tests for cointegration based on canonical correlation analysis

Tests for Cointegration Based on Canonical Correlation AnalysisAuthor(s): Ronald Bewley and Minxian YangSource: Journal of the American Statistical Association, Vol. 90, No. 431 (Sep., 1995), pp. 990-996Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2291335 .

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Tests for Cointegration Based on Canonical

Correlation Analysis

Ronald BEWLEY and Minxian YANG*

Critical values are provided for four new tests for cointegration based on the canonical correlations and variates of a development of the Box-Tiao procedure. It is found that in finite samples the power of three of these tests, unlike the power of Johansen's and Engle and Yoo's tests, is highly robust to the correlation between the disturbances in the cointegrating relationships and those generating the common trends. The proposed tests perform well against these alternatives, but neither set of tests dominates over the entire parameter space.

KEY WORDS: Maximum likelihood; Monte Carlo; Unit roots; Vector autoregression.

1. INTRODUCTION

For I(1) processes, the Box-Tiao procedure (Box and Tiao 1977) for estimating cointegrating vectors produces canonical variates that asymptotically are nonstationary for variates corresponding to unit canonical correlations and are stationary otherwise. The eigenvectors associated with small canonical correlations converge to cointegrating vectors in the sense of Engle and Granger (1987), but Box and Tiao did not provide a decision rule for classifying these canonical vectors in small samples.

In a Monte Carlo comparison of the Box-Tiao and Jo- hansen (1988) estimators of the cointegrating parameter based on a first-order bivariate model, Bewley, Orden, Yang, and Fisher (1994) found that the Box-Tiao estimator per- forms well when compared to Johansen's maximum likelihood estimator (MLE). In particular, they found that the distribution of the Box-Tiao estimator is relatively less dis- perse arid exhibits relatively less kurtosis when the disturbances generating the cointegrating equation are not strongly correlated with the disturbances generating the common trend, but that the reverse is true when this correlation is large. This article proposes methods of testing for cointegration to accompany a development of the Box-Tiao estimation procedure for vector autoregressive models and in- vestigates the power of these tests compared to the Johansen alternatives.

Bossaerts (1988) suggested that the Box-Tiao canonical variates could be directly tested for the presence of unit roots using the methods and critical values provided by Dickey and Fuller (1979) and Fuller (1976). Bewley and Orden (1994) conjectured that the critical values of these tests depend on the number of time series.

We show that the null hypothesis of no cointegration can be tested against the alternative of greater than or equal to one cointegrating vector in a first-order vector autoregression by applying one or more of four tests based on a development of the Box-Tiao estimator. Two of the tests are direct applications of the t test and the autocorrelation coefficient test, proposed by Dickey and Fuller (1979), to the most stationary canonical variate (i.e., the variate cor-

* Ronald Bewley is Professor of Econometrics and Minxian Yang is an Australian Research Council Research Associate, School of Economics, University of New South Wales, Sydney 2052, Australia. The authors thank Graham Elliott, David Giles, S0ren Johansen, David Orden, and the referees and editors for helpful comments.

responding to the smallest canonical correlation), whereas the other two tests, based on a minimal root and a trace statistic, have direct parallels with the Johansen ( 1988, 1991) maximal root and trace tests and the Stock and Watson ( 1988) test based on a principal component analysis. Each of the proposed statistics is shown to be asymptotically distributed as functionals of standard Brownian motions, and the testing procedure is extended to allow for multiple cointegrating vectors and higher-order vector autoregressions.

Critical values are provided for each of these tests for models with up to and including six variables. Bossaerts' conjecture-that Fuller's tables can be used for the autocorrelation coefficient test-is not substantiated by these results.

2. TEST STATISTICS Consider an n-dimensional vector process { Yt } that has

a first-order error correction representation AYt = -af'yt-i + et, (1)

where a and fi are full-rank n X r matrices (r < n) and the n-dimensional innovation { et } is iid with zero mean, positive definite covariance matrix Q, and finite moments up to the fourth order. It is assumed that In - (In - a')z I = 0 implies that either I z > 1 or z = 1 and that a' lfl, is of full rank, where a 1 and Il are full-rank n X (n - r) matrices orthogonal to a and ,B. Thus, as shown by Johansen ( 1992a), { Yt } is I( 1 ) with r cointegration relations among its elements; that is, {fl'Yt} is I(O).

