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Cointegration analysis using M estimators
Economics Letters 71 (2001) 149–154 www.elsevier.com / locate / econbase
Cointegration analysis using M estimators Ted Juhl* University of Kansas, Department of Economics, 213 Summerfield Hall, Lawrence, KS 66045, USA Received 12 June 2000; accepted 20 December 2000
Abstract Tests for cointegration are developed using multivariate M estimators. The tests are based on analyzing the singular values of the parameter estimates standardized by the covariance matrix and do not require a reduced rank estimator. 2001 Elsevier Science B.V. All rights reserved. Keywords: Cointegration; Robust inference JEL classification: C12; C22
1. Introduction The dominant strand of the cointegration literature is based on ordinary least-squares (OLS) and reduced rank regression. However, since economic data often exhibit non-Gaussian behavior, it would seem that other estimation strategies may provide an increase in efficiency, translating into increased power in testing. Lucas (1996, 1997, 1998) successfully implemented tests for cointegration using Wald tests, pseudo likelihood ratio tests, and LM tests based on a non-Gaussian likelihood that falls in the class of M estimators. This paper introduces a new test of cointegrating rank, one which is based on multivariate M estimation of a vector autoregression (VAR). The multivariate M estimator of the first VAR coefficent is standardized by its covariance matrix, and we perform a singular value decomposition. The singular values are tested directly in order to determine the rank of the cointegrating space. This test is called the M singular value test or MSV. In the remainder of the paper, we use the following standard notation. We denote convergence in d distribution by →. If W(s) is a Brownian motion with s [ (0, 1), then the Lebesgue integral e01W(s) ds is denoted by eW and the stochastic integral e01W(s) dW(s)¡ 5 eW dW ¡ . Finally, if K is a p 3 r matrix ¡ of full column rank, then K' is a p 3 ( p 2 r) matrix such that K ' K 5 0. *Corresponding author. Tel.: 11-785-864-3501; fax: 11-785-864-5270. E-mail address: [email protected] (T. Juhl). 0165-1765 / 01 / $ – see front matter PII: S0165-1765( 01 )00381-0
2001 Elsevier Science B.V. All rights reserved.
T. Juhl / Economics Letters 71 (2001) 149 – 154
150
2. M-singular value test We first consider a kth order VAR with iid errors. Let
O G DX
k 21
DXt 5 m 1 P Xt21 1
i
t2i
1 et ,
(1)
i 51
where the vectors Xt and et are p 3 1, P and Gi are p 3 p, and et has covariance matrix See . Testing the null of r cointegrating vectors involves testing the rank of the matrix P. One way to determine the rank of a matrix is to examine the singular values. The singular values of a matrix K are the eigenvalues of K ¡ K and the rank of a matrix is the number of nonzero singular values. Presumably, ˆ and calculate the singular values of the standardized we could estimate P, find a standard error for P ˆ P. Finally, we must have some statistical criteria for determining how many of the singular values are significantly different from zero. We pursue the strategy mentioned above using the class of multivariate M estimators for P. Let u 5 ( m, P, G, See ). We define the estimator of u as
FO S T
uˆM 5 arg min u
r S
O G DX )D 2 ]T2 loguS uG
k21 21 / 2 ee
(DXt 2 m 2 P Xt21 2
t51
i
t2i
ee
(2)
i51
for a function r :R p → R . Suppose that Xt was a stationary vector and that m 5 Gi 5 0. Then the (stationary) estimator of P, ˆ M would have an estimated covariance matrix given by which we denote P 21 21 21 S 11 ^ Cˆ 1 Sˆ cc Cˆ 1 , T T ˆ ˆ ¡ , with c 5 ≠r / ≠e and where S11 5 T 21 o tT51 Xt 21 X t¡21 , Cˆ 1 5 T 21 o t51 cˆ 9, and Sˆ cc 5 T 21 o t51 cc c 9 5 ≠c / ≠e ¡ . The matrices Cˆ 1 and Sˆ cc are estimates of C1 5 E(c 9) and Scc 5 E(cc ¡ ). Define the 21 21 matrix SˆC 5 Cˆ 1 Sˆ cc Cˆ 1 . Next, we construct the standardized matrix 21 / 2 P˜ M 5 Sˆ C Pˆ M S 111/ 2 , ¡ ˜ M where l˜ 1 . l˜ 2 . ? ? ? . l˜ p . The null hypothesis of r and let l˜ i denote the eigenvalues of P˜ M P cointegrating vectors can be tested using the statistic MSVr 5 T o pi5r11 l˜ i . The MSV statistic is asymptotically equivalent to Johansen’s (1988) likelihood ratio statistic if one uses OLS 1 . The asymptotic distribution of MSVr is given below.
