TESTING FOR GAUSSIANITY AND LINEARITY OF A STATIONARY TIME SERIES

Abstract

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 {cxxx(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→‐∞