Linearity Test

Abstract

A stationary Gaussian time series has the following properties: (i) the residual 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 [52] definition that the model is linear if the linear predictor is optimal. This assumption seems to be the minimum requirement. This means that the residual sequence e_t fulfils the following conditions:

$$ {\text{E}}\left( {e_t \left| {F_{t - 1} } \right.} \right) = 0,{\text{E}}\left( {e_t^2 |F_{t - 1} } \right) = \sigma ^2 , $$

((5.1))