For a given data set {Yt}T, let gt = Yt - -

and ht = Yt-I -Y-1, where y = y71: IYt and --1 = T`1 ? Yt-i, and denote G = [g1, . , gT], H' = [hl, . .. , hT]. Following Box and Tiao (1977), a levels canonical correlation analysis (LCCA), as opposed to the Johansen (1988, 1991) canonical correlation analysis that relates levels to first differences of Yt, solves

[XiG'G - G'H(HH)- H'G]ji = 0 (2)

for pairs [ ji, i = 1, . . ., n, where the eigenvalues are ranked such that Xn ? ? X*A. The first r eigenvectors form a cointegration space estimate P- = [h, , Pr] and are subject to the normalizations T-, AI(G'G)ji = 1 for i = 1, ...,r, and T-2pA,(G'G)pi = lfor i = r + 1,. n. Johansen ( 1992b) has shown that under (1), fI =

? 1995 American Statistical Association Journal of the American Statistical Association

September 1995, Vol. 90, No. 431, Theory and Methods

990



Bewley and Yang: Cointegration Based on Canonical Correlations 991

+ Op( T-1), with Q converging in probability to a full-rank constant r X r matrix Q.

Given Box and Tiao's original motivation for the canonical correlation analysis between Yt and Yt- -, it is natural to consider whether there is a unit root in the first canonical variate { I Y, } as a test for cointegration (r > 0). This is analogous to the ordinary least squares (OLS) residual-based tests ad- vocated by Engle and Granger (1987), Engle and Yoo (1987), and Phillips and Ouliaris (1990). Bossaerts (1988) proposed using the Za, statistic of Phillips (1987) and the critical values of Fuller (1976, p. 371) to test the LCCA variates for cointegration. On the other hand, Bewley and Orden ( 1994) suggested that the Dickey-Fuller t test could be used in sequence on the ordered canonical variates. As no asymptotic theory in this context was offered by Bossaerts or Bewley and Orden, these testing approaches remain to be justified. In addition to proposing two additional statistics, this section provides an asymptotic analysis for these LCCA- based tests.

Let the null hypothesis be that there is no cointegration in {yt4; that is, HPo: r = 0 (a = 8 = 0). Under the null, { Yt } in ( I ) is a vector random walk with a full-rank long- run covariance matrix Q, and no linear combination of its elements can result in a stationary process. It will be shown that under this null, the usual unit root tests (i.e., Dickey- Fuller's t and Phillips' Z) can be performed on the LCCA variates, but the asymptotic distributions and, consequently, the critical values of these statistics depend on the number of variables.

Lemma 1. Suppose that { Yt } is generated by ( 1 ) and that eigenpairs [,ui, pi ] solve

Ai BgB' ds -Q fPi = 0, i = 1, ... ., n, (3)

where BQ is an n-dimensional demeaned Brownian motion with covariance Q, and the eigenvalues ,A1, *--, - ,ln are distinct and ranked in descending order, with the corresponding eigenvectors normalized such that pi (fo BQBQ ds)pi = 1. Then under the null hypothesis HPO, the pairs [T( 1 X- i), Pi] from (2) converge in distribution to eigenpairs [Ai, pi].

It is worth noting that BQ can be written as BQ = Q 1/2(W -fo W ds), with W as a standard n-dimensional Brownian motion. Hence the pairs [Ai, Q '/2pi ]in (3) will depend only on some functionals of the standard n-dimensional Brownian motion (see also Ooms and Van Dijk 1992). Thus the statistic T( 1 - X1) can be used to test for cointegration. Note that in this case the asymptotic distribution of TI 1 -i ) is the same as that of Phillips and Ouliaris' P, statistic.

Let ct = I'lgt and dt = iilht; then ct = dt + et, where et =

I 'e, Performing usual unit root tests on the first canonical variate { Ct } T involves regressing ct on ct-I, including a constant. This can be done equivalently by regressing ct on dt with the constant suppressed. Thus Dickey-Fuller's t statistic is given by t,, = (~Tdt2) 12( -1 )/S, where p = (~Tdh1- (~Tdtct) and = (T -2)-:T(ct - pdt)2. Itis clear that both tp and p3 are invariant to the normalization of the ei-

genvector, because p'I appears on both numerator and de- nominator in the same fashion.

Lemma 2. Under the conditions stated in Lemma 1,

(a) T( -1) - p fQo/2[ W dW - W(l) fo W' ds] Q /2pl, and

(b) tp ==/ W dW' - W(1) fol W' ds] Q1/2 Pi,

where W is the standard n-dimensional Brownian motion and both T( - I) and 4p are asymptotically free of nuisance parameters.

In this context, the statistic T( I- ) is the equivalent of Phillips' Za, statistic used by Bossaerts. Clearly, the asymptotic distributions of T(j - 1) and tp depend on the dimension of Yt, and the critical values in the Fuller table are inappropriate in this context. Table 1 reports critical values of T(1 - X1), Ty2-r(1- i), T(p - 1), and 4p for six sample sizes and five dimensions, computed from 100,000 replications of independent vector random walks. Pseudonormal random numbers were generated by the algorithm of Kinderman and Ramage ( 1976), and the Jacobi algorithm of Press, Teukolsky, Vetterling, and Flannery (1992, pp. 456-462) was used to solve eigen problems.