Theorem 1. Suppose that assumptions given in Lucas (1998) hold. If rank(P ) 5 r so that P 5 ab ¡ with a and b both p 3 r, then p
MSVr 5 T
O l˜ →traceSEdF F¯ SEF¯ F¯ D EF¯ dF D. 21
d
i
1
¡ 2
2
¡ 2
2
¡ 1
(3)
i 5r11
F1 (s) and F2 (s) are standard Brownian motions. The correlation between F1 and F2 is a diagonal 1
The trace statistic given in Johansen (1988) is 2 T o pi 5r 11 log(1 2 l˜ i ).
T. Juhl / Economics Letters 71 (2001) 149 – 154
151
¡ 21 matrix with the canonical correlations between a ¡ ' et and a ' C 1 ct . If there is no intercept and one is ¡ not estimated, F¯ 2 5 F2 . If an intercept is present (and estimated) and a ' m 5 0, F¯ 2 5 F2 2 eF2 . If ¡ a ' ± 0, the first element of F¯ 2 is s and the remaining elements are the remaining elements of F2 2 eF2 .
The asymptotic distribution of MSVr is identical to that of the Wald and LM statistics proposed by Lucas (1996, 1998). There are nuisance parameters remaining in the limiting distribution in that the Brownian motions are imperfectly correlated, which changes the limiting distribution for each M estimator selected. However, we can find estimates of the correlations and use the Gamma approximation suggested in Boswijk and Doornik (1999) to obtain asymptotic critical values.
3. Monte Carlo We conduct a simple Monte Carlo experiment using the data generating process DXt 5
S
c 2] T 0
D
0 0
Xt 21 1 et .
If c 5 0, there are no cointegrating vectors and if c . 0, there is one (trivial) cointegrating vector. Several distributions are considered for et ; normal, t with 3 degrees of freedom, mixed normal 2 , and Cauchy 3 . We set r (e) 5 ln(1 1 e ¡ e / 5) so that we use a student-t likelihood with 5 degrees of freedom for the MSV and LM tests. The Gaussian likelihood ratio test of Johansen (1988) is included for comparison and denoted GLR. The sample size is 200 and 10 000 replications are performed. In order to find critical values for the MSV and LM tests, we must find an estimate of the nuisance parameters that affect the limiting distribution. That is, we estimate the canonical correlations between F1 and F2 . Using these estimates, critical values are generated for the tests in each replication by implementing the Gamma approximation proposed in Boswijk and Doornik (1999). The rejection percentage for the tests are listed in Table 1. The MSV test based on the t distribution has good size and power properties and effectively exploits heavy-tailed errors. The Gamma approximation works very well for both the MSV and LM tests.
4. Conclusion We have developed a new test for cointegrating rank using robust tests based on multivariate M estimators. There are several advantages of the MSV test. First, the test does not depend on the ordering of the variables in the system, a feature shared with the LM test of Lucas (1998). Second, we 2
The observation comes from a normal distribution with standard deviation 2.5 with probability 0.2 and from a normal distribution with standard deviation 0.5 with probability 0.8. 3 The Cauchy distribution does not satisfy the assumptions given in the theorem but is used as an example of extreme conditions.
T. Juhl / Economics Letters 71 (2001) 149 – 154
152 Table 1 Size and size adjusted power Test
Test
c
MSV
LM
GLR
Normal
0 3 6 9 12
0.06 0.07 0.11 0.17 0.27
0.04 0.07 0.11 0.17 0.27
0.05 0.07 0.11 0.19 0.30
t 3
0 3 6 9 12
0.06 0.15 0.33 0.54 0.74
0.05 0.15 0.33 0.54 0.73
0.05 0.07 0.11 0.20 0.31
c
MSV
LM
GLR
Mix
0 3 6 9 12
0.05 0.35 0.72 0.92 0.98
0.05 0.34 0.72 0.91 0.98
0.05 0.07 0.11 0.18 0.30
Cauchy
0 3 6 9 12
0.05 0.98 0.99 1.00 1.00
0.05 0.98 0.99 1.00 1.00
0.06 0.24 0.32 0.42 0.52
use a Monte Carlo experiment to show that the size and power are quite good. Finally, the computation of the MSV test is particularly simple in that we do not impose a reduced rank condition at the estimation stage, reducing computational complexity compared to the Wald, PLR, and LM tests in Lucas (1996, 1997, 1998). Given the non-linear optimization required for some M estimators, this is highly advantageous. The proposed MSV test could be easily extended to incorporate adaptively estimated models. Again, we need only obtain one estimate of P adaptively, and consider the singular values from the standardized estimate.
