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TESTING FOR GAUSSIANITY AND LINEARITY OF A STATIONARY TIME SERIES
Author(s) -
Hinich Melvin J.
Publication year - 1982
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.1982.tb00339.x
Subject(s) - bispectrum , mathematics , autoregressive model , test statistic , estimator , series (stratigraphy) , statistics , gaussian , statistical hypothesis testing , spectral density , paleontology , physics , quantum mechanics , biology
. 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 →‐∞