Let the alternative hypothesis be HPo+: r > 0 (r < n) for considering the consistency of the tests.

Lemma 3. Under the alternative HPO+, the tests are consistent with T(' - 1) and T( I - ) diverging at rate T and with tp diverging at T1'2.

When HPo is rejected by data, it is natural to consider testing r = 1 against r > 1 by performing the unit root tests on the second canonical variate. But it can be shown that the asymptotic distributions of such statistics depend on nuisance parameters. Thus certain modifications have to be made to use the results in Lemmas 1-3. In the case of testing for r against r + 1, it is clear from ( 1 ) that aclAy, = acl et, which implies that HPo still applies to { a I Yt }, except that the dimension is n - r rather than n. Although a, is un- known, an estimate &?l of a, can be constructed by solving a a, = 0, where a is an estimate of a obtained by regressing gt- ht on W'ht without an intercept.

Lemma 4. Suppose that data { Yt } o are generated by ( 1), where r ? 1. Let f - = [ P1 ... , Pr] be the LCCA estimate of ,8, which can be written as #BQ + Op(T-'), with Q converging to a full-rank matrix Q. Then a^' =-(fiHHO)-1fi'H'(G - H) = Qt-'a' + op 1 () -p Q-'a' holds.

For a given sample size Tand the estimate a', the columns of a' can always be uniquely rearranged by a given rule such that the first r columns are linearly independent; that is, there is an n X n matrix P, which is a product of some elementary matrices, such that a'P = [&, &I with the r X r submatrix &'" being invertible. Thus for a given c&, al can be constructed as a"' = [&2(c1 )-1,In-r]P', which is of full rank and bounded in T and satisfies &al = 0. It is noted that a' is defined as In when r = 0. As a ̂ aQ'1, P converges to P which is again a product of some elementary matrices and satisfies [a'1, aX'2] = Qtac'P. Hence al -p al = [a2(a1 )1 An-rI_]P' holds. It is useful to decompose a1 into



992 Journal of the American Statistical Association, September 1995

Table 1. Critical Values of the Cointegration Tests Based on Canonical Correlations Analysis Procedure

tp T( -1) T(1-A) TI(1 - 1)

n T 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%

2 75 4.26 3.65 3.34 27.85 21.64 18.60 45.03 36.50 32.08 54.66 44.41 39.29 100 4.21 3.62 3.32 28.75 22.03 18.90 48.70 38.74 33.72 58.20 46.76 41.12 150 4.16 3.59 3.30 29.46 22.53 19.32 52.34 41.01 35.55 62.10 49.33 43.13 200 4.14 3.58 3.29 29.86 22.77 19.42 54.61 42.32 36.38 64.22 50.42 43.99 250 4.13 3.57 3.28 30.21 22.93 19.61 56.15 43.11 37.02 65.51 51.17 44.58 500 4.10 3.56 3.28 30.86 23.21 19.75 59.05 44.69 38.15 68.58 53.08 46.01

3 75 4.92 4.30 4.00 35.78 29.09 25.87 54.05 46.36 42.29 79.71 68.61 63.07 100 4.85 4.25 3.96 37.07 29.32 26.51 59.89 50.34 45.39 86.05 73.26 66.89 150 4.75 4.19 3.92 38.06 30.68 27.14 65.90 54.39 48.73 92.55 77.99 70.86 200 4.70 4.17 3.89 38.96 31.05 27.44 69.66 56.63 50.47 96.60 80.92 73.06 250 4.69 4.16 3.89 39.34 31.40 27.67 71.63 58.24 51.70 98.70 82.37 74.33 500 4.66 4.13 3.86 40.13 31.94 27.98 76.18 61.11 53.81 103.77 85.79 77.00

4 75 5.49 4.90 4.59 42.62 35.89 32.59 60.62 54.09 50.44 108.74 96.97 90.68 100 5.41 4.81 4.52 44.36 37.11 33.59 68.56 59.87 55.26 118.46 104.25 97.14 150 5.29 4.74 4.46 46.34 38.43 34.64 77.81 66.32 60.64 128.97 112.53 104.28 200 5.26 4.71 4.43 47.20 39.03 35.10 82.54 69.84 63.47 134.41 116.71 107.69 250 5.21 4.68 4.42 47.96 39.52 35.50 86.04 72.14 65.36 138.14 119.64 110.01 500 5.14 4.64 4.38 49.27 40.33 36.12 93.10 76.74 69.11 146.44 125.45 115.22

