An Alternative Test for Regression Coefficient Stability - King, Maxwell L. (The Review of Economics and Statistics, 1987)
Recently, Watson and Engle (1985) considered the problem of testing for a constant regression coefficient against the alternative hypothesis that the coefficient follows a stationary first-order autoregressive process. This alternative is the return to normalcy model proposed by Rosenberg (1973). Watson and Engle observe that the unknown autoregressive parameter is not identified under their null hypothesis and they suggest the use of the test procedure proposed by Davies (1977) for such situations. Davies' approach involves applying Roy's Union-Intersection Principle to the class of test statistics one gets by assuming the non-identified parameter takes a known value. Unfortunately, Watson and Engle's test statistic has no closed form and is approximated by maximisation using a grid search. Furthermore, both its finite sample and asymptotic distributions are unknown under the null hypothesis although they do provide a method of calculating a critical value whose asymptotic size can be bounded from above. In this note we suggest a different approach that helps overcome these problems. Rather than testing for zero variance in the autoregressive process as Watson and Engle suggest, we propose testing for lack of variation in the regression coefficient over time. This allows the construction of a locally best invariant (LBI) test using the results of King and Hillier (1985). Because the resultant test statistic is a ratio of quadratic forms in normal variables, standard computational techniques can be used to calculate exact and approximate critical values. A further advantage of this alternative test is that it is also LBI against the hypothesis that the coefficient follows a random walk process.