1. The framework considered here may seem overly restrictive with the normality and independence assumptions for the error sequence and the fact that no constant nor deterministic time trend are included. The latter could be relaxed by using different statistics than the ones presented in the next sections. Much the same conclusions regarding the behavior of the power functions as the sampling frequency is changed would remain. The present study is simply illustrative of some phenomena that occur in a more general context. The normality assumption has been relaxed in Perron (1986) and the conclusions are basically the same. Our study, however, cannot generalize to the case where additional correlation is present in the errors since the test statistics based on the sequence Δy
t} are indeed constructed for testing the null hypothesis that Δy
t is uncorrelated.
2. See also Lepage and Zeidan (1981) and Girard (1983) who analyzed some of the statistics described here in a different context.
3. A comment about the determination of the critical values is in order. In this section all the tests considered have the property that
$$[J-E(J)]/\sqrt{Var(J)}$$
tends asymptotically to a N(0,1) variable as T →221E;, under the null hypothesis. For most statistics, the asymptotic approximation is very good even for quite small values of T. Nevertheless, there may be a significant discrepancy for values of T as small as 8, and even 16, studied in this paper. The exact distributions have been tabulated in most cases for such small sample sizes. However, due to the discreteness of the exact distributions we cannot get critical values for which tests of size 0,05 can be constructed. A possible resolution of this problem would be to use randomized test procedures which would create a test of exact size 0,05. Since the main concern of this paper is the behavior of the power function as h tends to 0 with T increasing, it is not worthwhile to carry such a procedure. We therefore use the asymptotic critical values for all sample sizes. The effect on the size of the tests can be evaluated and the estimated sizes are presented with the power results.