
79.2 231
94.5 255
113 282
117 289
127 307
1
100 300
24.3 91.2
28.3 98.4
30.6 102
39.1 116
49.6 131
52.0 135
56.7 142
2
100 300
0
100 300
113 366
123 384
128 393
147 424
168 457
174 467
184 482
1
100 300
43.8 164
49.4 173
53.0 179
63.9 197
76.0 214
78.1 221
85.0 230
2
100 300
0.00 0.00
0.01 0.01
0.63 0.59
0.02 0.03 (3) Full cointegration
0.00 0.00
0.01 0.00
0.03 0.02
3.35 3.05
4.56 4.20
7.51 7.18
test
0.64 0.57
3.30 3.26
4.43 4.46
7.16 6.94
LR statistics are reported in table 5. The result indicates the presence of unit roots at seasonal frequencies as well as at the zero frequency in both UN, and ZMM,. The seasonal cointegration tests at each frequency are then performed. As is shown in table 5, we can reject the null hypothesis of no seasonal cointegration at the biannual frequency (w = :) at the 5% significance level, but not at any other frequencies. The absence of cointegration at
26
H.S. Lee, Cointegration
and seasonal cointegration
tests
Table 4c Quantiles
in the distribution
of the LR statistics.
(n - r) = (3 - 1) = 2
True model: r, = WI, + W2, + EX,’
y, = w,, - 2W*, + FYl.
where
w,, and w2, are generated
2, = w>, - 2w,, + E;,, by (4.1)
r
T
1%
0
100 300
20.6 56.2
1
100 300
1.65 1.32
2.29 2.13
2.91 2.72
6.60 5.97
2
100 300
0.00 0.00
0.00 0.00
0.02 0.02
0.41 0.36
0
100 300
1
100 300
0.66 0.33
1.37 1.07
1.92 1.62
5.30 4.86
2
100 300
0.00 0.00
0.00 0.00
0.01 0.01
0.28 0.24
0
100 300
1
100 300
1.37 1.31
2.79 2.5 1
3.92 3.14
8.63 7.53
2
100 300
0.00 0.00
0.00 0.00
0.02 0.02
0.47 0.44
0
100 300
1
100 300
0.31 0.35
1.00 0.96
1.47 I .32
4.57 4.21
10.2 9.91
2
100 300
0.00 0.00
0.00 0.00
0.01 0.01
0.27 0.23
1.61 1.32
10%
5% (1) Seasonal 24.6 63.0
(2) Seasonal 33.7 102
Cf. seasonal 37.8 106
50%
cointegration
cointegration 41.9 116
cointegration 42.9 I15
36.0 80.2
60.6 186
52.3 97.8
58.5 105
12.1 11.6
13.8 13.2
19.8 17.7
2.08 I.64
64.6 193
2.72 2.13
4.13 3.84
test at w = f [with IJ4 = 0] 53.2 134
58.4 138
(3) Full cointegration 54.1 173
47.7 93.9
67.5 151
71.7 157
79.9 I68
13.2 12.6
18.4 17.8
11.3 10.2 1.68 1.65
test at w = $ [no assumption 46.2 121
99%-
test at w = 0,;
26.8 66.0
39.5 112
954:
90%
2.42 2.18
4.13 3.69
on II,]
73.7 157
78.5 161
86.2 173
16.2 14.7
19.2 18.0
23.2 22.2
2.47 2.34
3.39 3.27
5.76 5.70
test
79.2 214
96.0 217
101 251 12.5 11.8 2.18 2.10
Ill 262 17.9 15.9 4.12 3.62
the zero frequency might be expected from the plots of the data.x Hence we conclude that the data exhibit one seasonal cointegrating relationship at w = i only, which implies that DII, = Z13 = L14 = 0 and 17, = y2~; for 2 X 1 vectors ‘For comparison, the usual test for cointegration is also performed by applying ML inference in Johansen (1988). The result (not reported here) shows no cointegrating relationship, indicating that no information can be drawn from the standard cointegration approach on the comovement between unemployment and immigration rates.
H.S. Lee, Cointegration and seasonal cointegration tests
27
UN&Y 13:
::, :: ’ :::: !i :
12:
: :;: :: : :.i \ i \ : -,
11: 10:
:
9
Immigration
1 1960 Fig. 1. Immigration
(MM)
* 10
I
I
1
1970
1980
1990
and unemployment
rates (IIN);
1962.Ql-1985.Q4.
