TESTING FOR GAUSSIANITY AND LINEARITY OF A STATIONARY TIME SERIES

Abstract. 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 →‐∞