5 75 6.05 5.43 5.13 49.05 42.16 38.85 65.58 60.05 56.95 140.61 127.99 121.40 100 5.94 5.33 5.05 51.51 44.02 40.30 75.93 67.99 63.70 154.41 139.15 131.35 150 5.79 5.23 4.96 53.97 45.90 41.97 87.98 77.12 71.46 169.95 151.80 142.60 200 5.71 5.18 4.92 55.42 46.78 42.68 94.91 81.90 75.50 178.57 158.99 148.71 250 5.67 5.12 4.89 56.27 47.39 43.13 98.94 85.03 78.13 182.81 162.39 152.02 500 5.61 5.10 4.85 58.23 48.59 44.20 108.95 91.91 83.87 195.33 171.98 160.12

6 75 6.58 5.96 5.65 54.75 48.21 44.81 69.19 64.87 62.17 175.42 161.95 155.06 100 6.43 5.83 5.53 57.87 50.68 46.91 81.76 74.95 71.00 193.64 177.78 169.49 150 6.22 5.70 5.42 61.24 53.08 48.99 96.73 86.64 81.20 215.52 196.01 185.92 200 6.16 5.64 5.37 63.11 54.14 49.95 105.42 92.95 86.65 227.20 205.89 194.73 250 6.11 5.60 5.33 64.36 55.19 50.76 111.35 97.38 90.45 236.04 212.29 200.40 500 6.00 5.52 5.28 66.39 56.76 52.09 123.23 106.41 98.17 251.48 225.47 212.41

NOTE: Minus signs have been omitted for simplicity; n is the number of variables, and T is the number of observations.

1 =all+ aJ, (4)

where I -p Inr and J -p 0. Now let the null hypothesis be HPr: there are r cointegra-

tion relations, and let the alternative be HPr+: there are r+ cointegration relations (r+ > r). The statistics T( 1- TW( 1 - Xi), T(, - I), and tp are now extracted from the data set { a^' g, a`1 h, } 'by solving the n - r-dimensional eigen problem

[XiGrGr - rGHr(H'rHr)H-HrGr]G i = 0, (5)

where Gr = Ga& and Hr = Ha^. It is shown, in the sense of Lemma 5, that the statistics drawn from the data set { ci' Yt } are asymptotically equivalent to those from { a Yt } 0 under the null HPr. As the asymptotic distributions involve only the limits of the quantities listed in Lemma 5, Lemmas 1 and 2 still apply, although the dimension of the Brownian motion becomes n - r.

Lemma 5. Denote Ur = Gr-Hr. Under the null HPr,

(a) GG{Gr ='{ 'ai (St - S)(St - S)'a }i + o,(T2), (b ) H'H, = '{2ITaI(St_, - S_,)(St_l - S_,ya I

+ op(T2), (C) U'rUr = i'{ ai(e- )(St- -)'aj}i+?o(T),and

(d) U' Hr = F{ j2TaI (et -)(St-I -S-1)'a }I + op(T)

hold, where I p Infr, St = 2 ei, and S and S_I are sample means of St and St-,.

Under the alternative HPr?, the true cointegrating rank is r+ > r, so that the dimensions of a and f. are understood to be n X r+. Similar to Lemma 4, it can be shown that a. n X r, satisfies a^ = aQ+ + op( 1), with Q+ converging to a full-rank r+ X r matrx Q+. Consequently, a&, n X (n - r), will converge to (aQ+), and can be expressed as a, = fa + fl1b, where fi and IN, are full-rank matrices of n X r+ and n X (n - r+) and a and b converge to constant matrices a and b of r, X (n - r) and (n - r+) X (n - r); that is, &? -h a1 = a + 8l1b. As r+ > r and a, is of full rank by construction, there is an (n - r) X (r+ - r) matrix 6 such that 68 = 0 and A' is of full rank. As a and b are convergent, so is 5. Hence LCCA applied to { a Yt } o produces the first eigenvector p+ that can be written as p+ -jq+ + Op( T), where q+ is a convergent (r+ - r) vector. Therefore, the results in Lemma 3 still apply.

Consider the case where {Y,} is generated by a VAR(/ + 1) model AYt = -afl'yt_I + 21lIjAYt-j + et, where at, /5, and et are the same as defined in (1) and the Ili's are n X n coefficient matrices. It is assumed that I In -(I, - a')z - I (Ijz(I - z) = 0 implies that either jzI > 1 or z = 1 and that the matrices ol/S _Land a' (In - Ij)8L are invertible, a condition ensuring that { Yt } is I(1) and {18'Yt } is I(O) (Johansen 1992a).




To test for the hypotheses HPr against HPr+, the LCCA is first applied to { YI } Ty to produce the ordered eigenvectors [mI, . j. , m"], of which the first r constitute an estimate , for,i (, 0 when r = 0). One way to produce a supercon- sistent estimate of / is to apply the method described by Bewley and Orden ( 1994); that is, purging Yt and Yt-I of the influence of short-run dynamics by regressing each on [1, Ay,-,, * *.. , Ayt-1] to produce residuals gt and ht, respectively. These residual series are then used in (2) to produce eigenvectors Pi, which constitute the columns of i.