Table 5 LR statistics
for seasonal
cointegration
test. .~
Seasonal Frequency
Unemployment
w=o m=+ u=;
Immigration
0.431 3.301 0.227 Seasonal
Frequency
0.841 3.712 2.602
cointegration
Rank(Hk)
o=o @Et @Cf “Significant
unit root test
test
= 0
Rank(nk)
3.251 13.567” 8.184
= 1
0.556 3.837 0.252
at the 5% level.
and cy2. Under this hypothesis, the model (3.3) with p = 6 will now be reduced to yz
A4x, =L!2yz,r-1
+~,A~,x,-~
+LI~A~x,-~
+E~.
(4.5)
The seasonal cointegrating vector (Ye in (4.5) can be found from the canonical variate corresponding to the largest squared partial canonical
28
H.S. Lee, Cointegration
and seasonal cointegration
tests
correlation between the residuals from y2 , _ , and Aqx, corrected for the effect of the lagged differences A4x,_, and Aqx,_2. The estimate G2 = (0.135 1.548)’ can naturally be normalized by making the coefficient of UN, equal to 1. The equilibrium error process can then be formulated as z,=(l--B+B’--B3)(UN,+11.49ZMM,).
(4.6)
This indicates a negative relationship between unemployment and immigration rates. This result is expected from fig. 1 and is also consistent with earlier empirical results [see, e.g., Marr and Siklos (1988)l. Similarly f2 can also be normalized, giving us an error correction model of the form: AaUNt =
O.l29z,_,
+
(0.063) -
1.316A,UN,_, (2.303)
0.609A,UN,_,
A,IMM,
=
O.O15z,_, (0.008) +
0.894A,IMM,-,
,
(4.7)
(0.263) O.O33A,UN,_, (0.292)
O.O07A,UN,_, (0.300)
(0.215)
+
(2.369)
2.034A,IMM,_,
-
+
+
0.518A,ZMM,+, (0.027)
O.l33A,IMM,_,
.
(4.8)
(0.033)
In the error correction model (4.7), the t-statistics for the error correction term and the lagged values of A,ZMM, are all significant. This indicates that there is a highly significant relationship found for immigration causing unemployment. In the model (4.8), on the other hand, the t-statistic for z,_ I appears to be significant only at the 10% level and those for the lagged values of AbUN, are not. This suggests that the evidence for unemployment causing immigration is quite weak. This result seems to contradict earlier empirical work which found a relatively strong link between a current increase in the unemployment rate and a future decrease in immigration, and the absence of an effect of past immigration on current unemployment. The finding of seasonal cointegration relationship between unemployment and immigration rates, however, provides us with important information on the effectiveness of immigration policy in Canada. 5. Concluding
remarks
It has been widely observed that many economic seasonality and a few attempts have been made
time series exhibit strong to describe the seasonal
H.S. Lee, Cointegration and seasonal cointegration tests
29
patterns. As is shown in EGH (19891, the Engle-Granger type two-step procedure for cointegration test is not appropriate when the variables in the system are contaminated with unit roots at some seasonal frequencies other than zero. The use of seasonally filtered data, suggested by HEGY (1990) and EGHL (19901, is not recommended, because it needs pretesting for seasonal unit roots, the implication of which has not yet been investigated. This paper develops a general framework of the tests for cointegration and seasonal cointegration and proposes test statistics for nonstationary time series which have unit roots at seasonal frequencies as well as at the zero frequency. Since the maximum likelihood inference on cointegration and seasonal cointegration is developed on the basis of the asymptotic distribution theory, it is important to examine the finite sample properties of the testing procedure. The empirical results in the previous section seem promising. The actual size of the test is quite close to the nominal size for T larger than 100, and the power of the test in rejecting the false null hypothesis is very high (above 95%) at least in the simple structure considered. This implies that the asymptotic distribution theory provides reasonable approximations to the finite sample behavior of the testing procedure developed in this paper. The methods are applied to Canadian data on unemployment and immigration, and we can find a seasonal cointegration relationship at the biannual frequency, while there is no evidence for cointegration by applying the standard cointegration approach. The evidence for seasonal cointegrating relationship between unemployment and immigration rates indicates something about the effectiveness of Canada’s immigration policy. Although many important implications can be derived from the results on the seasonal cointegration test, much more interesting results might be expected by extending the seasonal cointegration model in this paper. First, seasonal cointegration at frequency w = d considers the unit roots at ki so that yjr =B(l -B2)x,. For monthly data, we have unit roots at other frequencies, hence we need to consider the case where y,, = I?(1 - 2~0s BB + B2)x, for 0 E (0, rrTT). The results in Chan and Wei (1988) and Beaulieu and Miron (1990) can be applied to answer this question. Second, it has been shown that inclusion of a constant term changes the distributional properties of the test statistics for cointegration, and hence is crucial for statistical and probabilistic analysis [see Johansen (198911. Based on this result, it is pointed out in Lee and Siklos (1990) that the distribution of the test statistics for cointegration at the zero frequency in the presence of other seasonal unit roots is also affected by including a constant term in our model. We can also show that inclusion of seasonal dummies will change the distribution of the test statistics for seasonal cointegration at seasonal frequencies other than zero [see Lee and Siklos (1991b)l. Third, although the empirical results in the previous section show some robustness to the assumption that II4 = 0 when cointegration is contempora-
J.Econ
B
30
H.S. Lee, Cointegration
and seasonal cointegration
tests
neous, we need to further investigate the influence of this assumption in a more general context. When the null hypothesis of interest is that IT, = U4 = 0 (i.e., no seasonal cointegration at frequency w = a>, the assumption that II4 = 0 might merely lower the power of the test against some alternatives. However, in the general case where we need to consider the polynomial cointegrating vector, a much more complicated testing procedure will be required for testing the joint hypothesis on U3 and II4, and hence it will affect the distribution of the testing procedure.
