Identification of Coefficients in a Quadratic Moving Average Process Using the Generalized Method of Moments

The output of a causal, stable, time-invariant nonlinear filter can be approximately represented by the linear and quadratic terms of a finite parameter Volterra series expansion. We call this representation the “quadratic nonlinear MA model” since it is the logical extension of the usual linear MA process. Where the actual generating mechanism for the data is fairly smooth, this quadratic MA model should provide a better approximation to the true dynamics than the twostate threshold autoregression and Markov switching models usually considered. As with linear MA processes, the nonlinear MA model coefficients can be estimated via least squares fitting, but it is essential to begin with a reasonably parsimonious model identification and non-arbitrary preliminary estimates for the parameters. In linear ARMA modeling these are derived from the sample correlogram and the sample partial correlogram, but these tools are confounded by nonlinearity in the generating mechanism. Here we obtain analytic expressions for the second and third order moments – the autocovariances and third order cumulants – of a quadratic MA process driven by i.i.d. symmetric innovations. These expressions allow us to identify the significant coefficients in the process by using GMM to obtain preliminary coefficient estimates and their concomitant estimated standard errors. The utility of the method for specifying nonlinear time series models is illustrated using artificially generated data.