A MULTIVARIATE TEST OF THE NULL OFCOINTEGRATION USING COVARIATES

TED JUHL

Abstract. A multivariate test for stationarity is proposed in this paper. We de-velop Lagrange Multiplier type tests based on using additional stationary covariates.We show that the tests can be used to determine the number of cointegrating vec-tors in a system, starting with the null hypothesis of r cointegrating vectors andtesting for less than r cointegrating vectors. A Monte Carlo experiment shows thatthe new tests have more power than standard stationarity tests if the covariates arehighly correlated with the stationary part of the data.

Keywords: stationarity, cointegration, unit roots, covariates.JEL Classification: C12, C32.

1. Introduction

The literature on cointegrated time series has generally focused on testing the null

of no cointegration against the alternative of cointegration. The original proposal of

Engle and Granger (1987) and Phillips and Ouliaris (1990) are widely used. More-

over, the proposed method of Johansen (1988, 1991) also takes the null hypothesis

of no cointegration and proceeds by testing for more cointegrating vectors under the

alternative. In the proceeding cases, the distribution theory is based on some variant

of the Dickey Fuller test, another case where the null hypothesis is nonstationarity.

In the univariate case, Kwiatkowski, Phillips, Schmidt, and Shin (1992), hereafter

KPSS, developed a test where the null hypothesis is stationarity. Based on this

Address correspondence to: Ted Juhl, University of Kansas, Department of Economics, 213Summerfield Hall, Lawrence, KS 66045; email: juhl@ku.edu. Tel: 785-864-2849.

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idea, Shin (1994) proposed a test where the null hypothesis is cointegration and the

alternative is no cointegration. Xiao and Phillips (2002) developed a similar idea using

a CUSUM type test on the cointegrating regression residuals. Other tests based on

the null hypothesis of cointegration include Harris (1997), Choi and Ahn (1995), and

Snell (1999) among others.

A mulitvariate generalizations of the KPSS test is given in Nyblom and Harvey

(2000) that can be used to test the null of more cointegration against the alternative

of less cointegration. Although designed for a different purpose, the test for common

deterministic trends proposed by Bierens (2000) is also a generalization of the KPSS

test to the multivariate setting.

We suggest a test that is similar in spirit to the tests of Bierens (2000) and Nyblom

and Harvey (2000). However, we include additional stationary covariates in the model.

The motivation for including stationary covariates in the model is two-fold. First, one

can envision the use of additional data providing better estimation of the parameters

in the model, thus leading to improved power. This motivation was originally provided

in Hansen (1995) for unit root testing in a Dickey Fuller framework and extended to

the case with the null hypothesis of no cointegration by Seo (1998). Second, one may

be interested in an economic system and want to test for cointegration among a subset

of the variables in the system. Traditionally, researchers ingnore those variables that

are stationary since they do not enter into the cointegration relationship. However,

3

it is possible to view the tests in this paper as using all the variables in the complete

system when specifying the model of cointegration.

The tests proposed in this paper have a Lagrange Multiplier interpretation when the

null hypothsesis is that all variables are stationary. We find the limiting distribution

when there are r ≤ p stationary variables in the system where p is the number of

series that are not known to be stationary in advance. The limiting distribution

contains several nuisance parameters which we must estimate. We provide a Monte

Carlo experiment to explore the performance of the test when using covariates. If

the covariates are highly correlated with existing cointegration relationships, the tests

provide a large power improvement relative to tests that ignore covariates.

2. Model and Test

Consider the multivariate model

yt = µy + γt + ut

γt = γt−1 + vt

where yt and γt are p× 1 vectors. Suppose that ut and vt are i.i.d. and independent

of each other. Moreover, suppose that ut ∼ N(0, Σuu) and vt ∼ N(0, Σvv). If Σvv = 0,

then yt is a stationary vector. Another way to view the hypothesis of interest is to

examine the roots of

(1) |Σvv − θjΣuu| = 0, j = 1, . . . p.

4

If yt is stationary, all of the roots of (1) are zero. If some of the roots are nonzero,

then yt contains a stochastic trend coming from the term γt. Nyblom and Harvey

(2000) suggest testing against the alternative that θ1 = θ2 = · · · = θp = θ > 0 so that

the test has power directed at alternatives with all roots equal. In other words, the

alternative is Σvv = θΣuu which is referred to as a homogeneous alternative.

