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 (sigma (t)) is normal. Assuming that E sigma (t) = 0, some of the third order cumulants sub c xxx (m,n) = Ex(t)x(t+m)x(t+n) will be non-zero if the sigma (t) are not normal and E sigma cube(t)=0. If the relationship between x(t) and sigma (t) is non-linear, then (x(t)) is non-Gaussian even if the sigma (t) are normal. This paper presents a simple estimator of the bispectrum, the Fourier transform of (sub 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. A related test statistic is then 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 approaches infinity.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 1981
- Accession Number
- ADA107548
Entities
People
- Melvin J. Hinich
Organizations
- Virginia Tech