The exact quasi-likelihood of time-dependent ARMA models

Abstract

Abstract. The purpose of the paper is to propose a simple and efficient algorithm to evaluate the exact quasi-likelihood of (possibly marginally heteroscedastic) ARMA models with time-dependent coefficients. The algorithm is based on the Kalman filter and is therefore simpler than a previous algorithm based on a Cholesky factorisation. Computational efficiency is obtained by taking the ARMA structure into account. Empirical evidence is given. It is also shown how the algorithm can be used as an approximation in the following non-linear models: conditionally heteroscedastic ARMA models (with GARCH errors) and threshold ARMA models, in order to improve the treatment of the initial observations when the parameters of these models are estimated.

Asymptotic Properties of Quasi-Maximum Likelihood Estimators for ARMA Models with Time-Dependent Coefficients

“…Contrarily to Dahlhaus (1997) assertions, the exact maximum likelihood method is not computationally intensive, not very much than for ARMA models with constant coefficients. Indeed, although the number of operations at each time is quadratic with respect to the model orders, the experiments made by Mélard (1982) and Azrak and Mélard (1998) show that the computation times remain reasonable. For example, if the Ansley (1979) Cholesky factorization algorithm is used, the number of operations at each time is barely multiplied by 2, plus the operations needed for computing the coefficients in terms of the parameters, of course.…”

“…The estimation method which is used in the examples of this section has been implemented in Time Series Expert (Mélard and Pasteels, 1997) the exact Gaussian likelihood is computed by the algorithm of Mélard (1982) and the optimum is obtained using a customised variant of Marquardt's (1963) non-linear least-squares optimisation procedure. The algorithm given independently by Dahlhaus (1996a) and Azrak and Mélard (1998) which provides a generalisation of Gardner et al (1980) combined with a good optimisation procedure could have been used instead. Contrarily to Dahlhaus (1997) assertions, the exact maximum likelihood method is not computationally intensive, not very much than for ARMA models with constant coefficients.…”

“…The advantage of that approach is that the Gaussian likelihood function can be computed exactly, with very efficient algorithms (e.g. Mélard, 1982;Azrak and Mélard, 1998).…”

Parameter estimation of an ARMA model for river flow forecasting using goal programming

“…There have been several attempts to estimate the parameters of ARMA models by different researchers (Ljung and Box, 1979;Ansely, 1979;Gardner et al, 1980;Pearlman, 1980;Melard, 1984;Azrak and Melard, 1998). Most of these methods are based on the Kalman filter which needs a great amount of computation.…”

“…Monte-Carlo experiments have shown that for ARMA models with relatively short lengths of data, e.g. 50 or 100 observations, exact maximum likelihood estimation is far superior to conditional maximum likelihood estimation or least-squares estimation (Azrak and Melard, 1998). Recently, artificial neural network techniques have been used to estimate ARMA parameters.…”