Using results of Yang ( 1994), it can be shown that , = + Op,( T- I) still holds for higher-order processes with Q converging to a full-rank constant matrix Q, although the proof is tedious. Then regressing AYt on [1, -Wyt-1, AYt-i, *., Ayt-1] produces the residuals ft and the estimated coefficient matrices [c, a, It, .I. ., II], and a & and i? are computed in the same fashion as described in the paragraph following Lemma 4 (i.e., a = = In, when r = 0). Further, define Hr = al [hi, ... ., hT] and G' = al1g1, . . ., T], recalling that ht are the residuals from a regression of yt-I on [1, AYt -,

Ayt-l and it = ht + DfI, with D = /l[&I(In - HIIj) il&i. On replacing Grby Gr, the statistics T( 1 -X ), Tj ln-r( I- X), T A - 1), and tp are extracted from the solution of(5) and data set { & t'L, & ht } by regressing

p+:lton p'+&'? ht, without a constant, as in the first-order case. It can be shown that their asymptotic distributions under the null HPr are the same as those described in Lemmas 1 and 2, with dimensions n -r. Under the alternative HPr+, it can be shown that &xl (In - lj)B converges in probability to a constant matrix C. The invertibility ofthis matrix C is required to show the consistency of the tests.

3. SIMULATION STUDY Power comparisons of the four tests based on the LCCA

procedure, Engie and Granger's test, and Johansen's maximal root and trace tests are reported using a simulation experiment based on a bivariate model and one sample size, T = 75. Results from a more limited experimentation with a five-variable data-generating process are also presented, and additional results are available from the authors on request.

The data-generating process used by Banerjee, Dolado, Hendry, and Smith (1986), Bewley et al. (1994), and others has been adopted for the bivariate experiment:

Yt - bxt = zt, zt = 4zt- + ezt

xt + ayt = wt, wt = wt-1 + ewt

with var(edt) = 1, var(eat) = m2, and cov(esa, ep) = so,. In all of the reported experiments, a = 1, b = 1, a = 1,

and T = 75. Because the LCCA and Johansen's procedures are systems methods, variations in the values assigned to a and b have no impact on any of the Johansen or LCCA statistics. The power of the Engle-Granger test is highly sensitive to changes in a and b but, given the normalization problems associated with this test, alterative values of these parameters were not considered. Furthermore, only the power of the Engie-Granger test was found to be sensitive to variations in v. Additional results for T= 150 are not reported, but they do not alter the conclusions reached from the results based on the smaller sample.

The empirical size and size-adjusted power comparison reported in columns 1-5 of Table 2 is based on five values of the correlation 6 { ?.5, ?.25, .0 } and three values of the autocorrelation parameter, 4 { 1, .9, .8 }; each experiment is based on 10,000 replications and a nominal 5% size. The critical values for Johansen's tests were taken from Johansen and Juselius ( 1990, table A2) and those for the LCCA tests from Table 1, in each case using the number of decimal places reported. The Engle-Granger critical value was derived, from the work of MacKinnon ( 1991, p. 275) and rounded to two decimal places.

It can be noted that there is a slight overrejection of the null for the two Johansen tests using the published asymptotic critical values, but this small-sample effect can be reduced by using the correction factor suggested by Reinsel and Ahn (1992) and evaluated by Cheung and Lai (1993). There is no major size distortion in any of the other tests for 0 # 0. Nevertheless, empirical critical values were used with the same set of ez, and ewt for each value of X, to size adjust the power comparison.

The power of three of the proposed tests is highly robust to variations in the disturbance correlation, 0. But the LCCA trace test, the Engle-Granger test, and the Johansen maximal root and trace tests critically depend on this parameter. Be- cause of this dependence, additional experiments were con- ducted to produce power curves over a wider range of values of 6 {?.95, +.9, +.8, +.65, +.5, +35, +.25, ?.1, .0} and4 - .7; these results are presented in Figure 1.

It can be shown that Johansen's maximal root can be expressed asymptotically in terms of 0 and 0 using Xmax = (1 - k)/[2 -(1 + k)62] so that as 161 1, the power of Johansen's maximal root test, -Tln( 1 - Xm,), and hence the trace test, approaches unity. This behavior is reflected in Figure 1. Intuitively, as 161 1-* , a certain linear transfor- mation of the bivariate VAR produces one equation with a residual variance that approaches zero. Thus the Johansen canonical correlation approaches unity for the cointegrating equation. On the other hand, the LCCA minimal root can be expressed asymptotically as Xmin = 02, which partly ex- plains why the power of that test, as well as the related T( p - 1 ) test, is insensitive to variations in 0 in finite samples.

