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First order conditions for the maximum likelihood estimation of an exact ARMA model

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CBM R i~ 7826 ~~ , ~~~ a~1993 ,~; ~,~ ~ 61 1 ~ ~ ~~ P~~s~x~` IIII~'iIIIIIIVUINIdIi~INIrII~~ÍNI , ' FIRST ORDER CONDITIONS FOR THE MAXIMUM LIlCrLIH00D ESTIMATION OF AN EXACT ARMA MODEL Jan van der Leeuw FEw 611 Communicated by Prof.dr. B.B. van der Genugten ~ïj'" ~.1.~.;~ . ~ ~,;.: ~r 4r'~;-~~ "-L-K~,-R.Tí L~~~.~:su;~,,~.. ...-,-.---------e-- loc4x.chi 11.08.93 FIRST ORDER CONDITIONS FOR THE MAXIMUM LIKELIHOOD ESTIMATION OF AN EXACT ARMA MODEL Jan van der Leeuwi Dept. of Econometrics Tilburg University P.O.Box 90153 NL - 5000 LE Tilburg Abstract Using the exact covariance matrix of ARMA(p,q) errors first order conditions for the parameters are derived and solved. This is done for the pure MA case, the pure AR case and the general ARMA model. Our approach applies both to maximum likeli- hood and minimum distance estimation. The exact covariance is written in the form of lag matrices, which can simply be differentiated. The resulting first order conditions have at least one solution. The difference between maximum likelihood and minimum distance estimation amounts to a function of the elements of the covariance matrix. This function is simple in case of the pure MA or AR case, but more complicated in the general AR:~fA case. Of course, the solutions for the AR and M.4 parameters are in general conditional. Only in the pure MA and AR case of a time series model without explanatory varia- bles direct solutions are found. lI am indebted to H.H. Tigelaar for his suggestions and comments on an earlierdraft. 1 1. Introduction In a well known article C.M. ~Beach and J. MacKinnon (1978) presented a maximurn likelihood procedure for estimating the parameters of a linear regression model with first-order autocorrelation. For a fixed value of the .AR(1) parameter they estimate the regression parameter, next they calculate the AR(1) parameter con- ditional on this estimate. J.Magnus (1978) showed in a more general way that such a procedure converges and that

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