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Abstract:
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This paper develops the theoretical background for a new quadratic prediction method for time series. First, we broaden the characterization of quadratic processes (i.e., those stochastic processes for which quadratic filters provide additional forecast accuracy over purely linear filters) by developing a more general theory of polyspectral factorization. New bijections between a restricted space of higher-dimensional cepstral coefficients (where the restrictions are induced by the symmetries of the polyspectra) and the auto-cumulants are derived. Second, the existence of the quadratic h-step ahead forecast filter (based on an infinite past) is derived and presented, thereby providing a theoretical framework for finite-sample quadratic prediction. This fundamental result is built upon a chassis of new factorization results for two- and four-dimensional arrays of Laurent series. Third, applications to model fitting, residual analysis, and simulation are developed; in particular, it is shown that semi-parametric nonlinear time series modeling can be accomplished by approximation of the cepstral representation of polyspectra.
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