Estimation of time series models using residuals dependence measures

We propose new estimation methods for time series models, possibly non-causal and/or non-invertible, using serial dependence information from the characteristic function of model residuals. This allows to impose the iid or martingale difference assumptions on the model errors to identify the unknown location of the roots of the lag polynomials for ARMA models without resorting to higher order moments or distributional assumptions. We consider generalized spectral density and cumulative distribution functions to measure residuals dependence at an increasing number of lags under both assumptions and discuss robust inference to higher order dependence when only mean independence is imposed on model errors. We study the consistency and asymptotic distribution of parameter estimates and discuss efficiency when different restrictions on error dependence are used simultaneously, including serial uncorrelation. Optimal weighting of continuous moment conditions yields maximum likelihood efficiency under independence for unknown error distribution. We investigate numerical implementation and finite sample properties of the new classes of estimates. distributional assumptions on model errors, Gaussian Pseudo Maximum Likelihood (PML) estimates based on least squares are typically prescribed. The Gaussian PML estimates try in fact to match data sample autocovariances with the model implied ones, or equivalently, minimize the magnitude of residuals autocorrelations to match the zero serial correlation white noise assumption, which only under Gaussianity is equivalent to serial independence. Conditional moments based models lead to unconditional moment restrictions using the uncorrelation of errors with past information described by instrumental variables (see e.g. the survey by Ana-tolyev, 2007). These instruments are constructed with lags of observations and/or residuals, though these alternative representations of past information are not equivalent in general, for instance, when the true model is non-invertible.