Appendix In order to investigate the distributional properties of the estimators and test statistics derived in section 3, we need to impose more precise assumptions on the process. The basic assumption is, first of all, that all roots of I@(z)1 = 0 satisfy Izl 2 1 where Q(z) = I - @,z - @,z* - . .. -@,,zp. This implies that the nonstationarity of the process comes from unit roots at some seasonal frequencies as well as at the zero frequency. Hence the nonstationarity can be removed by seasonal filtering combined with differencing. In particular, we shall assume that X, is seasonally integrated of order 1 at seasonal frequencies o = $ and $ as well as at the zero frequency (w = O), so that A4x, is stationary and has an MA representation
A4x, = C( B)E~ = f: Cj~,_j,
(A.1)
j-1
for some exponentially decreasing coefficients Cj’s. Moreover, we shall assume that the process X, has rk cointegrating vectors (Ye for rk < n (k = 1,2,3) at each frequency. We shall first provide the asymptotic behavior of the product moment matrices defined in (3.4)-(3.6) and find the asymptotic properties of Sij and D,j (i, j= 0 ,..., 4), and then apply these results to the test statistics derived in the theorems. These procedures generalize the methods in Johansen (19881, where the analysis is confined to integration and cointegration at the zero frequency. Much of the material in this appendix is drawn from Johansen (1988). Let the covariance function of the stationary process A4x, be such that
pij = E(A dx,-iA4x,-j)
= Icr(i -j),
i, j=O
,...,
P -
4,
H.S. Lee, Cointegration and seasonal cointegration tests
31
and define the matrices
PY3Y3
= -
f (-l)‘lil$(Zi). i= --m
Then we can derive the following relations between yk, , _ 1‘s and A4x,_j’s such that
P YII.=
.= CLY2J
=
lim cov(y,,,_,,
AdX,-j)=
i
t-m
for the covariance
function
4(i),
i=l-j
lim cov(y,,,_,,A,~,_~) t-m
i=T_j( -l)‘+(i)
or
E
(-l)i+l@(i),
i-1-j
P Y31 .=
lim
t-c.2
=
COV(Y~,~-,,A~X~-~)
E
(-1)‘$(2i)
or
or
5
or
(-1)‘$(2i+l)
i=[(2-j)/Zl
i = [(2 -j)/21
f ( - 1)‘+‘$(2i) r=[C2-j)/21
f
(-l)i+‘~(2i+
1).
i=[(2-j)/21
Note that the expression for pyZj depends on whether j is odd or even, and depends on [mod 41 of j. that The following lemmas extend the results in Johansen (1988) to the time series process of our interest that have unit roots at seasonal frequencies other than zero. Py,j
H.S. Lee, Cointegration
32
Lemma A. 1.
and seasonal cointegration
tests
As T -+ to, we have (A.2a)
T-1’2~lcrrI 5 C(l)B,(r), T-9
_ lfT’ly
T-‘MYY,,
:
2[Tr]
5
(A.2b)
C(-l)B,(r),
C(l)jo’B,B;
(A.3a)
drC(l)‘,
(A.3b)
T-‘MYY22-:C(-1)~o’B2B;drC(-l)‘,
lB,B: dr + _/‘B,B; dr 0
+c,
‘B,B: dr + /‘B,B;
+c,
‘B,B; dr - /‘B,B;
+c,
lB,B; dr - /‘B,B:
dr
0
dr
0
1 1,
dr C;
0
j=O ,...,
kEY,j~C(l)~‘B’dB~C(l)‘+p,,j,
IWKY,~5 ( - l)‘+‘C(
- l)jgLB2 dB;C(
p-4,
- 1)’ + kyJj,
(A.3c)
(A.4a)
(A.4b)
j=O ,...,P-4, ‘B,dB:+
MYX,j~(-l)“+2’/2~
+c,
‘B,dB:+
+c,
‘B,dB:-
+c,
‘B,dB;-
j’B,dBj 0
/lB,dB; 0
1
C;
1
+pYjj
(A.4c) for j even,
H.S. Lee, Cointegration
and seasonal cointegration
+c,
‘BSdB;-
/lB,dB;
+c,
‘B,dB:+
/lB,dB;
33
tests
0
(A.4d)
0
for j odd, where B,(r), B,(r), B,(r), and B,(r) are mutually independent Brownian motions in n dimensions with couan’ance matrix 0, and CR and C, denote the real and imaginary part of C(i), respectively. Notice first that the limiting distributions for MIX,, and Mkx,, depend on whether j is odd or even. We can also show that T-‘MYY,, has the same limiting distribution as T-‘MYY,, and that Myx,, has the same limiting distribution as MIx3, for i = j + 1. Proof.