Based on this representation, Nyblom and Harvey (2000) proposed a test of sta-

tionarity against the homogeneous alternative. The test (modified for possible serial

correlation in ut and vt) is a multivariate generalization of the stationarity test pro-

posed by Kwiatkowski, Phillips, Shin, and Schmidt (1992). Bierens (2000) proposed a

test for testing for common nonlinear trends. However, the test has the same limiting

distribution if applied to a vector time series with some unit roots.

Given the power gains from using stationary covariates in testing for unit roots and

cointegration found by Elliot and Jansson (2003), Hansen (1995), and Seo (1998), we

propose using covariates in tests where the null hypothesis is cointegration (station-

arity). To this end, we consider the inclusion of a q × 1 vector stationary covariate,

say zt, to the system so that we have

yt = µy + γt + ut

γt = γt−1 + vt

zt = µz + wt.

5

We define Σuz as the covariance matrix between u and z and let Y be a T × p

matrix containing y>t as the tth row and denote Z as the T × q matrix with rows z>t .

Let Vi be the T vector containing the ith elements of the vt vector. Define Ui and Wi

similarly and let

C =

1 0 . . .

.... . . 0

1 . . . 1

.

Then we write the T (p + q)× 1 vector x = vec(Y, Z) so that

x =

CV1 + U1 + µy11T

...

CVp + Up + µyp1T

W1 + µz11T

...

Wq + µzq1T

where

µy =

µy1

µy2

...

µyp

, µz =

µz1

µz2

...

µzp

,

and 1T is a T × 1 vector of 1’s. Under the assumption of normality, we have

6

x ∼ N

µy ⊗ 1T

µz ⊗ 1T

, Ω(θ)

where

Ω(θ) =

θΣuu ⊗ CC> + Σuu ⊗ IT Σuz ⊗ IT

Σzu ⊗ IT Σzz ⊗ IT

.

We formulate a Lagrange Multiplier test for null hypothesis of Σvv = 0 against the

homogeneous alternative hypothesis Σvv = θΣuu in the following theorem.

Theorem 2.1. The LM test of Σvv = 0 against the homogeneous alternative Σvv =

θΣuu rejects for large values of

trace

(ΘT−2

T∑t=1

[t∑

i=1

(ξi − ξ)

][t∑

i=1

(ξi − ξ)>])

where ξi = (y>i , z>i )> and

Θ = Σ−1

Σyy 0

0 0

Σ−1

with

Σ =1

T

T∑t=1

(ξt − ξ)(ξt − ξ)>

and Σyy is the upper p× p block of Σ.

The test has similar structure to the KPSS stationarity test and the tests of Bierens

(2000) and Nyblom and Harvey (2000). However, there are two distinct differences.

First, we include the stationary variable zt in the vector ξ since zt is part of the

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system. In addition, the matrix Θ is not just an estimate of the variance of ξ if it

were stationary. This is due to the fact that zt is already known to be stationary so

that we do not direct any power to test whether it is stationary. It is this feature

that gives rise to a different limiting distribution which we give in the next section.

3. Cointegration and Asymptotic Distribution Theory

The test in the last section was generated under the null hypothesis of Σvv = 0

and under joint normality of ut and vt as well as i.i.d. data. We find the limiting

distribution under more general conditions in this section. Moreover, we also treat

the case of cointegration which requires a modification of the existing test.

Recall the model

yt = µy + γt + ut

γt = γt−1 + vt

where var(vt) = Σvv. Suppose that Σvv is not equal to zero but that it has rank

p − r. Then there exists a p × r matrix β such that β>Σvvβ = 0, and an or-

thogonal complement to β is the (p − r) × p matrix β⊥ such that β>β⊥ = 0 and

β>⊥Σvvβ⊥ 6= 0. This implies that the β>yt does not have stochastic trends so that

β contains the r cointegrating vectors. The proposed LM test is the trace of the

matrix(ΘT−2

∑Tt=1

[∑ti=1(ξi − ξ)

] [∑ti=1(ξi − ξ)>

]). Note that the trace is equal to

the sum of the eigenvalues. Given the presence of cointegration, not all directions

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are stationary (β⊥) so that some of the eigenvalues may not converge. It is shown in

the appendix that the smallest r + q eigenvalues converge in distribution while the

largest p− r eigenvalues diverge. In fact, it is obvious that Θ is not full rank so that

the q smallest eigenvalues must be zero. Hence, a natural test for the null hypothesis

of r cointegrating vectors is to sum the r middle eigenvalues, starting with q + 1 and

summing to the (q + r)th eigenvalue. This will be our test statistic for the null of r

cointegrating vectors with the special case of the trace resulting if the null is r = p.