For approximately 1 0 1 < .5, each of the LCCA tests has greater power than either of the Johansen tests or the Engle- Granger test, whereas the Johansen test dominates for 161 > .5. The LCCA trace test is as powerful as the minimal root test for 0 near zero, but its power rapidly dissipates as 16 - 1.

Intuitively, the LCCA trace test is affected by large 6 for the same reason that Johansen's tests are affected. The larger root in the LCCA approaches 1 as 6 -- 1, even in small samples, and the LCCA trace test depends on both roots. There is seemingly little difference between the power of the T(i - ) and T( 1-X1) tests in samples of 75 observations, and these two tests have about 7 percentage points more power than the tp test over the entire range of 0 for 0 = .7.

From Table 2 and Figure 1, it can be noted that the Engle- Granger test monotonically increases in power with 6. A second power curve could have been defined for Figure 1 using the alternative normalization for the OLS Engle-




Granger regression, and this curve would be symmetric with that shown about 0 = 0.

It also follows from these results that the ranking of the power of the different tests depends on 0. To further inves- tigate this effect of 0 on power, additional experiments were computed but are not reported. It was found that for both Johansen and LCCA tests, the trace variant is more powerful than the single root alternative when 0 is large and the ranking is reversed for small 0. Both pairs of power curves cross near 0 = .75. Interestingly, Johansen's maximal root test and the LCCA minimal root test have almost identical power over the entire range of k when 0 = .5. It can be concluded from the results of these experiments that three of the LCCA tests perform well when 0 is small but that the reverse is true when 0 is large.

The model used for investigating the power of multivariate systems (n > 2) is that used for the bivariate model except that xt, a, b, wt, and ewt are each redefined to be vectors of dimension n - 1. The covariance matrix for the higher- dimensional model is specified to be

ezt var J

- [ 1 Oot'[(l - 3)/(n - [0O4[(1 - 6)/(n - )] 1/2 T2[I- bt'/(n- 1)]J

where t is a vector of n - 1 ones and I 6 I < 1. This definition specializes to that used in the bivariate design when n = 2

0.8 -

0.6

0.4

0.2

0 , 0 ',(1 -(1) *(3) ,K(4) --5) -(6) --(7, -0.75 -0.5 -0.25 0 0.25 0.5 0.75

6

Figure 1. Comparison of Power in the Bivariate Model With k = .7. - * - * -, Johansen's trace statistic; * *, Johansen's maximal root statistic; X X i, p; - x - x - x -, T(p -1); - - -, T2(l1-A,); y~ T(1 - A,); - - -, Engle-Granger (Dickey-Fuller) statistic.

and a = 0. The parameter 0 can be interpreted as the correlation between the disturbance generating the cointegrating equation and the average of the disturbances generating the common trends. The parameter 3 controls the correlations among the disturbances generating the common trends. Al-

Table 2. Size and Size-Corrected Power Comparison for a Nominal 5% Test

Bivariate model (n = 2) Multivariate model (n = 5)

0 0 Statistic -.50 -.25 .0 .25 .50 -.50 -.25 .0 .25 .50

0 = 1.0 0 = 1.0

Trace .0588 .0567 .0550 .0551 .0541 .1022 .0969 .1015 .0993 .0967 Max. root .0649 .0609 .0582 .0598 .0594 .0867 .0984 .0862 .0850 .0843 Engle-Yoo .0483 .0478 .0441 .0501 .0483 .0548 .0562 .0538 .0478 .0527 tp .0495 .0502 .0467 .0474 .0471 .0501 .0528 .0566 .0500 .0485 T(- 1) .0480 .0502 .0470 .0497 .0477 .0499 .0493 .0565 .0479 .0495 T (1 - X1) .0484 .0508 .0490 .0494 .0487 .0485 .0482 .0546 .0489 .0497 T (1-X,) .0519 .0497 .0496 .0472 .0471 .0526 .0503 .0522 .0512 .0510

,0= .9 = .7

Trace .1442 .1153 .1070 .1181 .1464 .2184 .1455 .1316 .1522 .2215 Max. root .1219 .0953 .0886 .0933 .1275 .2068 .1308 .1175 .1358 .2124 Engle-Yoo .0683 .0901 .1049 .1136 .1222 .1294 .1264 .1238 .1192 .1132 tp .1010 .1074 .1124 .1144 .1054 .1860 .1919 .1970 .1925 .1867 T(b -1) .1244 .1298 .1349 .1269 .1224 .2058 .2153 .2148 .2084 .2082 T (1 - X1) .1283 .1357 .1375 .1344 .1246 .2078 .2091 .2149 .2110 .2041 T(l- Xj) .1506 .1751 .1788 .1772 .1523 .2391 .2874 .2921 .2931 .2416