Acknowledgements I thank Bruce Hansen, Roger Koenker, Donald Lien, and Andre´ Lucas for comments on earlier versions of this paper.
Appendix A. Proof of Theorem 1 We prove the theorem when the data generating process has no intercept and one is not estimated. The other cases are similar. Given the conditions in the theorem,
OScb X D ⇒SBB (s)(s)D
[Ts]
T 21 / 2
c
t ¡
t51
t
2
where Bc and B2 are correlated Brownian motions. The variance matrices associated with Bc and B2 ¡ ¡ ¡ are Scc and (a ' b' )21 a ' See a' ( b ' a' )21 . Let H 5 ( b, b' ) be normalized such that H ¡ H 5 I. It is straightforward to show that
T. Juhl / Economics Letters 71 (2001) 149 – 154
vecsT
1/2
153
d 21 ˆ M 2 P )bd →N(0, S 11 ^ C 121 Scc C 121 ). (P
and d
SE
ˆ M 2 P )b'd → ( B2 B 2¡ )21 ^ C 121 vecsT( P
DE
B2 ^ dBc .
Let the distributions above (not vectorized) be denoted as J1 and J2 respectively. We write
Pˆ M 5 P 1 (J1 :J2 )A T 1 o p (A T ) with A T 5 (T
21 / 2
(A.1)
b :T 21 b' )¡ . The roots of
¡ 1 / 2 ˆ ¡ ˆ 21 ˆ 1/2 u lI 2 P˜ P˜ u 5 u lI 2 S 11 P M S C PM S 11 u
are equivalent to the roots of ¡ 21 ˆM ˆ M S11 u Sˆ C P u lS11 2 S11 P
since S11 is positive definite in finite samples. Define ¡ 21 ˆM ˆ M S11 . Sˆ C P S( l) 5 lS11 2 S11 P ¡
Note that S( l) shares the roots of H S( l)H. Using rules for partitioned determinants gives ¡ uH ¡ S( l)Hu 5 u b ¡ S( l)b uu b ' hS( l) 2 S( l)b ( b ¡ S( l)b )21 b ¡ S( l)j b' u
(A.2)
We will consider the asymptotic distribution of r 5 Tl since the eigenvalues are normalized by T in the MSVr statistic. Note that under the null hypothesis, P 5 ab ¡ . Using this fact and the result p b ¡ S11 b → S11 from Johansen (1988), we have p
b ¡ S( l)b → 2 S11 a ¡ C1 S 21 cc C 1 aS11 .
(A.3)
Next, 1 ¡ ˆ ¡ ˆ 21 ˆ b¡ b S b 2b¡ ' S( l)b 5 r ] ' S 11 P M S C PM S 11 b. T ' 11 p
¡ From Johansen (1988), b ¡ ' S 11 b 5 Op (1), b S 11 b → S11 , and
S
E
D
¡ ¡ ¡ 21 b¡ ' S( l)b ⇒ 2 Op (1)a 1 ( B 2 B 2 )J 2 C 1 S cc C 1 aS11 .
