- Email: [email protected]
Spurious regressions and residual-based tests for cointegration when regressors are cointegrated
Journal
of Econometrics
60 (1994) 313-320.
North-Holland
Spurious regressions and residual-based tests for cointegration when regressors are cointegrated In Choi* The Ohio State University, Received
September
Columbus,
OH 43210, USA
1991, final version
received
September
1992
We study a spurious regression with cointegrated regressors. Asymptotic properties of the OLS estimates and other conventional statistics are reported. Using these results, we analyze the asymptotic null distributions of such residual-based cointegration tests as Phillips’ 2, and the augmented Dickey-Fuller (ADF) tests, when regressors are cointegrated. Key words: Spurious
regression;
Cointegrated
regressors;
Residual-based
tests for cointegration
1. Introduction Granger and Newbold’s (1974) classic study of regressing one integrated process on a set of integrated processes has attracted a good deal of attention among theoretical econometricians and empirical researchers. This practice of using integrated processes as both regressand and regressors is now commonly called a spurious regression [unless the regressand and regressors are cointegrated in the sense of Engle and Granger (1987)]. Subsequently, Phillips (1986) provided the asymptotic analyses that can fully explain the points raised by Granger and Newbold. These asymptotic results of Phillips are useful beyond explaining Granger and Newbold’s simulation results, because residualbased tests for cointegration [see, for example, Engle and Granger (1987) Phillips and Ouliaris (1990) Stock (1988), and Choi (1990)] take the spurious
Correspondence to: In Choi, Department High Street, Columbus, OH 43210-1172,
of Economics, USA.
The Ohio
State University,
1945 North
*I would like to thank the Associate Editor and two anonymous referees for their helpful comments on an earlier version of this paper. At the suggestion of the Associate Editor, the proofs of all theorems are omitted, but can be obtained from the author upon request.
0304-4076/94/$06.00
c
1994-Elsevier
Science Publishers
B.V. All rights reserved
314
I. Choi, Spurious regressions with cointegrated regressors
regression as the null hypothesis, and hence their asymptotic distributions depend on those of coefficient estimates from the spurious regression. Remember, though, that it is assumed that regressors are not cointegrated in the aforementioned studies of the spurious regressions and residual-based tests for cointegration. But in some applications, it is not nonsensical to conjecture that regressors are cointegrated. For example, when we regress one price variable (e.g., exchange rates, interest rates, etc.) on other price variables for cointegration analysis, it is reasonably expected that the regressors are cointegrated. The asymptotic properties of the OLS coefficient estimates in cointegrating regressions with cointegrated regressors are studied in Park and Phillips (1989) and Wooldridge (1991). However, spurious regression is the focus of this paper, partly because the spurious regression with cointegrated regressors is of theoretical interest and partly because the null of residual-based tests is the spurious regression. To date, no systematic analysis has been given in the literature regarding the spurious regression and the asymptotic distributions of residualbased tests for cointegration when regressors are cointegrated, though it is not uncommon to hear empirical researchers express concerns about using residualbased tests in such cases. Thus, the first objective of this paper is to analyze the spurious regression with cointegrated regressors. We will study the asymptotic properties of the OLS estimates and other conventional statistics when regressors are cointegrated, but the regression itself is spurious in nature. To this end, we will use the weak convergence results obtained by Phillips and his associates [see Phillips (1988) and references therein] and methods similar to those of Choi (1991). The second objective of this paper is to analyze the asymptotic null distributions of residual-based cointegration tests, when regressors are cointegrated. We focus on Phillips’ A’, and the augmented Dickey-Fuller (ADF) tests, because these are most commonly used in applications. However, qualitatively similar results hold for other kinds of residual-based tests. These analyses will show that the asymptotic distributions of 2, and ADF with cointegrated regressors differ from those with noncointegrated regressors. The structure of this paper has been planned as follows. Section 2 introduces the model and assumptions. Section 3 derives the asymptotic distributions of the OLS estimates and various conventional statistics from the spurious regression with cointegrated regressors. Section 4 studies the asymptotic null distributions of 2, and ADF. Section 5 contains a summary and further remarks. A few words on our notation: All the limits are taken as T + co unless otherwise specified. B(r) denotes a multivariate Brownian motion, and B, s:B, s:BB’, and JiBdB’
are understood
to be B(r), and J:B(r)dr,
jiB(r)dB(r)‘, respectively. Convergence X denotes T-l~~=l~f.