We call the test the Covariate Cointegration Test with

CCT =

r+q∑i=q+1

λi

where λi are the ordered eigenvalues.

Before we proceed with the distribution theory, it is necessary to revisit the as-

sumption of normality and i.i.d. data. Clearly economic data is typically not i.i.d. so

we want to allow for possible serial correlation in ut, vt, and wt. To account for serial

correlation, we consider the long run variance estimator

Σ(bT ) = Γ(0) +T−1∑τ=1

k

(τ

bT

) (Γ(τ) + Γ(τ)>

)

where

Γ(τ) =1

T

T∑t=τ+1

(ξt − ξ)(ξt−τ − ξ)>,

9

bT is a bandwidth parameter, and k(·) is a kernel function. We then define

Θ(bT ) = Σ−1(bT )

Σyy(bT ) 0

0 0

Σ−1(bT ).

Now we construct the matrix

Θ(bT )T−2

T∑t=1

[t∑

i=1

(ξi − ξ)

] [t∑

i=1

(ξi − ξ)>]

upon which we will base our statistics.

Given the allowance for serial correlation, we provide the assumptions required to

obtain the asymptotic distribution theory.

Assumption 1. Let the vector st be defined as st = (u>t , v>t , w>t )>. Let st be a linear

process such that

(a) st = C(L)εt =∑∞

j=0 Cjεt−j,∑∞

j=0 ja‖Cj‖ < ∞, for some a > 1.

(b) εt is iid with mean zero, variance matrix Σε > 0 and finite fourth order cumulants.

(c) The long run covariance matrix of st is positive semi-defininte and is given by

C(1)ΣεC(1) =

Σuu Σuv Σuw

Σvu Σvv Σvw

Σwu Σwv Σww

where Σuu and Σww are full rank matrices.

Assumption 2. The kernel function k(·) : R → [−1, 1] is a twice continuously

differentiable even function with:

10

(a) k(0) = 1, k′(0) = 0, k′′(0) 6= 0

(b) k(x) = 0, |x| ≥ 1, with lim|x|→1k(x)

(1−|x|)2 .

The conditions in Assumption 1 are similar to Assumption EC given in Phillips

(1995), with the exception that we must allow the possibility that Σvv is less than full

rank. That is, we allow for cointegration in yt so that β>Σvvβ = 0. The assumption

of a linear process is required because we must estimate several nuisance parameters

in the limiting distribution. We will return to this point after the statement of the

theorem.

Assumption 2 is similar to Assumption KL of Phillips (1995). This assumption

again facilitates the estimation of several nuisance parameters in the limiting distri-

bution. Moreover, the use of this assumption allows us to employ results of Phillips

(1991) so that we can retain the same distribution theory as if the data were i.i.d.

We give the limiting distribution of the test in the following theorem.

Theorem 3.1. Suppose that Assumptions 1 and 2 hold and that bT = T c where

c ∈ (0, 0.5). Let λ1 ≤ λ2 ≤ · · · ≤ λp+q be the ordered eigenvalues of

Θ(bT )T−2

T∑t=1

[t∑

i=1

(ξi − ξ)

][t∑

i=1

(ξi − ξ)>]

.

Under the null hypothesis of r cointegrating vectors,

CCT =

r+q∑i=q+1

λid→ trace

[D(F11 − F12F

−122 F>

12)]

11

D =

(I −R1R>1 )−2 −(I −R1R

>1 )−2R1

−R>1 (I −R1R

>1 )−2 R>

1 (I −R1R>1 )−2R1

F11 =

∫ 1

0

B1(u)

B2(u)

(B1(u)> B2(u)>

)du

F12 =

∫ 1

0

B1(u)

B2(u)

[∫ u

0

(W3(r)−

∫ 1

0

W3(s)ds

)>]du

F22 =

∫ 1

0

[∫ u

0

(W3(r)−

∫ 1

0

W3(s)ds

)] [∫ u

0

(W3(r)−

∫ 1

0

W3(s)ds

)>]du

W (u) is a p + q vector Brownian Motion partitioned into vectors of length r, q and

p− r denoted by W1(u), W2(u), and W3(u). W (u) has covariance matrix given by

Ir R1 R3

R>1 Iq R2

R>3 R>

2 Ip−r

where R1, R2, and R3 are the long run correlation matrices between β>ut and zt,

β>⊥vt and zt, and β>ut and β>⊥vt respectively. B1(u) and B2(u) are the first order

vector Brownian bridges constructed from W1(u) and W2(u). If a deterministic linear

trend is estimated, then ξ is replaced with the estimated trend and B1(u) and B2(u)

are second order Brownian bridges.

