Testing structural hypotheses on cointegration relations with small samples

SU ZHOU [*] This study examines the finite-sample bias of Johansen's [1991] likelihood ratio tests for structural hypotheses on cointegration relations among economic variables through the Monte Carlo experiments. It is found that the Johansen tests with small samples are biased toward rejecting the null hypotheses more often than what asymptotic theory suggests, even after the test statistics are adjusted by Sims's correction. A bootstrap method for obtaining problem-specific critical values for the tests is proposed It is shown that using the bootstrap procedure may substantially reduce the small-sample bias. An empirical application of the procedure is demonstrated. (JEL C12, C22) I. INTRODUCTION Since Nelson and Plosser [1982] discovered that most macroeconomic variables have a nonstationary time series structure, cointegration analysis has been substantially developed and widely applied in empirical studies in various areas of economics. Cointegration analysis offers a natural way to test for the existence of long-run relationships among nonstationary economic variables. A set of variables, [X.sub. t], is said to be cointegrated if these variables are individually nonstationary, but at least one linear combination of them, [z.sub. t] = [beta]'[X.sub. t], exists that is stationary. The vector [beta] is referred to as the cointegration vector. If [X.sub. t] is a cointegrated system, we may say that the variables in [X.sub. t] do not drift "too far" apart and there exists one or more long-run equilibrium relationships among these variables. Cointegration analysis mainly consists of three segments: Tests for the rank of cointegration, estimation of cointegration vectors, and tests for structural hypotheses suggested by economic theory. One of the most popular methods of cointegration analysis is the multivariate maximum-likelihood approach of Johansen [1988, 1991]. The Johansen approach yields trace statistics and maximal eigenvalue statistics for identifying the number of cointegration vectors and provides consistent maximum-likelihood estimates of the coefficients of cointegration vectors. In addition, Johansen [1991] and Johansen and Juselius [1990, 1992] develop the likelihood ratio tests for structural hypotheses, that is, the hypotheses of linear restrictions on the cointegration relations, with a multivariate error correction model. Johansen and Juselius [1990, 1992] have tabulated the asymptotic critical values for both the maximal eigenvalue and the trace tests. Also, Johansen [1991] has derived the asymptotic distributions of the test statistics for structural hypotheses. Their critical values from the asymptotic distributions have been extensively employed in the cointegration studies. Unfortunately, empirical studies of long-run (cointegration) relationships are limited by finite sample sizes. It could be misleading to draw conclusions based on a small sample study while using the asymptotic critical values, which are appropriate only for large samples. Cheung and Lai [1993] investigate the accuracy of Johansen and Juselius's asymptotic approximations by computing the finite-sample distributions of the maximal eigenvalue and the trace statistics; they find that the asymptotic critical values are more liberal than the finite-sample critical values. That is, using the asymptotic critical values, one would reject the null of no cointegration too often when the sample size of the data used is rather small. There have been a number of other studies examining the small sample performance of several popular estimation and testing procedures designed for cointegrated systems, including the Johansen approach. [1] However, as far as I know, none of the existing studies has systematically investigated the small sample performance of any test for structural hypotheses. The present study intends to fill this gap and thus expand this body of literature. There is an obvious reason that researchers have overlooked the small sample problem of the tests for structural hypotheses on cointegration relations. …