The exact Gaussian likelihood estimation of time-dependent VARMA models

Alj, Abdelkamel; Jónasson, Kristján; Mélard, Guy

doi:10.1016/j.csda.2014.07.006

Cited by 5 publications

(4 citation statements)

References 30 publications

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“…Additionally, [16] provided a better foundation for the asymptotic theory for array processes, a theorem for a reduction of the order of moments from 8 to slightly more than 4 and tools for obtaining the asymptotic covariance matrix of the estimator. In [17], there was an example of vector tdAR and tdMA models on monthly log returns of IBM stock and the S&P500 index from January 1926 to December 1999, treated first in [18].…”

Section: Let φmentioning

confidence: 99%

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

Azrak¹,

Mélard²

2022

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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 -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).

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Section: Let φmentioning

confidence: 99%

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

Azrak¹,

Mélard²

2022

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“…The simulation experiment is performed in Matlab by using the program, which we call AJM, described in Alj et al (2015c) and based on a special case of tdVAR(1) process defined in (4.4)-(4.5), with A ′ 11 = 0.8 and A ′ 22 = −0.9 and (ǫ t1 , ǫ t2 ) T has a bivariate normal distribution with covariance matrix Σ = I 2 . A simulated series using these specifications is shown in Fig.…”

Section: Example 1: Tdvar(1) a Generalization Of Kwoun And Yajima (1986)mentioning

confidence: 99%

“…There is no assumption of stationarity but, although it is not illustrated in our examples, there is an adjustment in the asymptotic theory for allowing non-normal observations. One major advantage of QMLE is that the Gaussian likelihood function can be computed exactly, with an efficient algorithm, Alj et al (2015c), and this is very important for short time series. The main task in the Azrak-Mélard approach, hence also in our extension, consists in checking conditions from two crucial theorems in Klimko & Nelson (1978), which respectively ensure existence of an almost surely (a.s.) consistent estimator and prove asymptotic normality of that estimator, whilst providing the asymptotic covariance matrix.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Properties of QML Estimators for VARMA Models with Time‐dependent Coefficients

Alj

Azrak

Ley

et al. 2017

Scandinavian J Statistics

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This paper is about vector autoregressive-moving average (VARMA) models with time-dependent coefficients to represent non-stationary time series. Contrary to other papers in the univariate case, the coefficients depend on time but not on the series' length n. Under appropriate assumptions, it is shown that a Gaussian quasi-maximum likelihood estimator is almost surely consistent and asymptotically normal. The theoretical results are illustrated by means of two examples of bivariate processes. It is shown that the assumptions underlying the theoretical results apply. In the second example the innovations are marginally heteroscedastic with a correlation ranging from −0.8 to 0.8. In the 1 Preprint version of a paper that will appear in Scandinavian Journal of Statistics (2017) two examples, the asymptotic information matrix is obtained in the Gaussian case. Finally, the finite-sample behavior is checked via a Monte Carlo simulation study for n from 25 to 400. The results confirm the validity of the asymptotic properties even for short series and the asymptotic information matrix deduced from the theory.

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“…To cite few examples of nonlinear multivariate processes, let us mention the self-exciting threshold vector autoregressive [seeTsay (1998)], the smooth transition vector autoregressive(Camacho 2004), the random coefficient VARMA(Alj et al 2014).…”

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confidence: 99%

Modified Schwarz and Hannan–Quinn information criteria for weak VARMA models

Maïnassara

Kokonendji

2015

Stat Inference Stoch Process

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Numerous multivariate time series admit weak vector autoregressive movingaverage (VARMA) representations, in which the errors are uncorrelated but not necessarily independent nor martingale differences. These models are called weak VARMA by opposition to the standard VARMA models, also called strong VARMA models, in which the error terms are supposed to be independent and identically distributed (iid). This article considers the problem of order selection of the weak VARMA models by using the information criteria. It is shown that the use of the standard information criteria are often not justified when the iid assumption on the noise is relaxed. As a consequence, we propose the modified versions of the Schwarz or Bayesian information criterion and of the Hannan and Quinn criterion for identifying the orders of weak VARMA models. Monte Carlo experiments show that the proposed modified criteria estimate the model orders more accurately than the standard ones. An illustrative application using the squared daily returns of financial series is presented.

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The exact Gaussian likelihood estimation of time-dependent VARMA models

Cited by 5 publications

References 30 publications

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

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

Asymptotic Properties of QML Estimators for VARMA Models with Time‐dependent Coefficients

Modified Schwarz and Hannan–Quinn information criteria for weak VARMA models

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