To prove (A.2a) and (A.2b), we shall use the fact that for T + CQ, [Trl
T-l/*
c
ei 3 B,(r),
(A.5a)
i=l
[Trl
T-1/2iFI(
-1)‘~~ 5 B*(r).
Note that we can represent t-1
)‘I,
=
t-1 =
and
yZt
as
t
C Adx,-j = C A,xj, j=O
~2~
Y,,
(A.5b)
(A.6a)
j=l
f
jFo ( - l)‘A,x,_j = ( - 1)’ C ( j=l
l)‘A,Xj*
(A.6b)
34
Then
H.S. Lee, Cointegration
the result
yr,=
and seasonal cointegration
follows from the following
($j(
&;i
Cvl>‘Y*t=
+
j,
lg”(-l)‘c;
-
;
;fp’c,
(
_l)‘Ej I
(-l)‘&j
f: j=t-i+l
i
[~“c;jc&+,&J)~
il)‘i,)
I=-ii1
i
-
equations:
+ [,F;,Ci,&,E,j
( ~~~~l~Lci)(
tests
. i
The relations (A.3a) and (A.3b) follow from (A.2a) and (A.2b) and the continuous mapping theorem [see Billingsley (1968, theorem 5.1)]. To show (A.~c), note first that we can represent yxt as Kt Y 3,t+1
=
1)/21
c
(A.6c)
(-#4x,-*,.
j=O
This can be rewritten
Y3,1+1
in the form such that
y{(-l),d’xl]sin[(f +
=
l):]
Kt+ 1)/Z]
jF,
+
(-1)'A4x2j-l
lcos[(t + I);]
[
=C,
C,sin(t+l)t
-SS,cos(t+
[ -C,
C,cos(r+l)~+S,sin(t+I)~ [
1):
1 1
+o,(T’/‘),
H.S. Lee, Cointegration and seasonal cointegration tests
35
where t C,
=
f
7T
( 1
C cos -j
2
j=l
and
.cj
S,=
57.
Csin j=l
1
yjEj. i
Using the fact that for T -+ m,
Tp112 2
cos( tj)Ej
3 2-‘/*B,(
r),
(A.7a)
[rrl T-1/2 cq sin
(A.7b)
j=l
we can now derive (A.3c) by applying the trigonometric orthogonality relationships and the continuous mapping theorem. The independence of B,, B,, B,, and B, are shown in Chan and Wei (1988, theorem 2.1). Similarly, the relations (A.4a), (A.4b), (A.4~1, and (A.4d) can be obtained from the representations (A.6a), (A.6b), and (A.6~1 for y,,, y,,, and y3r, respectively, so that the details are omitted. Lemma A.2. Mxx,,
As T -+ 03, we hate i, j=O ,...,P-4,
=G pij,
P
kzl,
T- %4YY,, + 0,
k, I= 1,2,3 T-‘MYY,,
+ T-‘MYY,,
(A-8)
3 0.
(A.lOa) or
1,2,4,
(A.lOb)
Proof. The result (A.81 follows by noting that (A,x,} is stationary and ergodic, and the relation (A.91 follows from the stationarity of (YZ~~,~_1 (k = 1,2,3X The result (A.lOa) follows from the fact that they are asymptotically uncorrelated [see, e.g., Chan and Wei (1988)l. To prove (A.lOb), we first
36
H.S. Lee, Cointegration and seasonal cointegration tests
derive the asymptotic
distribution
such that
‘BsB; dr + jlB,B: 0
+c,
[see Chan and Wei (1988, theorem fact that T-‘MYY,, = (T-‘MYI’,,)‘. Lemma A.3.