Finally, 1 ¡ ¡ 21 ¡ ¡ ˆM ˆ M S11 b' , b' S( l)b' 5 r ] b ' S11 b' 2 b ' S11 P Sˆ C P T so that
1 ] T
¡ b' S11 b' ⇒ eB2 B 2¡ , so
(A.4)
T. Juhl / Economics Letters 71 (2001) 149 – 154
154
E
¡ 21 ¡ b¡ ' S( l)b' ⇒ r B 2 B 2 2 D 1 C 1 S cc C 1 D 1
(A.5)
follows similarly with D1 5 Op (1)a ¡ 1 (eB2 B 2¡ )J 2¡ . Inserting (A.3), (A.4), and (A.5) into (A.2) results in
E
u r B2 B 2¡ 2 D2 D3 D 2¡ u
(A.6)
21 21 ¡ 21 21 with D2 5 Op (1)a ¡ 1 eB2 dB ¡ and D3 5 C1 S 21 3 c C1 cc C 1 2 (C 1 S cc C 1 )a (a C 1 S cc C 1 a ) 21 a ¡ (C1 S cc C1 ). Following Lemma 10.1 in Johansen (1995), we can show that 21 21 21 ¡ D3 5 a' (a ¡ a' ' C 1 Scc C 1 a' )
which will eliminate the Op (1)a
E
¡
terms. Then A.6 becomes
E
E
21 ¡ 21 21 21 u r B2 B ¡ a' C 121 dBc B 2¡ u. 2 2 B 2 dBc C 1 a' (a ' C 1 Scc C 1 a' )
(A.7)
21 / 2 Pre and post multiplying A.7 by u(a ¡ (a ¡ ' See a' ) ' b' )u will not change the roots and it standardizes B2 (s) so that all the Brownian motions have unit variance. Then using the trace we have p
T
O l˜ ⇒ trSEdF F SEF F D EF dF D 21
i
1
¡ 2
2
¡ 2
2
¡ 1
(A.8)
i 5r 11
¡
21
21
where F1 5 (a ' C 1 Scc C 1 a' )
21 / 2
¡ ¡ a' C 121 Bc (s) and F2 5 (a ' See a' )21 / 2 (a ' b' )B2 (s).
h
References Boswijk, H., Doornik, J.A., 1999. Distribution approximations for cointegration tests with stationary exogenous regressors. Universiteit van Amsterdam. Johansen, S., 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12, 231–254. Johansen, S., 1995. Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press, New York. Lucas, A., 1996. Outlier robust unit root analysis. Ph.D. Thesis, Tinberen Institute, Amsterdam. Lucas, A., 1997. Cointegration testing using pseudolikelihood ratio tests. Econometric Theory 13, 149–169. Lucas, A., 1998. Inference on cointegrating ranks using LR and LM tests based on pseudo-likelihoods. Econometric reviews 17, 185–214.
Cointegration analysis using M estimators Ted Juhl* University of Kansas, Department of Economics, 213 Summerfield Hall, Lawrence, KS 66045, USA Received 12 June 2000; accepted 20 December 2000
Abstract Tests for cointegration are developed using multivariate M estimators. The tests are based on analyzing the singular values of the parameter estimates standardized by the covariance matrix and do not require a reduced rank estimator. 2001 Elsevier Science B.V. All rights reserved. Keywords: Cointegration; Robust inference JEL classification: C12; C22
1. Introduction The dominant strand of the cointegration literature is based on ordinary least-squares (OLS) and reduced rank regression. However, since economic data often exhibit non-Gaussian behavior, it would seem that other estimation strategies may provide an increase in efficiency, translating into increased power in testing. Lucas (1996, 1997, 1998) successfully implemented tests for cointegration using Wald tests, pseudo likelihood ratio tests, and LM tests based on a non-Gaussian likelihood that falls in the class of M estimators. This paper introduces a new test of cointegrating rank, one which is based on multivariate M estimation of a vector autoregression (VAR). The multivariate M estimator of the first VAR coefficent is standardized by its covariance matrix, and we perform a singular value decomposition. The singular values are tested directly in order to determine the rank of the cointegrating space. This test is called the M singular value test or MSV. In the remainder of the paper, we use the following standard notation. We denote convergence in d distribution by →. If W(s) is a Brownian motion with s [ (0, 1), then the Lebesgue integral e01W(s) ds is denoted by eW and the stochastic integral e01W(s) dW(s)¡ 5 eW dW ¡ . Finally, if K is a p 3 r matrix ¡ of full column rank, then K' is a p 3 ( p 2 r) matrix such that K ' K 5 0. *Corresponding author. Tel.: 11-785-864-3501; fax: 11-785-864-5270. E-mail address: [email protected] (T. Juhl). 0165-1765 / 01 / $ – see front matter PII: S0165-1765( 01 )00381-0