= .8 = .5

Trace .4222 .3083 .2829 .3194 .4394 .5696 .3998 .3610 .4155 .5789 Max. root .4092 .2940 .2649 .2953 .4436 .6975 .4836 .4279 .4878 .7149 Engle-Yoo .1953 .2680 .3283 .3407 .3858 .3520 .3529 .3489 .3442 .3173 tp .3329 .3466 .3612 .3514 .3428 .6596 .6848 .6788 .6794 .6648 T(p-1) .4059 .4150 .4297 .4184 .4023 .7108 .7307 .7206 .7111 .7132 T (1 - X1) .4169 .4231 .4378 .4318 .4121 .7031 .7222 .7217 .7109 .6982 T 2(1 - Xj) .3981 .4543 .4761 .4736 .4131 .4717 .5557 .5704 .5679 .4856

NOTE: Trace and Max. root refer to Johansen's trace and maximal root statistics; Engle-Yoo refers to either the bivariate Engle-Granger (1987) test or the multivariate Engle-Yoo (1987) generalization. The other four statistics are defined in the text.




though this design is not completely general, it does place the focus of attention on 0, a parameter judged to play an important role in the bivariate design. The choices of parameter values include n = 5, a' = ( -1 1 -1 1 ), and b' = t. Not surprisingly, 6 was found to have no impact on the power of the Johansen or the LCCA tests. As a result, only experiments pertaining to 6 = 0 are reported in Table 2, and, with additional results on 0, power curves relating to 6 = 0 and

= .5 are presented in Figure 2. The results for the five-equation model are broadly in line

with those for the bivariate case, but two interesting features emerge. First, although the T( 1 - X ) test has approximately a constant differential in power over the t, test, and a similar power to the TG3 - 1) test in the bivariate case, the power of T( 1- ) falls off for large I 0 1 such that its power is less than that for the t, test in the extreme cases. Second, the power of the Engle-Yoo test is dominated in the case of a $* 0 by all of the other tests considered for the entire range of 0; however, it should be stressed that the power of the Engle-Yoo test critically depends on the values of a, b, and a. Other experiments could be constructed that would in- dicate that the Engle-Yoo test is more powerful than the system alternatives, but in practice, one would rarely have sufficient a priori information to ensure that the Engle-Yoo test would indeed be more powerful.

4. CONCLUSIONS

This article was motivated by the desirable small-sample properties of the Box and Tiao estimator reported by Bewley et al. (1994). To complete the modeling strategy, methods for testing for cointegration needed to be developed and their properties investigated.

Four statistics for testing the null of no cointegration using canonical correlations and variates estimated within the LCCA procedure for VAR( 1) models have been introduced here. Two tests are direct applications of the Dickey-Fuller t test and autocorrelation coefficient test to the canonical variate associated with the smallest root; that is, the most stationary canonical variate. A third statistic is a transfor- mation of the minimal root, T( 1 - X,), which has an obvious correspondence to Johansen's maximal root test. The fourth is a trace statistic that corresponds to Johansen's trace statistic.

Each proposed test is shown to be asymptotically distributed as functionals of standard Brownian motions, and critical values have been tabulated for up to six variables. The power properties of each statistic are investigated in a Monte Carlo experiment using a bivariate and a five-variable data- generating process.

On the basis of the simulation experiment, the use of the LCCA trace statistic is not recommended, because of the dependence of its power on the correlation between the disturbances. But each of the other three proposed tests perform well compared to the Johansen trace and maximal root tests and the Engle-Granger, or Engle-Yoo, test, except when the disturbances generating the cointegrating equation are highly correlated with those generating the common trends. At such extremes, the power of the Johansen tests approach unity as one linear combination of the variables has a disturbance

0.8

0.6 _

0.4

0.2

o , , 1(t) ~ ~~-(Z)(3) *(4) t5) -(6) --(7) -0.75 -0.5 -0.25 0 0.25 0.5 0.75

0 Figure 2. Comparison of Power in the Five-Variable Model With X = .5.

- * - * -, Johansen's trace statistic; - * *, Johansen's maximal root statistic; X X i, p; - x - x - x -, T(p -1); - - -, T2;(l - i); - , T(l - A,); - - -, Engle-Granger (Dickey-Fuller) statistic.

variance that approaches zero. It is not difficult to find empirical cases where such correlations are low, and so there is potential for the LCCA tests to play a role not only in estimating the cointegrating parameters, but also in testing for the rank of the cointegrating matrix.

Given that the simulation experiment is limited to first- order models and a null hypothesis of no cointegration, no comparative evidence on the small-sample properties in extended situations (i.e., for higher cointegration rank or higher-order lags) is available. But some reduction in per- formance might be expected for the tests based on the LCCA estimators, due to the multiple estimation stages involved in such tests.

