Parameter estimation for all-pass time series models

Abstract

All-pass models are autoregressive-moving average models in which the roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa. They generate uncorrelated (white noise) time series, but these series are not independent in the non-Gaussian case. All-pass models can be used to simplify the process of fitting noncausal autoregressive and noninvertible moving average models and, in this dissertation, these procedures are used in the deconvolution of a simulated water gun seismogram and to fit stock market trading volume data. Because all-pass series are uncorrelated, estimation methods based on Gaussian likelihood, least-squares, or related second-order moment techniques cannot identify all-pass models. Consequently, least absolute deviations, maximum likelihood, and rank techniques are used to obtain parameter estimates. The rank estimator considered was first proposed by Louis Jaeckel for estimating linear regression parameters. Jaeckel's estimator minimizes the sum of model residuals weighted by a function of residual rank. The asymptotic properties of the three types of estimators are examined and their behavior is studied for finite samples via simulation. For all-pass series with finite variance, maximum likelihood and rank estimation are studied in-depth, and some extensions of previous results for least absolute deviations estimation are given in the Appendix. In the infinite variance case, least absolute deviations estimation is considered when the noise distribution is in the domain of attraction of a non-Gaussian stable distribution, and maximum likelihood estimation is considered when the noise distribution is non-Gaussian stable. Under general conditions, it is shown that the estimators are asymptotically normal in the finite variance case, and, in the infinite variance case, the estimators converge in distribution to nondegenerate maxima of stochastic processes.