2001 Elsevier Science B.V. All rights reserved.
T. Juhl / Economics Letters 71 (2001) 149 – 154
150
2. M-singular value test We first consider a kth order VAR with iid errors. Let
O G DX
k 21
DXt 5 m 1 P Xt21 1
i
t2i
1 et ,
(1)
i 51
where the vectors Xt and et are p 3 1, P and Gi are p 3 p, and et has covariance matrix See . Testing the null of r cointegrating vectors involves testing the rank of the matrix P. One way to determine the rank of a matrix is to examine the singular values. The singular values of a matrix K are the eigenvalues of K ¡ K and the rank of a matrix is the number of nonzero singular values. Presumably, ˆ and calculate the singular values of the standardized we could estimate P, find a standard error for P ˆ P. Finally, we must have some statistical criteria for determining how many of the singular values are significantly different from zero. We pursue the strategy mentioned above using the class of multivariate M estimators for P. Let u 5 ( m, P, G, See ). We define the estimator of u as
FO S T
uˆM 5 arg min u
r S
O G DX )D 2 ]T2 loguS uG
k21 21 / 2 ee
(DXt 2 m 2 P Xt21 2
t51
i
t2i
ee
(2)
i51
for a function r :R p → R . Suppose that Xt was a stationary vector and that m 5 Gi 5 0. Then the (stationary) estimator of P, ˆ M would have an estimated covariance matrix given by which we denote P 21 21 21 S 11 ^ Cˆ 1 Sˆ cc Cˆ 1 , T T ˆ ˆ ¡ , with c 5 ≠r / ≠e and where S11 5 T 21 o tT51 Xt 21 X t¡21 , Cˆ 1 5 T 21 o t51 cˆ 9, and Sˆ cc 5 T 21 o t51 cc c 9 5 ≠c / ≠e ¡ . The matrices Cˆ 1 and Sˆ cc are estimates of C1 5 E(c 9) and Scc 5 E(cc ¡ ). Define the 21 21 matrix SˆC 5 Cˆ 1 Sˆ cc Cˆ 1 . Next, we construct the standardized matrix 21 / 2 P˜ M 5 Sˆ C Pˆ M S 111/ 2 , ¡ ˜ M where l˜ 1 . l˜ 2 . ? ? ? . l˜ p . The null hypothesis of r and let l˜ i denote the eigenvalues of P˜ M P cointegrating vectors can be tested using the statistic MSVr 5 T o pi5r11 l˜ i . The MSV statistic is asymptotically equivalent to Johansen’s (1988) likelihood ratio statistic if one uses OLS 1 . The asymptotic distribution of MSVr is given below.
Theorem 1. Suppose that assumptions given in Lucas (1998) hold. If rank(P ) 5 r so that P 5 ab ¡ with a and b both p 3 r, then p
MSVr 5 T
O l˜ →traceSEdF F¯ SEF¯ F¯ D EF¯ dF D. 21
d
i
1
¡ 2
2
¡ 2
2
¡ 1
(3)
i 5r11
F1 (s) and F2 (s) are standard Brownian motions. The correlation between F1 and F2 is a diagonal 1
The trace statistic given in Johansen (1988) is 2 T o pi 5r 11 log(1 2 l˜ i ).
T. Juhl / Economics Letters 71 (2001) 149 – 154
151
¡ 21 matrix with the canonical correlations between a ¡ ' et and a ' C 1 ct . If there is no intercept and one is ¡ not estimated, F¯ 2 5 F2 . If an intercept is present (and estimated) and a ' m 5 0, F¯ 2 5 F2 2 eF2 . If ¡ a ' ± 0, the first element of F¯ 2 is s and the remaining elements are the remaining elements of F2 2 eF2 .
The asymptotic distribution of MSVr is identical to that of the Wald and LM statistics proposed by Lucas (1996, 1998). There are nuisance parameters remaining in the limiting distribution in that the Brownian motions are imperfectly correlated, which changes the limiting distribution for each M estimator selected. However, we can find estimates of the correlations and use the Gamma approximation suggested in Boswijk and Doornik (1999) to obtain asymptotic critical values.
3. Monte Carlo We conduct a simple Monte Carlo experiment using the data generating process DXt 5
S
c 2] T 0
D
0 0
Xt 21 1 et .
