Summary: Maximum Likelihood Estimation
for All-Pass Time Series Models
Richard A. Davis1,2
and F. Jay Breidt2
Colorado State University
November 2, 2005
An autoregressive-moving average model in which all roots of the autoregressive polyno-
mial are reciprocals of roots of the moving average polynomial and vice versa is called
an all-pass time series model. All-pass models generate uncorrelated (white noise) time
series, but these series are not independent in the non-Gaussian case. An approximate
likelihood for a causal all-pass model is given and used to establish asymptotic normality
for maximum likelihood estimators under general conditions. Behavior of the estimators
for finite samples is studied via simulation. A two-step procedure using all-pass models
to identify and estimate noninvertible autoregressive-moving average models is developed
and used in the deconvolution of a simulated water gun seismogram.
Supported in part by NSF Grants DMS9972015 and DMS0308109.