in distribution
JiB(r)B(r)‘dr,
is denoted
by j.
and Last,
2. Preliminaries WC are concerned .r’,
=
with the multiple
x;jr + II,,
1’ +
t=l’
regression
,-1
.
with an intercept
(1)
7;
where 7; = (yt. x;) is a vector 1(l) process of dimension ~1 + I (nt L 2) [see Angle and Granger (1987) for the definitions of 1(l) and I(O)]+ which is generated by z, = s,-
1
+
E,,
where z, = r(O). Once we assume that the error sequence {u!] is I( 1) and that the regressor X, is not cointegrated, eq. (1) is the prototgpe of a spurious regression studied by Granger and Newbold (1974) and Phillips (19X6). The case that the or noncoerror sequence jll,i ’ Is I(O) and Ihc regressor X, is either cointegrated integrated is studied in Park and Phillips (1989). In this paper, we assume that U, = I(I) and that there exists y (m - 1 2 p 2 I) linearly independent cointegrating vectors for the regressor s,, which are denoted as ;‘,, . y,,. Letting
\ve have that C’:s, = 18,is weakly stationary by the definition of cointegration. We standardize the cointegrating vectors such that y:y, = 1 (i = 1. , p). Further, we assume that ;‘i # 0 (i = I, . ..~ p), so that trivial cointograting relations for the regressor s, are excluded. We assume that the innovation sequcnct: $ = (&vi) is strictly stationary and ergodic with zero mean and finik variance. Further. WC require the partial sum invariance principle. process constructed from [ rl,!t to satisfy the multivariate That is,
where
is an (1 + 111 + p)-vector
Rrownian
motion
with covariancc
matrix
I. Choi, Spurious regressions
316
with cointegrafed
regressors
where
Cij and Aij for i, j = 1,2,3,4 are defined to be submatrices of C and A corresponding to Rij. The reader is referred to Phillips (1988) and the references therein for the multivariate invariance principle.
3. Spurious regressions with cointegrated
regressors
In this section we study the asymptotic properties of the OLS estimates, R2, F-statistics, t-ratios, and the Durbin-Watson statistic (DW) for the regression equation (l), when the regression is spurious in the sense of Granger and Newbold (1974) and the regressors involved are cointegrated. The main results of this section are presented in the following theorem. The null hypotheses for tp, and F, in the following theorem are, respectively, H,,: pi = 0 (i = 1, . . . , p) and Ho: p = 0. Theorem
1.
Suppose
section 2, we have
(4
(b) (4 (4
(4
(f)
that {ut} is I(1). Then, under the relevant
conditions
in
I. Choi, Spurious regressions
with cointegrated
317
regressors
where & = Bi - pi,
823, = CB2, Bj],
dij
=
Tij
+
/lij,
m-p 0
1 ..
0’ = (1, - b’), f 123= E(E,E;),
c2 = [
0
1 m,
.1
and Cl is dejined in section 2. Remarks (i) This theorem shows that i&s has a nondegenerate limiting distribution to and the distribution of j&s diverges. Moreover, R2 converges in distribution a well-defined random variable and DW converges to zero in probability. We expect from these results that the regressions of this type will yield moderate values of R2 and low values of D W, which are regarded as evidence of a spurious regression by Granger and Newbold (1974). We also observe that the distribution of the F-statistic diverges, while those of the t-ratios converge to welldefined random variables. Diverging t-ratio is an important characterization of the spurious regression as shown in Phillips (1986, p. 320), but we observe arises because here that the r-ratios do not diverge. This dissimilarity { C(xf - X)(x, - 2,’ } - ’ = O,( r- ‘) when the regressors are cointegrated, while {x(x( - X)(x, - I?)‘}-’ = 0,(Tp2) when they are not. However, it is important to notice that the assumption yi # 0 (i = 1, . . . , p) is essential for the weak convergence of the t-ratios. When yk = 0 for some k, the t-ratio corresponding to the regressor &k diverges, as one of the referees pointed out. Except for the t-ratios, the orders of magnitude for the OLS estimates and the statistics we study correspond with these reported in Phillips (1986, theorem 2). (ii) When regressors are not cointegrated, the limiting distributions of and ,i& reduce to those obtained by Phillips (1986). BOLS (iii) The fact that the asymptotic distribution of Do,, is different from that of the spurious regression with noncointegrated regressors has an important practical consequence for residual-based cointegration tests, because the null distributions of residual-based cointegration tests depend on the asymptotic distribution of ?oLs. Thus, different null distributions of residual-based cointegration tests will result as the asymptotic distribution of BoLs changes with cointegrated regressors. This point will be studied more specifically in the next section.