of T-‘MYY,,
dr
CL I
3.3.4)]. The result
1
now follows from the
As T + 00, we haue
T-‘(Skk
-MYI’-,,)
:
0,
k = 1,2,3,
(A.ll) (A.12)
k = 1,2,3,
(A.13)
k = 1,2,3,
(A.14)
kzl, T-l& where
+ T-‘S
43
’
--)
k,l=
Proof. The asymptotic properties can be derived by expressing Sk, in as follows: =
mk,
Sk, = Myx,,
~k~,=bkl.+k,p-4)
of Sk, (k, 1= 0,. . . ,4) defined in (3.12) of the mk,, and M-XX,,,
terI’IIS
MIxkj,
- MIX, * MXX, :MIx;, , - mk
(A.15a) (A.15b)
(k,l=0,yl,Yz,y3)> isthematrixwithelementspii(i,j=l,...,~-4).
sk,
or 1,2,4,
0,
~k,=Pkl-Pk*P&-&c
andp.,,
1,2,3
* MXX, : M.&X;, *,
S,, = MXX,, - MXX,, MXX, : MH;
*.
k,l=l,..., k = 1,...,4,
4,
H.S. Lee, Cointegration
and seasonal cointegration
tests
37
Then all the above relations follow from Lemma A.1 and A.2 by applying the results on MYI’,,, MIX,,, and MXXij. That is, (A.ll) follows from (A.31, (A.121 from (A.8), (A.13) from (A.9), and (A.15) from (A.lO). Since cu;C(l) = 0, Q!$< - 1) = 0, and c-u;C(i) = 0, (A.14) also follows from (A.4). Lemma A.4.
For T + 03, it holds that
T-‘( D,, - MYY,,)
:
0,
k = 1,2,3,
(A.16)
Q&,,
(A.17)
P
4 4, + &ZkO, T-‘D,,
+ T-‘D,,
k = 1,2,3,
(A.18)
k = 1,2,3,
(A.19)
5 0.
(A.20)
Proof. To prove the lemma, it suffices to show that D,, has the same distributional properties as S,,. For this, we shall first prove that as T + m, we have T-‘( D,, - S,,)
(Doe-S,,,J
T-‘(D34-S34)
S,,
(A.16’)
: 0,
(A.17’) k = 1,2,3,
(A.18’)
k = 1,2,3,
(A.19’)
1:O.
(A.20’) of D,,
(A.16’) follows from the definition
D,, = T-’ where
k = 1,2,3,
:oo,
(.y;(D,, - S,,)q
The relation
-T: 0,
5
Qk,Q;r
I=1
= [Si,, i,j+k,
such that
=S,, - Sk&&#,
i,j=l,...,
41 and
Sk,=[Skj,
j+k,
j=I
,...,
41.
H.S. Lee, Cointegration and seasonal cointegration tests
38
Now we can show that
T-y&s,, -D,,)
=
(T~‘S,,)(T~‘S#,)~‘(T_‘S,,)’
The first and third terms go to zero in probability as T + ~0by (A.l5a), while the second term is O,(l) by (A.ll). Hence the result follows from the product rule. The relation (A.20’) can also be shown by the same argument. Similarly using the definition of D,,(,, we get
so that we have
s,,,,-
D,,,, = (T-~‘*S,,)(T-‘S,,)-~(T-~‘*S,,,)‘.
Since the first and third terms vanish in probability by (A.4), we obtain (A.17’). The relations (A.18’) and (A.19’) can also be shown in similar ways and the details are omitted. With Lemma A.3, the relations (A.16’) through (A.20’) prove the relations (A.16) through (A.20), respectively. Lemma A.5
The ordered eigenvalues of the equation
IAD,, - Dk,P,, I 4kI = 07
k = 1,2,3,
converge in probability to (A,, . . . , A,, 0, . . . , O), where h,, . . , A,. are the ordered eigenvalues of the equation IAa;-Zkk~k - a;, &,,&,‘&,a, Proof. This follows from Lemma Johansen (1988, lemma 4). Lemma A.6.
I = 0,
k = 1,2,3.
A.4 and direct application
(A.21) of the result
in
We have the following relations:
zoo=c ~k~k,,+fl~
(A.22)
k=l
Yk
=
&k(Yk( a;z,&ak)
The relations (A.23) 114 = 0.
-‘,
and (A.24)
k = 1,2,
(A.23)
k = 1,2.