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There are several interesting points to note. First, the presence of R1, R2, and

R3 complicate the limiting distribution of the test. However, if each of these terms

is zero, the distribution simplifies to the distributions found in Bierens (2000) and

Nyblom and Harvey (2000). There are several possible ways to proceed when nui-

sance parameters remain in the limiting distribution. Hansen (1995) and Seo (1998)

simulate critical values for a variety of values of the nuisance parameters and provide

tables for several possible cases. In the univariate case, this is simple but such a

procedure becomes extremely complicated in a multivariate context, especially in our

case where there are three matrices of nuisance parameters.

The method we pursue here is to simulate critical values for each realization of the

statistic. This is the same method as followed in the above references but we do not

attempt to anticipate all of the possible cases. We illustrate the procedure in a series

of steps.

• Estimate β using the canonical correlation estimator β from Harris (1997).

• Calculate the long run correlation between β>yt and zt as an estimate of R1.

• Calculate the long run correlation between β>⊥∆yt and zt as an estimate of

R2.

• Calculate the long run correlation between β>⊥∆yt and β>yt as an estimate of

R3.

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• Simulate the p+q dimensional Brownian Motion W (u) with covariance matrix

as described in Theorem 3.1. Calculate test and repeat many times to calculate

critical values.

It is worth noting why this is a reasonable way to estimate nuisance parameters.

For example, in the calculation of our estimator of R2 we use

β>⊥∆yt = β>⊥(ut − ut−1 + vt)

= β⊥(ut − ut−1 + vt) + Op(T−1).

Given the conditions in Assumptions (1) and (2), Phillips (1995) shows that the

long run covariance estimator for (ut − ut−1) and zt is zero even if zt and ut are

correlated. Hence, the estimated long run correlation matrix will converge to the

long run correlation matrix of β>⊥vt and zt, which is R2.

For purposes of simulations in this paper, we simulate Brownian Motion with 1000

increments and simulate the test statistic 1000 times in order to calculate critical

values. All calculations and simulations take about one second for each time the

model is estimated. As a by product of this procedure, it is simple to obtain estimated

p-values. We illustrate the size and power of the procedure in the next section.

4. Monte Carlo

We explore and power through the use of the following model with p = 2 and q = 1,

so that there is one covariate, zt.

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yt = µy + δyt + γt + ut

γt = γt−1 + vt

zt = µz + δzt + wt.

There are four main cases and several subcases. Without loss of generality, we set

µy = δy = µz = δz = 0 in all cases. In Case 1, the null hypothesis is that both

series in yt are stationary and we estimate µy and µz. Following Jansson (2004), we

generate data with long run unit variances but with nonzero covariances. That is, we

set

ut

gt

wt

= (I5 − aI5)T∑

i=0

(aI5)iLiht

where ht has covariance matrix

1 0 0 0 r1

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

r1 0 0 0 1

so that r1 is the correlation between zt and the first element of ut. Note that the

parameter a controls the amount of serial correlation in the data. We define vt as

15

follows

vt =

0.02× d 0

0 0

gt.

Note that if d = 0, yt is stationary, and every vector is a cointegrating vector. How-

ever, if d 6= 0, then there is one cointegrating vector and one stochastic trend in the

system. We allow the parameter a to take values 0, 0.50, 0.70 and we allow d to take

values 0, 0.2, 0.4, 0.6, 0.8, 0.10.

The bandwidth parameter for the test is a truncated version of the data driven

method of Andrews (1991). As noted in Xiao and Phillips (2002), the use of a data

driven bandwidth renders some tests powerless since such a bandwidth will increase

at too fast of a rate. We employ the Andrews AR(1) univariate approximation, but

we truncate the AR(1) parameters at 0.3, 0.4, 0.5, and 0.6. We report only the results

for the truncation at 0.3 as the other results are similar yet less powerful for all tests

considered.

