Numerical distribution functions for unit root and cointegration tests

TLDR: In this article, the authors used response surface regressions based on simulation experiments to calculate distribution functions for some well-known unit root and cointegration test statistics, which can be used to calculate both asymptotic and finite sample critical values and P-values for any of the tests.

Abstract: SUMMARY This paper employs response surface regressions based on simulation experiments to calculate distribution functions for some well-known unit root and cointegration test statistics. The principal contributions of the paper are a set of data files that contain estimated response surface coefficients and a computer program for utilizing them. This program, which is freely available via the Internet, can easily be used to calculate both asymptotic and finite-sample critical values and P-values for any of the tests. Graphs of some of the tabulated distribution functions are provided. An empirical example deals with interest rates and inflation rates in Canada. Tests of the null hypothesis that a time-series process has a unit root have been widely used in recent years, as have tests of the null hypothesis that two or more integrated series are not cointegrated. The most commonly used unit root tests are based on the work of Dickey and Fuller (1979) and Said and Dickey (1984). These are known as Dickey-Fuller (DF) tests and Augmented Dickey-Fuller (ADF) tests, respectively. These tests have non-standard distributions, even asymptotically. The cointegration tests developed by Engle and Granger (1987) are closely related to DF and ADF tests, but they have different, non-standard distributions, which depend on the number of possibly cointegrated variables. Although the asymptotic theory of these unit root and cointegration tests is well developed, it is not at all easy for applied workers to calculate the marginal significance level, or P-value, associated with a given test statistic. Until a few years ago (MacKinnon, 1991), accurate critical values for cointegration tests were not available at all. In a recent paper (MacKinnon, 1994), I used simulation methods to estimate the asymptotic distributions of a large number of unit root and cointegration tests. I then obtained reasonably simple approximating equations that may be used to obtain approximate asymptotic P-values. In the present paper, I extend the results to allow for up to 12 variables, instead of six, and I correct two deficiencies of the earlier work. The first deficiency is that the approximating equations are considerably less accurate than the underlying estimated asymptotic distributions. The second deficiency is that, even though the simulation experiments provided information about the finite-sample distributions of the test statistics, the approximating equations were obtained only for the asymptotic case. The key to overcoming these two deficiencies is to use tables of response surface coefficients, from which estimated quantiles for any sample size may be calculated, instead of equations to