If c 5 0, there are no cointegrating vectors and if c . 0, there is one (trivial) cointegrating vector. Several distributions are considered for et ; normal, t with 3 degrees of freedom, mixed normal 2 , and Cauchy 3 . We set r (e) 5 ln(1 1 e ¡ e / 5) so that we use a student-t likelihood with 5 degrees of freedom for the MSV and LM tests. The Gaussian likelihood ratio test of Johansen (1988) is included for comparison and denoted GLR. The sample size is 200 and 10 000 replications are performed. In order to find critical values for the MSV and LM tests, we must find an estimate of the nuisance parameters that affect the limiting distribution. That is, we estimate the canonical correlations between F1 and F2 . Using these estimates, critical values are generated for the tests in each replication by implementing the Gamma approximation proposed in Boswijk and Doornik (1999). The rejection percentage for the tests are listed in Table 1. The MSV test based on the t distribution has good size and power properties and effectively exploits heavy-tailed errors. The Gamma approximation works very well for both the MSV and LM tests.
4. Conclusion We have developed a new test for cointegrating rank using robust tests based on multivariate M estimators. There are several advantages of the MSV test. First, the test does not depend on the ordering of the variables in the system, a feature shared with the LM test of Lucas (1998). Second, we 2
The observation comes from a normal distribution with standard deviation 2.5 with probability 0.2 and from a normal distribution with standard deviation 0.5 with probability 0.8. 3 The Cauchy distribution does not satisfy the assumptions given in the theorem but is used as an example of extreme conditions.
T. Juhl / Economics Letters 71 (2001) 149 – 154
152 Table 1 Size and size adjusted power Test
Test
c
MSV
LM
GLR
Normal
0 3 6 9 12
0.06 0.07 0.11 0.17 0.27
0.04 0.07 0.11 0.17 0.27
0.05 0.07 0.11 0.19 0.30
t 3
0 3 6 9 12
0.06 0.15 0.33 0.54 0.74
0.05 0.15 0.33 0.54 0.73
0.05 0.07 0.11 0.20 0.31
c
MSV
LM
GLR
Mix
0 3 6 9 12
0.05 0.35 0.72 0.92 0.98
0.05 0.34 0.72 0.91 0.98
0.05 0.07 0.11 0.18 0.30
Cauchy
0 3 6 9 12
0.05 0.98 0.99 1.00 1.00
0.05 0.98 0.99 1.00 1.00
0.06 0.24 0.32 0.42 0.52
use a Monte Carlo experiment to show that the size and power are quite good. Finally, the computation of the MSV test is particularly simple in that we do not impose a reduced rank condition at the estimation stage, reducing computational complexity compared to the Wald, PLR, and LM tests in Lucas (1996, 1997, 1998). Given the non-linear optimization required for some M estimators, this is highly advantageous. The proposed MSV test could be easily extended to incorporate adaptively estimated models. Again, we need only obtain one estimate of P adaptively, and consider the singular values from the standardized estimate.
Acknowledgements I thank Bruce Hansen, Roger Koenker, Donald Lien, and Andre´ Lucas for comments on earlier versions of this paper.
Appendix A. Proof of Theorem 1 We prove the theorem when the data generating process has no intercept and one is not estimated. The other cases are similar. Given the conditions in the theorem,
OScb X D ⇒SBB (s)(s)D
[Ts]
T 21 / 2
c
t ¡
t51
t
2
where Bc and B2 are correlated Brownian motions. The variance matrices associated with Bc and B2 ¡ ¡ ¡ are Scc and (a ' b' )21 a ' See a' ( b ' a' )21 . Let H 5 ( b, b' ) be normalized such that H ¡ H 5 I. It is straightforward to show that
T. Juhl / Economics Letters 71 (2001) 149 – 154
vecsT
1/2
153
d 21 ˆ M 2 P )bd →N(0, S 11 ^ C 121 Scc C 121 ). (P
and d
SE
ˆ M 2 P )b'd → ( B2 B 2¡ )21 ^ C 121 vecsT( P
DE
B2 ^ dBc .