4. Residual-based
tests for cointegration
with cointegrated
In this section we derive the null distributions tegration tests as z^, of Phillips and Ouliaris
regressors
of such residual-based coin(1990) and the augmented
318
I. Choi, Spurious regressions
with cointegrated
regressors
Dickey-Fuller (ADF) test suggested in Engle and Granger (1987), when regressors are cointegrated. The reader is referred to Phillips and Ouliaris (1990) for the definitions of 2, and ADF. The results are reported in the following theorem. Theorem
2.
Suppose
that the conditions for
that the order of the autoregression p = o(T’13).
Theorem
1 hold. Further,
in the ADF is such that p +
co as T +
assume 03 and
Then, we have
(4
W where B’23’ = [B;,
g;,
B’23’ = [B;,
@;I,
and 8 is as dejined in Theorem
B;, Bj],
3.1.
Remarks
(i) We find that the null distributions of 2a and ADF are different from those derived and tabulated and Phillips and Ouliaris (1990). Hence, using the critical values tabulated in this article will not yield correct asymptotic sizes, when regressors are cointegrated. Moreover, these null distributions depend on many nuisance parameters, which include the number of cointegrating vectors for the regressors and the cointegrating vectors. The corrections which make these statistics free of nuisance parameters in the limit appear to be technically complicated, and will require prior information regarding the cointegrating structure of the regressors. A practical implication of these considerations is that we need to test not only the presence of a unit root for individual series but also the presence of cointegrating vectors for the regressors prior to performing residual-based tests for cointegration. When the regressors are cointegrated, using the critical values which were already tabulated in the literature [e.g., Engle and Granger (1987) Engle and Yoo (1987, 1989) Phillips and Ouliaris (1990)] would not yield correct asymptotic size. (ii) The model we study includes an intercept term. But we can obtain similar results for models without time polynomials or with a linear time trend, and hence the conventional critical values for these models cannot be applied, when regressors are cointegrated. (iii) We report results for 2, and ADF here, but qualitatively the same results can be obtained for other residual-based tests studied in the literature [see, for
I. Choi. Spurious
regressions
with coinfegrated
regressors
319
example, Engle and Granger (1987), Phillips and Ouliaris (1990), Stock (1988) and Choi (1991)]. This is because the null distributions of residual-based cointegration tests depend on the asymptotic distribution of the coefficient estimate under the null of no cointegration, which undergoes changes with cointegrated regressors.
5. Summary
and further remarks
We have studied the spurious regression and residual-based tests for cointegration when regressors are cointegrated. The asymptotic results reported in section 3 show that the fundamental nature of the spurious regression does not change with cointegrated regressors, but that the asymptotic distributions of the coefficient estimates and such statistics as t, F, R2, and DW are different from those of the surious regression with noncointegrated regressors. An important practical consequence of these results is that the asymptotic distributions of residual-based tests for cointegration undergo changes with cointegrated regressors, as reported in section 4. It is a common practice these days to perform unit root tests for individual series prior to residual-based tests for cointegration. Analytic and experiment results reported in this paper show that not only unit root tests but also cointegration tests for regressors are required before mounting residual-based tests for cointegration. If there is statistical evidence for cointegrated regressors, using residual-based tests for cointegration with conventional critical values already tabulated cannot be justified. Unfortunately, it seems complicated to correct residual-based tests such that their asymptotic distributions are free of nuisance parameters when regressors are cointegrated. The main source of difficulty for this lies in that the asymptotic distributions depend on the cointegrating vectors for regressors, which are typically unknown. Considering this difficulty, it seems necessary and useful to devise testing procedures that are robust to cointegrated regressors. One such procedure is proposed in Wooldridge (1991).
