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 →‐∞