Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches
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, w...
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MDPI AG
2022-08-01
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| Online Access: | https://www.mdpi.com/2571-905X/5/3/46 |
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| author | Rajae Azrak Guy Mélard |
| author_facet | Rajae Azrak Guy Mélard |
| author_sort | Rajae Azrak |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-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). |
| first_indexed | 2024-03-09T22:30:47Z |
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| institution | Directory Open Access Journal |
| issn | 2571-905X |
| language | English |
| last_indexed | 2024-03-09T22:30:47Z |
| publishDate | 2022-08-01 |
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| spelling | doaj.art-4900c0c37cfb44ddbec519441a0109072023-11-23T18:57:53ZengMDPI AGStats2571-905X2022-08-015378480410.3390/stats5030046Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several ApproachesRajae Azrak0Guy Mélard1Faculté des Sciences Juridiques Economiques et Sociales, Université Mohammed V—Rabat, Salé Route Outa Hssain, Sala Al Jadida, Salé B.P. 5295, MoroccoSolvay Brussels School of Economics and Management and ECARES, Université Libre de Bruxelles, CP 114/04, Avenue Franklin Roosevelt, 50, B-1050 Brussels, BelgiumAutoregressive-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-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).https://www.mdpi.com/2571-905X/5/3/46nonstationary processtime seriestime-dependent modeltime-varying model<i>ρ</i>-mixing propertylocally stationary processes |
| spellingShingle | Rajae Azrak Guy Mélard Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches Stats nonstationary process time series time-dependent model time-varying model <i>ρ</i>-mixing property locally stationary processes |
| title | Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches |
| title_full | Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches |
| title_fullStr | Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches |
| title_full_unstemmed | Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches |
| title_short | Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches |
| title_sort | autoregressive models with time dependent coefficients a comparison between several approaches |
| topic | nonstationary process time series time-dependent model time-varying model <i>ρ</i>-mixing property locally stationary processes |
| url | https://www.mdpi.com/2571-905X/5/3/46 |
| work_keys_str_mv | AT rajaeazrak autoregressivemodelswithtimedependentcoefficientsacomparisonbetweenseveralapproaches AT guymelard autoregressivemodelswithtimedependentcoefficientsacomparisonbetweenseveralapproaches |