References Choi, I., 1990, Durbin-Hausman tests for cointegration, Unpublished manuscript (Ohio State University, Columbus, OH). Choi, I., 1991, Asymptotic normality of the least squares estimates for higher-order autoregressive integrated processes with some applications, Econometric Theory, forthcoming. Engle, R.F. and C.W.J. Granger, 1987, Co-integration and error correction: Representation, estimation and testing, Econometrica 55, 251-276. Engle, R.F. and S. Yoo, 1987, Forecasting and testing in cointegrated systems, Journal of Econometrics 35, 143-159. Engle, R.F. and S. Yoo, 1989, Cointegrated economic time series: A survey with new results, Working paper no. 8-89-13 (Pennsylvania State University, University Park, PA).
320
I. Choi, Spurious regressions
with coiniegrated
regressors
Granger, C.W.J. and P. Newbold, 1974, Spurious regressions in econometrics, Journal of Econometrics 2, 11 l-120. Park, J.Y. and P.C.B. Phillips, 1989, Statistical inference in regressions with integrated processes: . Part 2, Econometric Theory 5, 95-131. 33. 3 1 l-340. Phillins. P.C.B.. 1986. Understandina sourious repressions. Journal of Econometrics Phillips; P.C.B., 1988, Multiple regre&ibn with in&grated time series, Contemporary Mathematics 80, 79-105. Phillips, P.C.B. and S. Ouliaris, 1988, Testing fr cointegration using principal component methods, Journal of Economic Dynamics and Control 12, 205-230. Stock, J., 1988, A class of tests for integration and cointegration, Unpublished manuscript (Kennedy School of Government, Harvard University, Cambridge, MA). Wooldridge, J., 1991, Notes on regression with difference-stationary data, Unpublished manuscript (MIT, Cambridge, MA).
of Econometrics
60 (1994) 313-320.
North-Holland
Spurious regressions and residual-based tests for cointegration when regressors are cointegrated In Choi* The Ohio State University, Received
September
Columbus,
OH 43210, USA
1991, final version
received
September
1992
We study a spurious regression with cointegrated regressors. Asymptotic properties of the OLS estimates and other conventional statistics are reported. Using these results, we analyze the asymptotic null distributions of such residual-based cointegration tests as Phillips’ 2, and the augmented Dickey-Fuller (ADF) tests, when regressors are cointegrated. Key words: Spurious
regression;
Cointegrated
regressors;
Residual-based
tests for cointegration
1. Introduction Granger and Newbold’s (1974) classic study of regressing one integrated process on a set of integrated processes has attracted a good deal of attention among theoretical econometricians and empirical researchers. This practice of using integrated processes as both regressand and regressors is now commonly called a spurious regression [unless the regressand and regressors are cointegrated in the sense of Engle and Granger (1987)]. Subsequently, Phillips (1986) provided the asymptotic analyses that can fully explain the points raised by Granger and Newbold. These asymptotic results of Phillips are useful beyond explaining Granger and Newbold’s simulation results, because residualbased tests for cointegration [see, for example, Engle and Granger (1987) Phillips and Ouliaris (1990) Stock (1988), and Choi (1990)] take the spurious
Correspondence to: In Choi, Department High Street, Columbus, OH 43210-1172,
of Economics, USA.
The Ohio
State University,
1945 North
*I would like to thank the Associate Editor and two anonymous referees for their helpful comments on an earlier version of this paper. At the suggestion of the Associate Editor, the proofs of all theorems are omitted, but can be obtained from the author upon request.
