Computing moments of ratios of quadratic forms in normal variables

Abstract

The accuracy and speed of numerical methods for computing the moments of a ratio of quadratic forms in normal variables is examined, with particular application to the sample autocorrelation function. Methods based on a saddlepoint approximation are demonstrated to be not only superior to existing approximations, but are numerically reliable and virtually as accurate as the method suitable for exact computations, while taking only a fraction of the time to compute. The new method also maintains its accuracy for time series models near the nonstationary border, which is of significant interest for unit-root inference and also a case for which first-order mean and variance approximations break down. As a wide variety of test statistics and their power functions arising in econometric models are expressible in the general form considered, the method should prove very useful for data analysis and model building.