Let the distributions above (not vectorized) be denoted as J1 and J2 respectively. We write
Pˆ M 5 P 1 (J1 :J2 )A T 1 o p (A T ) with A T 5 (T
21 / 2
(A.1)
b :T 21 b' )¡ . The roots of
¡ 1 / 2 ˆ ¡ ˆ 21 ˆ 1/2 u lI 2 P˜ P˜ u 5 u lI 2 S 11 P M S C PM S 11 u
are equivalent to the roots of ¡ 21 ˆM ˆ M S11 u Sˆ C P u lS11 2 S11 P
since S11 is positive definite in finite samples. Define ¡ 21 ˆM ˆ M S11 . Sˆ C P S( l) 5 lS11 2 S11 P ¡
Note that S( l) shares the roots of H S( l)H. Using rules for partitioned determinants gives ¡ uH ¡ S( l)Hu 5 u b ¡ S( l)b uu b ' hS( l) 2 S( l)b ( b ¡ S( l)b )21 b ¡ S( l)j b' u
(A.2)
We will consider the asymptotic distribution of r 5 Tl since the eigenvalues are normalized by T in the MSVr statistic. Note that under the null hypothesis, P 5 ab ¡ . Using this fact and the result p b ¡ S11 b → S11 from Johansen (1988), we have p
b ¡ S( l)b → 2 S11 a ¡ C1 S 21 cc C 1 aS11 .
(A.3)
Next, 1 ¡ ˆ ¡ ˆ 21 ˆ b¡ b S b 2b¡ ' S( l)b 5 r ] ' S 11 P M S C PM S 11 b. T ' 11 p
¡ From Johansen (1988), b ¡ ' S 11 b 5 Op (1), b S 11 b → S11 , and
S
E
D
¡ ¡ ¡ 21 b¡ ' S( l)b ⇒ 2 Op (1)a 1 ( B 2 B 2 )J 2 C 1 S cc C 1 aS11 .
Finally, 1 ¡ ¡ 21 ¡ ¡ ˆM ˆ M S11 b' , b' S( l)b' 5 r ] b ' S11 b' 2 b ' S11 P Sˆ C P T so that
1 ] T
¡ b' S11 b' ⇒ eB2 B 2¡ , so
(A.4)
T. Juhl / Economics Letters 71 (2001) 149 – 154
154
E
¡ 21 ¡ b¡ ' S( l)b' ⇒ r B 2 B 2 2 D 1 C 1 S cc C 1 D 1
(A.5)
follows similarly with D1 5 Op (1)a ¡ 1 (eB2 B 2¡ )J 2¡ . Inserting (A.3), (A.4), and (A.5) into (A.2) results in
E
u r B2 B 2¡ 2 D2 D3 D 2¡ u
(A.6)
21 21 ¡ 21 21 with D2 5 Op (1)a ¡ 1 eB2 dB ¡ and D3 5 C1 S 21 3 c C1 cc C 1 2 (C 1 S cc C 1 )a (a C 1 S cc C 1 a ) 21 a ¡ (C1 S cc C1 ). Following Lemma 10.1 in Johansen (1995), we can show that 21 21 21 ¡ D3 5 a' (a ¡ a' ' C 1 Scc C 1 a' )
which will eliminate the Op (1)a
E
¡
terms. Then A.6 becomes
E
E
21 ¡ 21 21 21 u r B2 B ¡ a' C 121 dBc B 2¡ u. 2 2 B 2 dBc C 1 a' (a ' C 1 Scc C 1 a' )
(A.7)
21 / 2 Pre and post multiplying A.7 by u(a ¡ (a ¡ ' See a' ) ' b' )u will not change the roots and it standardizes B2 (s) so that all the Brownian motions have unit variance. Then using the trace we have p
T
O l˜ ⇒ trSEdF F SEF F D EF dF D 21
i
1
¡ 2
2
¡ 2
2
¡ 1
(A.8)
i 5r 11
¡
21
21
where F1 5 (a ' C 1 Scc C 1 a' )
21 / 2
¡ ¡ a' C 121 Bc (s) and F2 5 (a ' See a' )21 / 2 (a ' b' )B2 (s).
h
References Boswijk, H., Doornik, J.A., 1999. Distribution approximations for cointegration tests with stationary exogenous regressors. Universiteit van Amsterdam. Johansen, S., 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12, 231–254. Johansen, S., 1995. Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press, New York. Lucas, A., 1996. Outlier robust unit root analysis. Ph.D. Thesis, Tinberen Institute, Amsterdam. Lucas, A., 1997. Cointegration testing using pseudolikelihood ratio tests. Econometric Theory 13, 149–169. Lucas, A., 1998. Inference on cointegrating ranks using LR and LM tests based on pseudo-likelihoods. Econometric reviews 17, 185–214.