(A.24)
hold for k = 3 when it is assumed that
H.S. Lee, Cointegration and seasonal cointegration tests
Proof. From the normal eqs. (3.7) or (3.8) for the OLS regression, derive a system of equations:
=
we can
4
p-4 PO0
39
C
Ail*iO
+
C
nktLykO
+
(A.25)
n?
k=l
i=l
p-4 POj
=
C i=
Aipij
+
I?
p-4 pOykak
j=l
flk~~y~l’
,...,p-4,
(A.26)
k=l
1
=
c
4 Aipiyk
1=1
+
c
k=1,...,4.
nkp~lyk’
(A.27)
I=1
The relation (A.221 follows by solving (A.261 for the matrices (A,, . . . , A,_,) and inserting into (A.25). For (A.23), we insert (A,, . . . , A,_,) into (A.27) uncorrelated. The and use the fact that yk,r ‘s (k = 1,2,3) are asymptotically relation (A.24) follows from (A.23) since ii’, = Y~LY~. Lemma A.% As T+m, Ti,,,+ ,,..., ordered eigenualues of the equation
Ti,,,
A, ‘WW’dr-L’WdW’l’dWW’i I /0
concerge
in distribution to the
=O,
(A.28)
where W( r ) is a standard Brownian motion in (n - r ) dimensions. Proof. Note equation
first that
‘Y;DuWT
for any value
of T the ordered
eigenvalues
of the
a;DkkP/T
1 dJ',co&,'Dod 4Dk0D00'D0k~k -CL P'40hi'Do,d [NhoDi;Do~,c II
I[ P’D,,aU,/T
@D&/T
=
where
0,
(A.29)
p is an n X r matrix of full rank (n - r) such that p’ak = 0, are given
by /i1 = (T&n)-‘, It follows
from Lemma
c, = (Tlik,n_l)p1 A.4 that
,...,
;,
= (T&l)-‘.
pi, . . . , fi,, converge
in distribution
to the
40
ordered
H.S. Lee, Cointegration
eigenvalues
=
and seasonal cointegration
tests
of the equation
0,
(A.31)
where Mkk denotes the limiting distribution of TelMYY,, the weak limit of D,,p. Following the work in Johansen factor of (A.31) can be reduced to
in (A.3) and L, is (19881, the second
where 6, is an n X (n - r> matrix such that 6;~~ = 0. Since 6, is chosen that 6by, = 0, we have for k = 1
S’,D,,
-I:S;l’dB,
B;C(l)‘.
such
(A.33a)
0
Thus (A.321 now becomes
which shows that the limiting distribution of the (n - r) largest fi’s is given as the distribution of the ordered eigenvalues of (A.34). Since the null space of C(1)’ is the space spanned by the columns in (Y,, we have the representation: C(1) = PT~‘, for some nonsingular matrix T. This implies that
‘BIB; dr6,
-
yS;/‘B,dB; 0
S,#,/‘dB,
B;6,
= 0.
(A.35)
0
Since W, = 6;B, is a Brownian motion with variance 6’rR6, = I, the result follows from the fact that the solutions to (A.28) are the reciprocal values of the solutions to (A.35).
H.S. Lee, Cointegration and seasonal cointegration tests
41
Based upon the lemmas, we can now complete the proof of Theorem 3.1 and hence Theorem 3.2 as a corollary to Theorem 3.1. Proof of Theorem 3.1
To derive (3.24), we need to note that the solutions to (3.23) are the first r eigenvectors of D,,D,‘D,, with respect to D,,. The eigenvalues are the squared partial canonical correlations of R, with respect to R,, eliminating R,, R,, and R,. By normalizing a^, such that cG;Dllc?l = I, we obtain from (3.21) and (3.22), respectively, Tr =D,lcE,(&;DllcSl)-l
(A.36)
= DOlcGl,
(A.37)
fi = D,, - D,,cE,ci; D,, = D,, - T,+;.
Hence the maximized likelihood function is given by (A.38) while the maximized likelihood function without the constraint H, is given by L,,‘,/‘=
ID,1 fi
(1 - i,,i).
(A.39)
i=l
The LR test statistic for the hypothesis H, is then given as the ratio of (A.38) and (A.39), which is just eq. (3.24). To prove (3.25), we expand the test statistic (3.24) to obtain -2ln(Q)=
-T
i
ln(l-iI,i)=
2
Ti,,i+o,(l).
i=r+l
i=r+l
Now the result follows from Lemma A.7. Proof of Theorem 3.2
From (A.5.1) and (A.5.2), it is easy to see that this case is parallel to Theorem 3.1, since the distribution derived here is a mirror image of the zero-frequency case. Specifically, we can first show that /,,r dB, Bi e 10’dB, B; [see Chan and Wei (1988, theorem 3.2.111. Note also that, for S, such that 6;y, = 0, we have 6;D,,
jd -6; /‘dB,
B;C( -l)‘,
(A.33b)
0
and the representation: Cc-- 1) = p~8z. Hence the result follows as a simple corollary to Theorem 3.1.