0304-4076/94/$06.00
c
1994-Elsevier
Science Publishers
B.V. All rights reserved
314
I. Choi, Spurious regressions with cointegrated regressors
regression as the null hypothesis, and hence their asymptotic distributions depend on those of coefficient estimates from the spurious regression. Remember, though, that it is assumed that regressors are not cointegrated in the aforementioned studies of the spurious regressions and residual-based tests for cointegration. But in some applications, it is not nonsensical to conjecture that regressors are cointegrated. For example, when we regress one price variable (e.g., exchange rates, interest rates, etc.) on other price variables for cointegration analysis, it is reasonably expected that the regressors are cointegrated. The asymptotic properties of the OLS coefficient estimates in cointegrating regressions with cointegrated regressors are studied in Park and Phillips (1989) and Wooldridge (1991). However, spurious regression is the focus of this paper, partly because the spurious regression with cointegrated regressors is of theoretical interest and partly because the null of residual-based tests is the spurious regression. To date, no systematic analysis has been given in the literature regarding the spurious regression and the asymptotic distributions of residualbased tests for cointegration when regressors are cointegrated, though it is not uncommon to hear empirical researchers express concerns about using residualbased tests in such cases. Thus, the first objective of this paper is to analyze the spurious regression with cointegrated regressors. We will study the asymptotic properties of the OLS estimates and other conventional statistics when regressors are cointegrated, but the regression itself is spurious in nature. To this end, we will use the weak convergence results obtained by Phillips and his associates [see Phillips (1988) and references therein] and methods similar to those of Choi (1991). The second objective of this paper is to analyze the asymptotic null distributions of residual-based cointegration tests, when regressors are cointegrated. We focus on Phillips’ A’, and the augmented Dickey-Fuller (ADF) tests, because these are most commonly used in applications. However, qualitatively similar results hold for other kinds of residual-based tests. These analyses will show that the asymptotic distributions of 2, and ADF with cointegrated regressors differ from those with noncointegrated regressors. The structure of this paper has been planned as follows. Section 2 introduces the model and assumptions. Section 3 derives the asymptotic distributions of the OLS estimates and various conventional statistics from the spurious regression with cointegrated regressors. Section 4 studies the asymptotic null distributions of 2, and ADF. Section 5 contains a summary and further remarks. A few words on our notation: All the limits are taken as T + co unless otherwise specified. B(r) denotes a multivariate Brownian motion, and B, s:B, s:BB’, and JiBdB’
are understood
to be B(r), and J:B(r)dr,
jiB(r)dB(r)‘, respectively. Convergence X denotes T-l~~=l~f.
in distribution
JiB(r)B(r)‘dr,
is denoted
by j.
and Last,
2. Preliminaries WC are concerned .r’,
=
with the multiple
x;jr + II,,
1’ +
t=l’
regression
,-1
.
with an intercept
(1)
7;
where 7; = (yt. x;) is a vector 1(l) process of dimension ~1 + I (nt L 2) [see Angle and Granger (1987) for the definitions of 1(l) and I(O)]+ which is generated by z, = s,-
1
+
E,,
where z, = r(O). Once we assume that the error sequence {u!] is I( 1) and that the regressor X, is not cointegrated, eq. (1) is the prototgpe of a spurious regression studied by Granger and Newbold (1974) and Phillips (19X6). The case that the or noncoerror sequence jll,i ’ Is I(O) and Ihc regressor X, is either cointegrated integrated is studied in Park and Phillips (1989). In this paper, we assume that U, = I(I) and that there exists y (m - 1 2 p 2 I) linearly independent cointegrating vectors for the regressor s,, which are denoted as ;‘,, . y,,. Letting
\ve have that C’:s, = 18,is weakly stationary by the definition of cointegration. We standardize the cointegrating vectors such that y:y, = 1 (i = 1. , p). Further, we assume that ;‘i # 0 (i = I, . ..~ p), so that trivial cointograting relations for the regressor s, are excluded. We assume that the innovation sequcnct: $ = (&vi) is strictly stationary and ergodic with zero mean and finik variance. Further. WC require the partial sum invariance principle. process constructed from [ rl,!t to satisfy the multivariate That is,
where
is an (1 + 111 + p)-vector
Rrownian
motion
with covariancc
matrix
I. Choi, Spurious regressions
316
with cointegrafed
regressors
where
Cij and Aij for i, j = 1,2,3,4 are defined to be submatrices of C and A corresponding to Rij. The reader is referred to Phillips (1988) and the references therein for the multivariate invariance principle.
