Testing for Gaussianity and Linearity of a Stationary Time Series

This is the authors' abstract. We don't add key points for this paper.

Stable autoregressive (AR) and autoregressive moving average (ARMA) processes belong to the class of stationary linear time series. A linear time series { } is Gaussian if the distribution of the independent innovations {ε( t )} is normal. Assuming that E ε( t ) = 0, some of the third‐order cumulants c xxx = Ex ( t ) x ( t + m ) x ( t + n ) will be non‐zero if the ε( t ) are not normal and E ε 3 ( t )≠O. If the relationship between { x ( t )} and {ε( t )} is non‐linear, then { x ( t )} is non‐Gaussian even if the ε( t ) are normal. This paper presents a simple estimator of the bispectrum, the Fourier transform of { c xxx ( m, n )}. This sample bispectrum is used to construct a statistic to test whether the bispectrum of { x ( t )} is non‐zero. A rejection of the null hypothesis implies a rejection of the hypothesis that { x ( t )} is Gaussian. Another test statistic is presented for testing the hypothesis that { x ( t )} is linear. The asymptotic properties of the sample bispectrum are incorporated in these test statistics. The tests are consistent as the sample size N →‐∞