# Maximum Likelihood Estimation for All-Pass Models

Monday, November 12, 2001 - 11:00am - 12:00pm

Keller 3-180

Richard Davis (Colorado State University)

In the analysis of returns on financial assets such as stocks, it is common to observe lack of serial correlation, heavy-tailed marginal distributions, and volatility clustering. Typically, nonlinear models with time-dependent conditional variances, such as ARCH and stochastic volatility models, are suggested for such time series. It is perhaps less well known that linear, non-Gaussian models can display exactly this behavior. The linear models which we will consider are all-pass models: autoregressive-moving average models in which all of the roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa. All-pass models generate uncorrelated (white noise) time series, but these series are not independent in the non-Gaussian case. If the process is driven with heavy-tailed noise, then its marginal distribution will also have heavy tails, and the process will exhibit volatility clustering.

All-pass models are widely used in the engineering literature, and usually arise by modeling a series as an invertible moving average (all the roots of the moving average polynomial are outside the unit circle) when in fact the true model is noninvertible. The resulting series in this case can then be modeled as an all-pass of order r, where r is the number of roots of the true moving average polynomial inside the unit circle.

Estimation methods based on Gaussian likelihood, least-squares, or related second-order moment techniques are unable to identify all-pass models. Instead, method of moments estimators using moments of order greater than two are often used to estimate such models (Giannakis and Swami, 1990; chi and Kung, 1995). Breidt, Davis, and Trindade (2000) consider a least absolute deviations approach, motivated by approximating the likelihood of the all-pass model in the case of Laplace (two-sided exponential) noise. Under general conditions, the least absolute deviation estimators are asymptotically normal.

In this paper, we consider estimation based on an approximation to the likelihood. Asymptotic normality for the maximum likelihood estimator is established under smoothness conditions on the density function of the noise. Behavior of the estimators in finite samples is studied via simulation and estimation procedure is applied to problem of fitting noninvertible moving averages. (This is joint work with F. Jay Breidt and Beth Andrews.)

All-pass models are widely used in the engineering literature, and usually arise by modeling a series as an invertible moving average (all the roots of the moving average polynomial are outside the unit circle) when in fact the true model is noninvertible. The resulting series in this case can then be modeled as an all-pass of order r, where r is the number of roots of the true moving average polynomial inside the unit circle.

Estimation methods based on Gaussian likelihood, least-squares, or related second-order moment techniques are unable to identify all-pass models. Instead, method of moments estimators using moments of order greater than two are often used to estimate such models (Giannakis and Swami, 1990; chi and Kung, 1995). Breidt, Davis, and Trindade (2000) consider a least absolute deviations approach, motivated by approximating the likelihood of the all-pass model in the case of Laplace (two-sided exponential) noise. Under general conditions, the least absolute deviation estimators are asymptotically normal.

In this paper, we consider estimation based on an approximation to the likelihood. Asymptotic normality for the maximum likelihood estimator is established under smoothness conditions on the density function of the noise. Behavior of the estimators in finite samples is studied via simulation and estimation procedure is applied to problem of fitting noninvertible moving averages. (This is joint work with F. Jay Breidt and Beth Andrews.)