Approximate distributions for the various serial correlograms

Abstract

Saddlepoint methods are used to approximate the joint density of the serial correlogram up to lag m. Jacobian transformations also lead to approximations for the related partial correlogram and inverse correlogram. The approximations consider non-circularly and circularly defined models in both the null and the non-null settings. The distribution theory encompasses the standard non-circularly defined correlogram computed from least-squares residuals removing arbitrary fixed regressors. Connections of the general theory to the approximations given by Daniels and by Durbin in the circular setting are indicated. The double-saddlepoint density and distribution approximations are given for the conditional distribution of the non-circular lag m serial correlation given the previous lags from order 1 to m-1. This allows for the computation of p values in conditional inference when testing that the model is AR(m-1) versus AR(m). Numerical comparisons with the tests of Daniels and of Durbin suggest that their tests based on circularity assumptions are inadequate for short non-circular series but are in close agreement with the non-circular tests for moderately long series.