Of the tests proposed in this article, the test based on the smallest canonical correlation was found to perform well and is marginally simpler to compute. But there is very little difference between that test and one based on the Dickey- Fuller autocorrelation coefficient test using the associated canonical variate, and the latter is slightly more robust to variations in certain parameters. Both of these tests dominate the Dickey-Fuller t test performed on the Box-Tiao canonical variate.

APPENDIX: PROOFS OF LEMMAS

Proof of Lemma I

Denote ut = t- , where e = T-l2Tet, and U' = [ul, ... , UT].

Then, under the null, the relation G = H + U holds, which in turn implies that G'H(H'H)-1H'G = G'G - U'U + U'H(H'H)-1H'U. Therefore, (2) is equivalent to

{T(1 - Xj)[T-2G'G] - [T-1U'U]

+ [T-1U'H(H'H)-1H'U] } Pi = 0. (A.1)

Corollary (2.2) of Phillips and Durlauf (1986), or Donsker's invariance principle discussed by Billingsley (1968, p. 68), indicates that T-1/2g[TS] ;~ BQ(S), S E [0, 1]. Further, T-2(G'G)

=J f BQBQ ds and U'H(H'H)-'H'U = Op(1) by lemma 3.1 of




Phillips and Durlauf, and T-'(U'U) -*~p Q by Khinchin's weak law of law numbers. Applying these limits to (A. 1 ) leads to the desired results.

Proof of Lemma 2

Following the results of Lemma 1 and the normalization on eigenvectors, T-2 Tdt = T2p 1(H'H)P1 p'l(f1 BnBn ds)p1 = 1. Lemma 3.1 of Phillips and Durlauf implies that T`2 Tdte,

- T1PI (H'U)P p I 12f1l W dW - W(1) f4l W' ds]Q 12p and proves (a). Then (b) follows because 52 = T`I(c-pt)2 - T lTe+ Op(T-1) = T- (U'U) p P', UPI =ju1 . The statistic's independence of nuisance parameters comes from the fact that u 1 and Q l/2p, are only functionals of the standard n-dimensional Brownian motion W.

Proof of Lemma 3

Under HPO+, the first eigenvector can be written as P= I- + Op( T-), where q -*p q, a constant r-dimensional vector. It follows that T-' Td -*,, py(O) and T 2Tc1d1 p y( 1 ), where zy(j) - E(q'a'y1y_' aq), actually y(O) = 1 by normalization. This gives P )p PI = y( 1 )f/y(O) with I pI I < 1 and, consequently, T( p-1) - T(P1- 1) + T(&p-1) = T( - )+ OpO( T 2) diverges at rate T. Similarly, s2 = (T- 2) 1 (c - td1)2 _ y(0) -y( 1 )2/y(O), implying that t, diverges at rate T1'2. Further, noting that X1 asymptotically becomes the square of the autocorrelation between the first canonical variate and its lag, which under (9) is strictly less than 1, T( 1 - X1 ) also diverges at rate T, as the same argument for T(A - 1) applies.

Proof of Lemma 4

Note that G - H = -Hf,a' + U, where the notation is the same as used in the proof of Lemma 1. Then ,'H'(G - H) = -Q'('WH#t)a' + Q'(,H'U) + Op(T-')H'(U - HIa'). As Yt-i is independent of et, T-1 (WH'U) = op( 1) holds according to Markov's weak law of large numbers. Lemma 3.1 of Phillips and Durlauf (1986) implies that H'U and H'HK are of Op(T). Thus ,fH'(G - H) = -Q'(/B'H'H,8)a' + op(T). Because T-'(WHIWM)

p Ir by the normalization rule, it follows that a' - a - [(,l'HU'O Q'('H'O/-)Q - Ir] Q-'a' + op(1) = op(p).

Proof of Lemma 5

According to Johansen ( 1992a), the solution to (1) can be expressed as yt = . ?('I ?f ')-','Iyo + 6L (al.8?) aIl St + vt, where vt = 2;C Cit-j, with Ci exponentially decreasing in j. Then, using (4), aI gt = I'al (St - S) + i'al (vt - v) + J'a'gt, which together with the lemma 3.1 of Phillips and Durlauf implies that G' Gr

=-2 tTalgtgA = I 2 'a' (St - ;)(St - S)'a } I + op(T2). The same argument also proves (b). Then (c) is proven by noting that aIut = & (gt - ht) = I'al(et - e) - J'a'a,8vt-1 and U'Ur - 2 'T& utut&a. Finally, the expressions for al' ut and al ht, whose expression is similar to that of aI gt, lead to (d).

[Received September 1993. Revised October 1994.]

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tests for cointegration based on canonical correlation analysis

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