We compare the covariate test with the test of Nyblom and Harvey (2000), denoted

NH, with both tests using the truncated bandwidth selection procedure. The results

appear in Table 1. In general both tests have the same size properties. Notice that

both tests are subject to the well known size problems as a increases. In general, the

covariate test is slighly less powerful than NH when the covariates are not correlated

with the stationary part of the data (R21 = 0). As R1 increases, the covariate test has

more power than the test not employing covariates.

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For Case 2, the only change in the data generating process is

vt =

0.02× d 0

0 1

gt

so that there is one stochastic trend when d = 0, but two stochastic trends when

d 6= 0. The null hypothesis is one cointegrating vector and one stochastic trend.

The results appear in Table 2. In this case, although there are still gains from using

covariates, the power gains are not nearly as large.

For Case 3, the data generating process is the same as in Case 1. However, here

we estimate the linear trend through the parameters δy and δz. In this case, the size

distortions are largest for both tests. However, notice that as R21 increases, the size

distortions decrease for the covariate test. The power gains are very large when using

covariates for all configurations of a. The experiment for Case 4 is similar to that

for Case 2 with the exception that a trend is estimated and the results are similar to

Case 2. Simulations not reported here show that R2 and R3 have very little effect on

size and power for both tests.

5. Conclusion

We have proposed using covariates when testing the null of cointegration in a

multivariate system. The test is based on a Lagrange Multiplier test when the null is

that the entire system is stationary. When the null hypothesis is that there are some

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stochastic trends in the system, the test is calculated using eigenvalues and functions

of the estimated long run covariance matrices.

We show that the new tests can have dramatic power increases relative to tests

that do not make use of the additional stationary data. However, in order to reap the

benefits of these gains, we must deal with a fairly complicated limiting distribution.

We show that it is possible to estimate the nuisance parameters and simulate the

critical values for the test statistic based on the estimates.

Appendix A

Proof of Theorem 2.1: Let M = IT − 1T1T 1>T . If θ is zero, then the MLE for

Ω(0) =

Σuu Σuz

Σzu Σzz

⊗ IT

is

Ω(0) =

Y >MYT

Y >MZT

Z>MYT

Z>MZT

⊗ IT

= Σ⊗ IT .

Denote ζ = vec(Ω, µy, µz) and let ζ be the estimated value under θ = 0. Let

log L(θ, ζ) be the log likelihood. Then

log L(θ, ζ) ∼

x−

µy ⊗ IT

µz ⊗ IT

>

Ω(θ)−1

x−

µy ⊗ IT

µz ⊗ IT

18

and

∂ log L(θ, ζ)

∂θ

∣∣∣∣θ=0,ζ=ζ

= vec[x(Ip+q ⊗M)x>]>(Ω(0)−1 ⊗ Ω(0)−1

)vec

Σyy ⊗ CC> 0Tp×Tq

0Tq×Tp 0Tq×Tq

= trace

Ω(0)−1

Σyy ⊗ CC> 0

0 0

Ω(0)−1(I ⊗M)xx>(I ⊗M)

= trace[(Θ⊗ CC>)(I ⊗M)xx>(I ⊗M)

]

= x>(I ⊗M)(Θ⊗ CC>)(I ⊗M)x

= vec(MY, MZ)>(Θ⊗ CC>)vec(MY, MZ)

= trace[Θ(MY,MZ)>CC>(MY, MZ)]

= trace

(Θ

T∑t=1

[t∑

i=1

(ξi − ξ)

][t∑

i=1

(ξi − ξ)>])

,

where ξi = (y>i , z>i )>.

Appendix B

We prove Theorem 3.1 in this appendix. We first note that under Assumption 1,

we have the standard result

(2)

T−1/2∑[Tu]

t=1 β>ut

T−1/2∑[Tu]

t=1 (zt − µz)

T−1/2∑[Tu]

t=1 β>⊥vt

d→

Q1/21 0 0

0 Q1/22 0

0 0 Q1/23

W (u)

19

where W (u) is defined in Theorem 3.1. Let

D2 =

Q−1/21 β> 0

0 Q−1/22

b−1/2T T−1Q

−1/23 β>⊥ 0

Lemma B.1.

(D2T

−2

T∑t=1

[t∑

i=1

(ξi − ξ)

][t∑

i=1

(ξi − ξ)>]

D>2

)d→

F11 b−1/2T F12

b−1/2T F>

12 b−1T F22

where F11, F12, and F22 are defined in Theorem 3.1.

