ignificance test may then be performed. a more convenient alternative test statistic is given by the ratio of the variances, J, G J2N(0,2) Although the variance estimator a% is based on the differences of every other observation, alternative variance estimators may be obtained by using the differences of every qth observation. Suppose that we obtain nq 1 observations Xo, X1 ng, where q is any integer greater than 1. Defin the estimators (X-X-1)=(Xn-X (8a) q一q )2 J(q)≡0(q)-62J(q)≡ oi(q) (8d) The specification test may then be performed using Theorem 1.5 Theorem 1. Under the null hypothesis H, the asymptotic distributions of Jaq and(a are given by VnqJ(q)思M(0,2(q-1)) (9a) Vng ga N(o, 2(q-1)) (9b) Two further refinements of the statistics Ja and result in more desirable nite-sample properties. The first is to use overlapping qth differences of X, in estimating the variances by defining the following estimator of o 2 a2 ( q (X-X-q-)2 (10) This differs from the estimator ab(q) since this sum contains ng-q+1 terms, whereas the estimator ab(q contains only n terms. By using over lapping gth increments, we obtain a more efficient estimator and hence a Note that if(@a)2 is used to estimate o then the standard -test of /,-0 will yield inferences identical nose obtained from th esponding test of ,=0 for the ratio, since S Proofs of all the theorems are given in the Appendix