Testing of linearity in weak sense for time series based on the bispectrum

A stationary Gaussian time series has the following properties: (i) the innovation series of the moving average representation is a sequence of independent (and Gaussian) series and, (ii) the best predictor, i.e. the conditional expectation of the observation according to the past is linear. Both properties lead to the notion of the linearity of a time series. We follow Hannan's (1986) definition that the model is linear if the linear predictor is optimal. This assumption seems to be the minimum requirement. This notion of linearity will be referred to linearity in a weak sense. In view of our definition of weak linearity, testing the linearity is equivalent to checking the optimality of the linear predictor (Terdik and Math, 1998). The alternative is that the optimal predictor is of a quadratic type. The test is based on an additive property of the bispectrum of the innovation process which characterizes our hypothesis. The distribution of the test statistics is determined from the asymptotic distribution of the estimated bispectrum. The smoothed biperiodogram is used for the bispectrum estimation. The additive property of the bispectrum requires a particular set of frequencies and these will be used in the construction of the test statistics. Some example are included and the procedure is illustrated with simulated data used in the paper of Barnett at al (1997).