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.
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.
Several extensions to autoregressive integrated moving average (ARIMA) models have been considered in the recent years. Many of them are special cases of the extended ARIMA model treated in this paper. The main features are time-dependent coefficients in the autoregressive and moving average polynomials, various types of interventions (including the usual Box and Tiao form and the innovation interventions but also interventions acting on the scale), trends on the level or on the scale, built-in deterministic seasonal components and variable transformations. In the past, estimation procedures were limited to least squares although the evaluation of the likelihood function is available for special cases, including the ARMA model with time dependent coefficients. This paper deals with maximum likelihood estimation when several features of the extended ARIMA model are taken together. INTRODUCTIONSeveral extensions of ARIMA models have been considered in the recent years, including (a) the use of time-dependent coefficients in the autoregressive and moving average polynomials (Quenouille, 1957;Whittle, 1965;Abdrabbo and Priestley, 1967;Miller, 1968 and1969;Subba Rao, 1970; Mélard and Kiehm, 1981;Tyssedal and Tjøstheim, 1982;Grillenzoni, 1990), (b) various types of interventions, including the usual Box and Tiao (1975) formulation and the innovational interventions (Fox, 1972) but also interventions acting on the scale (Mélard, 1981a;Tsay, 1988), (c) additive (level) or multiplicative (scale) trend (Mélard, 1977), (d) built-in deterministic seasonal components on the variable (Abraham and Box, 1978) or on the innovation (Mélard, 1981b), (e) variable transformations (Box and Cox, 1964). 1 We are grateful to the referee for his/her comments, and more especially for the summary that we have used at the end of Section 1. 2The purpose of that model is to encompass several deterministic variations with respect to time in the framework of the usual stochastic ARIMA models. Other extensions not explicitly considered in this paper are ARMA models with GARCH errors (Bollerslev, 1986), threshold AR models (Tong, 1983), bilinear models (Subba Rao, 1981), fractional differencing ARIMA models (Granger and Joyeux, 1980). It should be noted, however, that some of these extensions can be handled using the same approach. For instance, threshold ARMA models (Mélard and Roy, 1988) can be seen as time-dependent ARMA models. Other approaches for time-dependent models include spectral density estimation (Priestley 1981(Priestley , 1988, recursive estimation (Ljung and Söderström, 1983;Young, 1984) and models with random coefficients (Nicholls and Quinn, 1982;Bougerol, 1992).Motivations for the extended ARIMA model which is used here have already been discussed elsewhere (Mélard, 1982a(Mélard, , 1985a). An illustration has already been provided (Mélard, 1985b). The estimation procedure was however limited to the conditional least squares approach, generalizing the approach of Box and Jenkins (1976). In this paper, an algorithm f...
An alternative central limit theorem for martingale difference arrays is presented. It can be deduced from the literature but it is not stated as such. It can be very useful for statisticians and econometricians. An illustration is given.
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