TIME SERIES RESIDUALS WITH APPLICATION TO PROBABILITY DENSITY ESTIMATION

P. M. Robinson

Journal of Time Series Analysis, 1987, vol. 8, issue 3, 329-344

Abstract: Abstract. A linear stationary and invertible process yt models the second‐order properties of T observations on a discrete time series, up to finitely many unknown parameters θ. Two estimators of the residuals or innovations ɛt of yt are presented, based on a θ estimator which is root‐T consistent with respect to a wide class of ɛt distributions, such as a Gaussian estimator. One sets unobserved yt equal to their mean, the other treats yt as a circulant and may be best computed via two passes of the fast Fourier transform. The convergence of both estimators to ɛt is investigated. We apply the estimated ɛt to estimate the probability density function of ɛt. Kernel density estimators are shown to converge uniformly in probability to the true density. A new sub‐class of linear time series models is motivated.

Date: 1987
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