The approximate distribution of partial serial correlation coefficients calculated from residuals from regression on Fourier series

Abstract

SUMMARY Approximations are found to the joint distributions of noncircular and circular partial serial correlation coefficients calculated from residuals from regression on Fourier series. Results for coefficients calculated from deviations from the true mean and from the sample mean are obtained as special cases. It is shown that when the observations aro independent the partial coefficients are approximately independently distributed in beta distributions that are the same for all odd-order coefficients and the same for all even-order coefficients. The approximations are of third-order accuracy in the sense that the error is of order n-312. They were obtained by the technique developed in another paper (Durbin, 1980). Daniels (1956) obtained the approximate distribution of the successive circular serial correlation coefficients and partial serial correlation coefficients, with and without mean correction, for the case of normally distributed observations generated by a circular autoregression. He also obtained an approximation to the distribution of the lag one noncircular statistic with and without mean correction. The approximations were obtained by means of the saddlepoint approximation method and are of third-order accuracy in the sense that the error committed is of order n-3/2. In this paper this work is extended to include noncircular and circular statistics calculated from residuals from regression on Fourier series. It is hoped that the results will serve as the basis for developing tests of serial correlation of successively higher order calculated from the residuals from least squares regression on slowly changing regressors. The approximations are derived by a technique of Durbin (1980) for deriving approximations for the densities of sufficient estimators which for this problem is technically simpler than the saddlepoint method. First we consider the appropriate choice of definition of the lag one noncircular coefficient. For a set of values zl, ... , zn a number of alternatives are open to us including