3. Spurious regressions with cointegrated
regressors
In this section we study the asymptotic properties of the OLS estimates, R2, F-statistics, t-ratios, and the Durbin-Watson statistic (DW) for the regression equation (l), when the regression is spurious in the sense of Granger and Newbold (1974) and the regressors involved are cointegrated. The main results of this section are presented in the following theorem. The null hypotheses for tp, and F, in the following theorem are, respectively, H,,: pi = 0 (i = 1, . . . , p) and Ho: p = 0. Theorem
1.
Suppose
section 2, we have
(4
(b) (4 (4
(4
(f)
that {ut} is I(1). Then, under the relevant
conditions
in
I. Choi, Spurious regressions
with cointegrated
317
regressors
where & = Bi - pi,
823, = CB2, Bj],
dij
=
Tij
+
/lij,
m-p 0
1 ..
0’ = (1, - b’), f 123= E(E,E;),
c2 = [
0
1 m,
.1
and Cl is dejined in section 2. Remarks (i) This theorem shows that i&s has a nondegenerate limiting distribution to and the distribution of j&s diverges. Moreover, R2 converges in distribution a well-defined random variable and DW converges to zero in probability. We expect from these results that the regressions of this type will yield moderate values of R2 and low values of D W, which are regarded as evidence of a spurious regression by Granger and Newbold (1974). We also observe that the distribution of the F-statistic diverges, while those of the t-ratios converge to welldefined random variables. Diverging t-ratio is an important characterization of the spurious regression as shown in Phillips (1986, p. 320), but we observe arises because here that the r-ratios do not diverge. This dissimilarity { C(xf - X)(x, - 2,’ } - ’ = O,( r- ‘) when the regressors are cointegrated, while {x(x( - X)(x, - I?)‘}-’ = 0,(Tp2) when they are not. However, it is important to notice that the assumption yi # 0 (i = 1, . . . , p) is essential for the weak convergence of the t-ratios. When yk = 0 for some k, the t-ratio corresponding to the regressor &k diverges, as one of the referees pointed out. Except for the t-ratios, the orders of magnitude for the OLS estimates and the statistics we study correspond with these reported in Phillips (1986, theorem 2). (ii) When regressors are not cointegrated, the limiting distributions of and ,i& reduce to those obtained by Phillips (1986). BOLS (iii) The fact that the asymptotic distribution of Do,, is different from that of the spurious regression with noncointegrated regressors has an important practical consequence for residual-based cointegration tests, because the null distributions of residual-based cointegration tests depend on the asymptotic distribution of ?oLs. Thus, different null distributions of residual-based cointegration tests will result as the asymptotic distribution of BoLs changes with cointegrated regressors. This point will be studied more specifically in the next section.
4. Residual-based
tests for cointegration
with cointegrated
In this section we derive the null distributions tegration tests as z^, of Phillips and Ouliaris
regressors
of such residual-based coin(1990) and the augmented
318
I. Choi, Spurious regressions
with cointegrated
regressors
Dickey-Fuller (ADF) test suggested in Engle and Granger (1987), when regressors are cointegrated. The reader is referred to Phillips and Ouliaris (1990) for the definitions of 2, and ADF. The results are reported in the following theorem. Theorem
2.
Suppose
that the conditions for
that the order of the autoregression p = o(T’13).
Theorem
1 hold. Further,
in the ADF is such that p +
co as T +
assume 03 and
Then, we have
(4
W where B’23’ = [B;,
g;,
B’23’ = [B;,
@;I,
and 8 is as dejined in Theorem
B;, Bj],
3.1.