Proof: The result follows from (2) and the Continuous Mapping Theorem.

Lemma B.2.

D2Σ(bT )D>2 =

A11 A12 A13

A>12 A22 A23

A>13 A>

23 A33

20

where

A11 = Q−1/21 β>Σyy(bT )βQ

−1/21

A12 = Q−1/21 β>Σyz(bT )Q

−1/22

A13 = b−1/2T T−1Q

−1/21 β>Σyy(bT )β⊥Q

−1/23

A22 = Q−1/22 Σzz(bT )Q

−1/22

A23 = b−1/2T T−1Q

−1/22 Σzy(bT )β⊥Q

−1/23

A33 = b−1T T−2Q

−1/23 β>⊥Σyy(bT )β⊥Q

−1/23 .

Then under Assumptions 1 and 2,

A11d→ I

A12d→ R1

A13d→ b

1/2T T−1Op(1)

A22d→ I

A23d→ b

1/2T T−1Op(1)

A33d→ T−1

∫ 1

0

(W3(u)−

∫W3

)(W3(u)−

∫W3

)>du

Proof: The results for A11, A12, and A22 are standard. The results for A13 and A23

follow from equation (A.12) of Phillips (1991) while A33 follows from (A.11) of Phillips

(1991).

21

Proof of Theorem 3.1:

Note that the roots of

(3)

∣∣∣∣∣λI − Θ(bT )T−2

T∑t=1

[t∑

i=1

(ξi − ξ)

][t∑

i=1

(ξi − ξ)>]∣∣∣∣∣ = 0

are the same as the roots of

∣∣∣∣∣∣λ

(D2T

−2

T∑t=1

[t∑

i=1

(ξi − ξ)

][t∑

i=1

(ξi − ξ)>]

D>2

)−1

− (D>2 )−1Θ(bT )D−1

2

∣∣∣∣∣∣= 0

Consider the term (D>2 )−1Θ(bT )D−1

2 . Using Lemma B.2 and the formula for parti-

tioned inverses, we have

(D>2 )−1Θ(bT )D−1

2 =(D2Σ(bT )D>

2

)−1

D2

Σyy(bT ) 0

0 0

D>

2

(D2Σ(bT )D>

2

)−1

d→

(I −R1R>1 )−2 −(I −R1R

>1 )−2R1

−R>1 (I −R1R

>1 )−2 R>

1 (I −R1R>1 )−2R1

b

1/2T Op(1)

b1/2T Op(1) TS−1

11

=

D b1/2T Op(1)

b1/2T Op(1) TS−1

11

where S11 =∫ 1

0

(W3(u)− ∫

W3

) (W3(u)− ∫

W3

)>du and D is defined in Theorem

3.1. Then the roots of (3) converge in distribution to the roots of

∣∣∣∣∣∣∣∣

λ(F11 − F12F−122 F>

12)−1 λb

1/2T Op(1)

λb1/2T Op(1) λbT (F22 − F>

12F−111 F12)

−1

−

D b1/2T Op(1)

b1/2T Op(1) TS−1

11

∣∣∣∣∣∣∣∣= 0.

22

Consider the lower right block. Since bT

T→ 0, any of the p − r roots of the term

|λbT (F22 − F>12F

−111 F12)

−1 − TS−111 | = 0 must diverge. Now consider the case where λ

does not diverge. We see that the remaining roots are the roots of

|λ(F11 − F12F−122 F>

12)−1 −D| = 0

which are the roots of

|λIr+q −D(F11 − F12F−122 F>

12)| = 0.

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24

Table 1.

Size and Power: Null=All Stationary

d NH CCT

R21 = 0 R2

1 = 0.50 R21 = 0.70 R2

1 = 0.90

0.00 0.049 0.048 0.050 0.042 0.048

0.02 0.165 0.161 0.316 0.462 0.687

a=0.0 0.04 0.423 0.426 0.616 0.725 0.883

0.06 0.603 0.597 0.731 0.834 0.947

0.08 0.690 0.690 0.827 0.905 0.953

0.10 0.763 0.762 0.870 0.917 0.967

0.00 0.099 0.100 0.084 0.069 0.072

0.02 0.262 0.255 0.390 0.462 0.663

a=0.5 0.04 0.480 0.464 0.595 0.706 0.830

0.06 0.663 0.616 0.746 0.803 0.885

0.08 0.703 0.688 0.793 0.830 0.891

0.10 0.741 0.736 0.825 0.854 0.911

0.00 0.197 0.199 0.184 0.160 0.140

0.02 0.404 0.385 0.495 0.550 0.728

a=0.7 0.04 0.619 0.584 0.719 0.795 0.879

0.06 0.736 0.732 0.801 0.856 0.902

0.08 0.771 0.748 0.845 0.888 0.909

0.10 0.825 0.809 0.890 0.904 0.923

25

Table 2.

