Vector Arma Models*

The problems of evaluating and maximizing the exact likelihood function of vector ARMA models are considered separately. A new and efficient procedure far evaluating the exact likelihood function is presented. This method puts together a set of useful features which can only be found separately in currently available algoritluns. A procedure for maximizing the exact likeliliood function, which takes full advantage of the properties offered by the evaluation algorithm, is also considered, Combining these two procedures, a new algorithm for exact maximum likelihood estimation of vector ARMA models is obtained. Comparisons with existing procedures, in tenns of both analytical arguments and a numerical example , are given in order to show that the new estimation algorithm performs at least as well as existing ones, and that relevant real situations occur in which it do es better. RESUMEN En este trabajo se estudian por separado los problemas asociados con la evaluación y la maximización de la función de verosimilitud exacta de modelos ARMA multivariantes. Por un lado, se diseña un nuevo algoritmo para evaluar dicha función eficientemente, que reúne una serie de propiedades que se encuentran por separado en los procedimientos actualmente disponibles. Por otro lado, se propone un mecanismo para maximizar la función de verósimilitud, que aprovecha las ventajas ofrecidas por el algoritmo de evaluación. Combinando ambos procedimientos, se obtiene un nuevo algoritmo de estimación por máxima verosimilitud exacta de modelos ARMA multivariantes, cuyo funcionamiento en la práctica resulta superior, en muchos aspectos, al de otros procedimientos alternativos. * This paper summarizes the author's doctoral dissertation Evaluación y Maxirnización de la Función de Verosimilirud de Procesos ARMA Multivariantes {1993, Madrid: Universidad Complutense de Madrid), and was partiaUy financed under the project fura Forecast and Monitor Service for the Spanish Economy, sponsored by Caja de Madrid (Spain), and by ! he DGICYT project PB89-0129 (Spain). The author is gratefu\ to Arthur B. Treadway, under whose supervision this research was carried ont, and to Emilio Cerdá, Josi Luis Gallego, VíctorM. Gonzaln, Mignel Jerez, Alfonso Novales, Daniel Peña and Femando Tusse\ for he! pful suggestions. }\/9C :. ::::. :~';¡ ·22'l(l.i39'-€ "~""--'~ l. INTRODUCTION This article propases a new procedure for exact maximum likelihood estimation of vector ARMA models. A sharp distinction between evaluating and maximizing the likelihood function is made. This permits a detailed analysis of all problems that arise in the estimation process. Tbe solutions obtained through this analysis can then be integrated into a complete estimation procedure that takes advantage of sorne properties of the likelihood function which have not been fully exploited in previous papers. Although there has been abundant research on the evaluation of the likelihood function of vector ARMA models (e.g. Hillmer and Tiao 1979; Nicholls and Hall 1979; Hall and Nicholls 1980; Shea 1987), only a few authors have paid attention to the problem of its subsequent maximization. Furtheunore, such attention has usually been restricted to suggesting, in a few lines, the use of a standard optimization algorithm to maximize the likelihood function, evaluated as extensively described in the cited papers. An interesting exception can be found in Shea (1984, pp. 99-100). With regard to the computation of the likelihood function, none of the existing methods can be taken as fully satisfactory. This is dueto the fact that many ofthe properties a method of evaluating the likelihood function should ha ve, are scattered among the existing procedures. Thus, although each of many existing algoritbms has sorne useful properties, it also lacks others that can be found in altemative procedures. For instance, the algorithm of Shea (1984, 1987) can be considered, from a computational viewpoint, the most efficient among the existing ones. However, its use does not permit the automatic detection of non-invertible models. This task is easily handled with the algorithm of Hall and Nicholls (1980) and with an extension to the multivariate context of the algoritlun of Ljung and Box (1979). However, the fonner involves a high computational cost (in many cases) and the latter both sorne computational inefficiency and a loss of numerical precision, due to the requirement for an explicit matrix. inversion. Finally, the algorithm ofHillmer and Tiao (1979) pennits the computation of neither the exact lik~lihood function nor an appropriate residual vector, except in the case of pure moving average models; this fact may become an important drawback when the m_odel considered has an autoregressive part and the sample contains extreme values among the initial observations. A thorough analysis of currently available procedures allows one to discover and fully exploit new possibilities ignored in the papers cited above. Thus, in Section 2, a new method of evaluating the exact likelihood function of vector ARMA models is described in detail. The new algorithm puts together the advantages which caribe found separately in existing procedures and does not suffer from any of their drawbacks. In Section3, computational techníques for maxirnizing the likelihood function are considered. These techníques take fμll _advantage of the properties offered by the evaluation algorithm. Section 4 gives an illustrative example of an actual situation in which the new estimation procedure perfomis better than one of the most frequently used procedures. Finally, in Section 5, conclusions are summarized. 2. EV ALUATION OF THE EXACT LIKELIHOOD FUNCTION Let w, be an m-dimensional vector~valued time series. It is supposed that w1 follows the vector ARMA(p,q) modo! <fi(B)W1 = 0(B)at, (2.1) where <fi(B) = 1 <fi1B ••• <fi~, 0(B) = 1 0 1B ··· 0/fl, Bis the back shift operator, W1 = w,-μ,, <f? ¡ (i = 1, •. , p), 0¡ (i = 1, •• ., q) and μ,are mxm, mxm and mxl parameter matrices, respectively, and the a¡s are mx 1 random vectors identically and independently distributed as N(O, a2Q), with a2 > O and Q (mxm) symmetric and positive definite. This decomposition of E(a,a~]. although not unique, is useful for obtaining maximum likelihood estimates by maximizing a concentrated log likelihood as a function of <fi¡ (i = 1, . , p), 9¡ (i = 1, . ., q) and Q only (Section 3). For stationarity, it is required that the zeros of l'1>(B)l lie outside the unit circle. Furthermore, (2, 1) is assumed to satisfy the conditions derived by Hannan (1969) in arder that the model be identified. Considera sample of size n and Jet W = (Wf, . , wr)T (mean-corrected observations), a = (af, . , a~)T (white noise perturbations), and u,. = <wf-p, . , wi), af-q, . , a¡))T (unknown presample values). Then, equation (2.1) may be written as