Remarks
(i) We find that the null distributions of 2a and ADF are different from those derived and tabulated and Phillips and Ouliaris (1990). Hence, using the critical values tabulated in this article will not yield correct asymptotic sizes, when regressors are cointegrated. Moreover, these null distributions depend on many nuisance parameters, which include the number of cointegrating vectors for the regressors and the cointegrating vectors. The corrections which make these statistics free of nuisance parameters in the limit appear to be technically complicated, and will require prior information regarding the cointegrating structure of the regressors. A practical implication of these considerations is that we need to test not only the presence of a unit root for individual series but also the presence of cointegrating vectors for the regressors prior to performing residual-based tests for cointegration. When the regressors are cointegrated, using the critical values which were already tabulated in the literature [e.g., Engle and Granger (1987) Engle and Yoo (1987, 1989) Phillips and Ouliaris (1990)] would not yield correct asymptotic size. (ii) The model we study includes an intercept term. But we can obtain similar results for models without time polynomials or with a linear time trend, and hence the conventional critical values for these models cannot be applied, when regressors are cointegrated. (iii) We report results for 2, and ADF here, but qualitatively the same results can be obtained for other residual-based tests studied in the literature [see, for
I. Choi. Spurious
regressions
with coinfegrated
regressors
319
example, Engle and Granger (1987), Phillips and Ouliaris (1990), Stock (1988) and Choi (1991)]. This is because the null distributions of residual-based cointegration tests depend on the asymptotic distribution of the coefficient estimate under the null of no cointegration, which undergoes changes with cointegrated regressors.
5. Summary
and further remarks
We have studied the spurious regression and residual-based tests for cointegration when regressors are cointegrated. The asymptotic results reported in section 3 show that the fundamental nature of the spurious regression does not change with cointegrated regressors, but that the asymptotic distributions of the coefficient estimates and such statistics as t, F, R2, and DW are different from those of the surious regression with noncointegrated regressors. An important practical consequence of these results is that the asymptotic distributions of residual-based tests for cointegration undergo changes with cointegrated regressors, as reported in section 4. It is a common practice these days to perform unit root tests for individual series prior to residual-based tests for cointegration. Analytic and experiment results reported in this paper show that not only unit root tests but also cointegration tests for regressors are required before mounting residual-based tests for cointegration. If there is statistical evidence for cointegrated regressors, using residual-based tests for cointegration with conventional critical values already tabulated cannot be justified. Unfortunately, it seems complicated to correct residual-based tests such that their asymptotic distributions are free of nuisance parameters when regressors are cointegrated. The main source of difficulty for this lies in that the asymptotic distributions depend on the cointegrating vectors for regressors, which are typically unknown. Considering this difficulty, it seems necessary and useful to devise testing procedures that are robust to cointegrated regressors. One such procedure is proposed in Wooldridge (1991).
References Choi, I., 1990, Durbin-Hausman tests for cointegration, Unpublished manuscript (Ohio State University, Columbus, OH). Choi, I., 1991, Asymptotic normality of the least squares estimates for higher-order autoregressive integrated processes with some applications, Econometric Theory, forthcoming. Engle, R.F. and C.W.J. Granger, 1987, Co-integration and error correction: Representation, estimation and testing, Econometrica 55, 251-276. Engle, R.F. and S. Yoo, 1987, Forecasting and testing in cointegrated systems, Journal of Econometrics 35, 143-159. Engle, R.F. and S. Yoo, 1989, Cointegrated economic time series: A survey with new results, Working paper no. 8-89-13 (Pennsylvania State University, University Park, PA).
320
I. Choi, Spurious regressions
with coiniegrated
regressors
Granger, C.W.J. and P. Newbold, 1974, Spurious regressions in econometrics, Journal of Econometrics 2, 11 l-120. Park, J.Y. and P.C.B. Phillips, 1989, Statistical inference in regressions with integrated processes: . Part 2, Econometric Theory 5, 95-131. 33. 3 1 l-340. Phillins. P.C.B.. 1986. Understandina sourious repressions. Journal of Econometrics Phillips; P.C.B., 1988, Multiple regre&ibn with in&grated time series, Contemporary Mathematics 80, 79-105. Phillips, P.C.B. and S. Ouliaris, 1988, Testing fr cointegration using principal component methods, Journal of Economic Dynamics and Control 12, 205-230. Stock, J., 1988, A class of tests for integration and cointegration, Unpublished manuscript (Kennedy School of Government, Harvard University, Cambridge, MA). Wooldridge, J., 1991, Notes on regression with difference-stationary data, Unpublished manuscript (MIT, Cambridge, MA).