Size and Power: Null=One Stochastic Trend

d NH CCT

R21 = 0 R2

1 = 0.50 R21 = 0.70 R2

1 = 0.90

0.00 0.045 0.036 0.032 0.026 0.017

0.02 0.129 0.098 0.139 0.123 0.144

a=0.0 0.04 0.306 0.232 0.312 0.364 0.353

0.06 0.476 0.389 0.458 0.484 0.506

0.08 0.527 0.418 0.565 0.575 0.628

0.10 0.616 0.513 0.628 0.647 0.650

0.00 0.054 0.054 0.043 0.033 0.047

0.02 0.153 0.143 0.156 0.200 0.200

a=0.5 0.04 0.325 0.311 0.389 0.414 0.435

0.06 0.458 0.430 0.495 0.523 0.565

0.08 0.530 0.515 0.585 0.604 0.662

0.10 0.608 0.585 0.666 0.666 0.694

0.00 0.100 0.092 0.092 0.089 0.105

0.02 0.255 0.235 0.229 0.293 0.315

a=0.7 0.04 0.412 0.394 0.442 0.492 0.512

0.06 0.530 0.512 0.571 0.631 0.669

0.08 0.626 0.594 0.654 0.686 0.689

0.10 0.666 0.644 0.691 0.705 0.726

26

Table 3.

Size and Power for Trend Model: Null=All Stationary

d NH CCT

R21 = 0 R2

1 = 0.50 R21 = 0.70 R2

1 = 0.90

0.00 0.047 0.046 0.052 0.045 0.043

0.02 0.085 0.074 0.129 0.207 0.439

a=0.0 0.04 0.191 0.177 0.389 0.552 0.785

0.06 0.365 0.343 0.583 0.742 0.905

0.08 0.519 0.498 0.716 0.821 0.961

0.10 0.602 0.583 0.802 0.894 0.969

0.00 0.120 0.097 0.097 0.096 0.092

0.02 0.138 0.153 0.224 0.272 0.477

a=0.5 0.04 0.310 0.272 0.420 0.582 0.758

0.06 0.444 0.406 0.583 0.717 0.856

0.08 0.552 0.521 0.721 0.789 0.881

0.10 0.657 0.639 0.789 0.852 0.928

0.00 0.266 0.269 0.236 0.199 0.184

0.02 0.359 0.330 0.374 0.400 0.623

a=0.7 0.04 0.513 0.444 0.602 0.675 0.838

0.06 0.602 0.576 0.711 0.834 0.913

0.08 0.731 0.702 0.808 0.862 0.929

0.10 0.777 0.778 0.861 0.907 0.948

27

Table 4.

Size and Power for Trend Model: Null=One Stochastic Trend

d NH CCT

R21 = 0 R2

1 = 0.50 R21 = 0.70 R2

1 = 0.90

0.00 0.041 0.036 0.034 0.031 0.035

0.02 0.087 0.078 0.081 0.076 0.079

a=0.0 0.04 0.196 0.162 0.170 0.212 0.171

0.06 0.334 0.272 0.327 0.351 0.313

0.08 0.430 0.334 0.408 0.444 0.447

0.10 0.545 0.454 0.539 0.538 0.534

0.00 0.083 0.073 0.065 0.062 0.065

0.02 0.093 0.094 0.127 0.128 0.123

a=0.5 0.04 0.206 0.187 0.267 0.282 0.271

0.06 0.359 0.349 0.421 0.424 0.369

0.08 0.458 0.429 0.502 0.515 0.486

0.10 0.548 0.515 0.608 0.625 0.585

0.00 0.147 0.138 0.131 0.133 0.132

0.02 0.187 0.185 0.205 0.251 0.199

a=0.7 0.04 0.346 0.311 0.360 0.402 0.335

0.06 0.473 0.450 502 0.530 0.502

0.08 0.557 0.533 0.569 0.587 0.599

0.10 0.612 0.586 0.637 0.651 0.658