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

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Abstract

For about thirty years, time series models with time-dependent coefficients have sometimes been considered as an alternative to models with constant coefficients or non-linear models. Analysis based on models with time-dependent models has long suffered from the absence of an asymptotic theory except in very special cases. The purpose of this paper is to provide such a theory without using a locally stationary spectral representation and time rescaling. We consider autoregressive-moving average (ARMA) models with time-dependent coefficients and a heteroscedastic innovation process. The coefficients and the innovation variance are deterministic functions of time which depend on a finite number of parameters. These parameters are estimated by maximising the Gaussian likelihood function. Deriving conditions for consistency and asymptotic normality and obtaining the asymptotic covariance matrix are done using some assumptions on the functions of time in order to attenuate non-stationarity, mild assumptions for the distribution of the innovations, and also a kind of mixing condition. Theorems from the theory of martingales and mixtingales are used. Some simulation results are given and both theoretical and practical examples are treated.

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Authors and Affiliations

  1. Faculté des Sciences juridiques, économiques et sociales, Université Mohammed V-Souissi, Salé, Marocco

    Rajae Azrak

  2. ECARES and ISRO, Université Libre de Bruxelles, Belgium

    Guy Mélard

  3. Institut de Statistique et de Recherche Operationnelle CP 210, Campus Plaine U.L.B., Bd du Triomphe, B-1050, Brussels, Belgium

    Guy Mélard

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  1. Rajae Azrak
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Correspondence to Guy Mélard.

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Received 2004; Final version 23 December 2004

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Azrak, R., Mélard, G. Asymptotic Properties of Quasi-Maximum Likelihood Estimators for ARMA Models with Time-Dependent Coefficients. Stat Infer Stoch Process 9, 279–330 (2006). https://doi.org/10.1007/s11203-005-1055-6

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