Tests for Cointegration Based on Canonical Correlation Analysis

Article excerpt

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 Johansen (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 performs 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 disperse and 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 investigates 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 auto-regression 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 corresponding 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 {[y.sub.t]} that has a first-order error correction representation

[Delta][y.sub.t] = -[Alpha][Beta][prime][y.sub.t-1] + [[Epsilon].sub.t], (1)

where [Alpha] and [Beta] are full-rank n x r matrices (r [less than] n) and the n-dimensional innovation {[[Epsilon].sub.t]} is iid with zero mean, positive definite covariance matrix [Omega], and finite moments up to the fourth order. It is assumed that [absolute value of [I.sub.n] - ([I.sub.n] - [Alpha][Beta][prime])z] = 0 implies that either [absolute value of z] [greater than] 1 or z = 1 and that [Mathematical Expression Omitted] is of full rank, where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are full-rank n x (n - r) matrices orthogonal to [Alpha] and [Beta]. Thus, as shown by Johansen (1992a), {[y.sub.t]} is I(1) with r cointegration relations among its elements; that is, {[Beta][prime][y.sub.t]} is I(0).

For a given data set [Mathematical Expression Omitted], let [Mathematical Expression Omitted] and [Mathematical Expression Omitted], where [Mathematical Expression Omitted] and [Mathematical Expression Omitted], and denote G[prime] = [[g. …