H.S. Lee, Cointegration
42
and seasonal cointegration
tests
Proof of Theorem 3.3 The LR test statistic (3.34) can be derived by using arguments similar to those applied in the proof of Theorem 3.1. To derive the asymptotic distribution (3.35), we need to modify Lemma A.7. Lemma ...,T&n
A.7’. As T + cc), we haue that Ti.l,r+,, . . . , Ti,,, and concerge in distribution to the eigencalues of the equation
/0 /0
-
I
I ‘(w,dW,‘+
W,y)dr
‘( W,Wl +
W,W’;) dr
W,dW’,‘)
0
/
x
‘(W,w,‘-
Ti++,,
[i
‘(W,
0
dW,’ - WCdW’,‘)
ol(dWc W,l+ dW’, W,)
‘(dW, W,-dW, /0
W,)
1= 0,
(A.40)
where W,(r) and W,(r) are independent standard Brownian motions in (n - r> dimensions. Proof. that
Consider
the eigenvalues
of the equation
of the form (A.29)
II
= 0,
such
(A.41)
43
H.S. Lee, Cointegration and seasonal cointegration tests
where
p, 1 O
p=
[
0
p,
is a 2n x 2r matrix such that P’a = 0 (or ordered eigenvalues are given by
/& =
where
M,,
&a,
= 0). The
(T&,,)-l,b,=(Tfi,,2,,-,)-1 ,...>bzn = (T”q.,)-’
which, from Lemma of the equation
=
equivalently,
A.4, converge
in distribution
to the ordered
eigenvalues
(A.42)
0, denotes
the limiting
distribution
of
T- ‘MYY,, T- ‘MYY,, _Zoq= (_%a, &), and L, is the weak limit of (Da, Do4)p. We shall now choose 6, of dimension n x (n - r) such that 6by, = 6:~~ = 0 and 6,Sb = fl-‘[IPYqUI)], where PYq(O> = y,(Q2-‘y,)-‘y~.Cn-‘. From the relations (A.24), we can also show that
By arguments similar to those in the proof of Lemma of (A.421 is reduced to
A.7, the second
factor
44
H.S. Lee, Cointegration
and seasonal cointegration
Since 6by, = aby4 = 0 by construction, (A.33b) that
+
+
+
tests
we can derive as in (A.33a) and
‘(dB, B:. - dB, B;)C; /n
/0
/0
‘(dB,B;-dZ?,B:)C:,
1
r(dB,B:+dB,B;)C;
.
Since abC(i) = 0, implying that a’,C, = 0 and (Y’$, = 0, the space spanned by the columns of (Ye is the null space of both Ck and Cl. Using (A.3c) for the limit distribution of M,,, we can derive from (A.43) that
l(dB, B: + dB, B;)6,
“$j’(dB, 0
= 0.
1 (A.44)
Now the result follows from the fact that WC= 6iB, completes the proof of Lemma A.7’. The asymptotic distribution Lemma A.7’.
B; - dB, B:)6,
and W, = SbBs. This
(3.35) can now be obtained by applying the
45
H.S. Lee, Cointegration and seasonal cointegration tests
Proof of Corollary 3.4
The asymptotic distribution of the LR statistic (3.38) can similarly be derived by using eq. (A.40 In this case, M,, and _XqO will be reduced to M,, and &, respectively, and L, will be replaced by L, denoting the weak limit of D&. By choosing 6, such that 6jy, = 0, we have iS;D,, :
6;( - ;)
+
/0
/‘(de, 0
B; + dB, B;)C;
‘(dB,B:-dB,B;)C;
(A.33~)
. I
The asymptotic distribution (3.39) can now be derived by noting that ‘(B,B:+B,B;)drc&
-_ILiYj+l’(B,dB:+B,dBj)S,G;~~‘(dB,B:+dB,B:)6, 0
0
=O.
This completes the proof of Corollary 3.4. Proof of Theorem 3.5
Noting that 2 1+B2 we can represent
1 ’
X, as
x f = &C(B)“,
= $(~,,l+~2,1+
2~3,r+i).
Hence the asymptotic properties of X, can also be extracted from those of yk.,‘s (k = 1,2,3). Denoting yAt =x,_~, it is easy to show that T-‘MYY,,
5 &T-l (MYY,,
+ MYY,,
+ 4MYYsJ.
46
H.S. Lee, Cointegration
and seasonal cointegration
tests
We can now derive that T-‘MYY,,
&C(l)/lB,B; 0
:
+ &C( -
d&(l)’
l)(B,B; drC( ‘B,B;
- 1)’
dr + j’B,B:,
dr
0
+c1
‘B,B;
+c,
lB,Bkdr
dr + jlB,B:,
dr
0
- j’B,B;dr 0
C; i
1 .
