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.

Maxwell L. King, Thomas S. Shively, Locally Optimal Testing When a Nuisance Parameter is Present Only Under the Alternative, The Review of Economics and Statistics, Vol. 75, No. 1 (Feb., 1993), pp. 1-7

This paper examines testing for AR(1) disturbances against MA(1) disturbances in the linear regression model. A Monte Carlo experiment compares the small-sample properties of the Cox test, some linearized Cox tests, and an approximate point optimal test, as well as a Lagrange multiplier test of AR (1) disturbances against ARM A (1,1) disturbances. The main findings are that the true sizes of the asymptotic non-nested tests can differ considerably from their nominal sizes, the Lagrange multiplier test's sizes are reasonably accurate and the point optimal test is generally more powerful than the other tests when appropriate critical values are used. When sizes are controlled at an arbitrary value of the AR (1) parameter, the relative power of the Cox test is increased substantially.