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Article

Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches

by
Rajae Azrak
1 and
Guy Mélard
2,*
1
Faculté des Sciences Juridiques Economiques et Sociales, Université Mohammed V—Rabat, Salé Route Outa Hssain, Sala Al Jadida, Salé B.P. 5295, Morocco
2
Solvay Brussels School of Economics and Management and ECARES, Université Libre de Bruxelles, CP 114/04, Avenue Franklin Roosevelt, 50, B-1050 Brussels, Belgium
*
Author to whom correspondence should be addressed.
Academic Editor: Wei Zhu
Stats 2022, 5(3), 784-804; https://doi.org/10.3390/stats5030046
Received: 20 June 2022 / Revised: 20 July 2022 / Accepted: 31 July 2022 / Published: 12 August 2022
(This article belongs to the Section Time Series Analysis)
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Abstract

Autoregressive-moving average (ARMA) models with time-dependent (td) coefficients and marginally heteroscedastic innovations provide a natural alternative to stationary ARMA models. Several theories have been developed in the last 25 years for parametric estimations in that context. In this paper, we focus on time-dependent autoregressive (tdAR) models and consider one of the estimation theories in that case. We also provide an alternative theory for tdAR processes that relies on a ρ-mixing property. We compare these theories to the Dahlhaus theory for locally stationary processes and the Bibi and Francq theory, made essentially for cyclically time-dependent models, with our own theory. Regarding existing theories, there are differences in the basic assumptions (e.g., on derivability with respect to time or with respect to parameters) that are better seen in specific cases such as the tdAR(1) process. There are also differences in terms of asymptotics, as shown by an example. Our opinion is that the field of application can play a role in choosing one of the theories. This paper is completed by simulation results that show that the asymptotic theory can be used even for short series (less than 50 observations). View Full-Text
Keywords: nonstationary process; time series; time-dependent model; time-varying model; ρ-mixing property; locally stationary processes nonstationary process; time series; time-dependent model; time-varying model; ρ-mixing property; locally stationary processes
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MDPI and ACS Style

Azrak, R.; Mélard, G. Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches. Stats 2022, 5, 784-804. https://doi.org/10.3390/stats5030046

AMA Style

Azrak R, Mélard G. Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches. Stats. 2022; 5(3):784-804. https://doi.org/10.3390/stats5030046

Chicago/Turabian Style

Azrak, Rajae, and Guy Mélard. 2022. "Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches" Stats 5, no. 3: 784-804. https://doi.org/10.3390/stats5030046

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