(A.45)
For the choice of 8, such that &14yA = 0, we can show from (A.33aHA.33c) that
+ /‘(dB,
B; + dB, B;)C:,
+ j!-$dB,
Bj - dB, B;)C;].
0
(A.33d) Therefore, by similar arguments to the previous cases, the limiting distribution of the test statistic (3.41) can be derived from the result (A.45) and (A.33d). References Beaulieu, J.J. and J.A. Miron, 1990, Seasonal unit roots in aggregate U.S. data, Manuscript (Department of Economics, Boston University, Boston, MA). Bell, W.R. and SC. Hillmer, 1984, Issues involved with the seasonal adjustment of economic time series (with comments), Journal of Business and Economic Statistics 2, 291-349. Bell, W.R. and D. Wilcox, 1990, A note on seasonal models, seasonal adjustment, and nonseasonal unit roots, Manuscript (Statistical Research Division, Bureau of the Census, Washington, DC). Billingsley, P., 1968, Convergence of probability measures (Wiley, New York, NY).
H.S. Lee, Cointegration
and seasonal cointegration
tests
41
Chart, N.H. and C.Z. Wei, 1988, Limiting distributions of least squares estimates of unstable autoregressive processes, Annals of Statistics 16, 367-401. Dickey, D.A., H.P. Hasza, and W.A. Fuller, 1984, Testing for unit roots in seasonal time series, Journal of the American Statistical Association 79, 355-367. ’ Engle, Robert F., 1987, On the theory of cointegrated economic time series, Discussion paper(Department of Economics, University of California at San Diego, CA). Engle, R.F. and C.W.J. Granger, 1987, Cointegration and error correction: Representation, estimation and testing, Econometrica 55, 251-276. Engle, R.F., C.W.J. Granger, and J. Hallman, 1989, Merging short and long run forecasts: An application of seasonal cointegration to monthly electricity sales forecasting, Journal of Econometrics 40, 45-62. Engle, R.F., C.W.J. &anger, S. Hylleberg, and H.S. Lee, 1990, Seasonal cointegration: The Japanese consumption function, Journal of Econometrics, forthcoming. Fuller. W.A.. 1976. Introduction to statistical time series (Wilev. New York. NY). Ghysels, Eric, 1988, A study toward a dynamic theory of seasbnality for economic time series, Journal of the American Statistical Association 83, 168-172. Ghysels, E., H.S. Lee, and P.L. Siklos, 1992, On the specification of seasonality: A survey of seasonal unit roots and seasonal cointegration, Manuscript (Department of Economics, Tulane University, New Orleans, LA). Granger, Clive W.J., 1981, Some properties of time series data and their use in econometric model specification, Journal of Econometrics 16, 121-130. Granger, Clive W.J., 1986, Developments in the study of cointegrated economic variables, Oxford Bulletin of Economics and Statistics 48, 213-228. Granger, Clive W.J. and Paul Newbold, 1986, Forecasting economic time series (Academic Press, New York, NY). Hylleberg, S., R.F. Engle, C.W.J. Granger, and B.S. Yoo, 1990, Seasonal integration and cointegration, Journal of Econometrics 44, 215-238. Johansen, Soren, 1988, Statistical analysis of cointegration vectors, Journal of Economic Dynamics and Control 12, 231-254. Johnsen, Soren, 1989, Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models, Econometrica, forthcoming. Lee, H.S. and P.L. Siklos, 1990, The influence of seasonal adjustment on unit roots and cointegration, Paper presented at the 1990 WEI international conference (Department of Economics, University of California at San Diego, CA). Lee, H.S. and P.L. Siklos, 1991a, Unit roots and seasonal unit roots in macroeconomic time series: Canadian evidence, Economics Letters 35, 273-277. Lee, H.S. and P.L. Siklos, 1991b, Seasonality in economic time series: Money-income causality in U.S. data revisited, Manuscript (Department of Economics, Tulane University, New Orleans, LA). Marr, W.L. and P.L. Siklos, 1988, An exploratory empirical analysis of the link between immigration and unemployment: Canada, Manuscript (Department of Economics, Wilfrid Laurier University, Waterloo, Ont.). Phillips, P.C.B., 1987, Time series regression with a unit root, Econometrica 55, 277-301. Phillips, P.C.B. and S. Ouliaris, 1988, Testing for cointegration using principal components methods, Journal of Economic Dynamics and Control 12, 205-230. Reinsel, G., 1983, Some results on multivariate autoregressive index models, Biometrika 70, 145-156. Stock, J.H. and M.W. Watson, 1988, Testing for common trends, Journal of the American Statistical Association 83, 1097-l 107.