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EXACT MAXIMUM LIKELIHOOD ESTIMATION IN AUTOREGRESSIVE PROCESSES
Author(s) -
Miller James W.
Publication year - 1995
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.1995.tb00258.x
Subject(s) - mathematics , likelihood function , autoregressive model , estimating equations , estimation theory , restricted maximum likelihood , likelihood ratio test , statistics , expectation–maximization algorithm , gaussian , series (stratigraphy) , maximum likelihood , paleontology , physics , quantum mechanics , biology
. The purpose of this paper is to complement the theory of exact maximum likelihood estimation in pure autoregressive processes by differentiating the exact Gaussian likelihood function with respect to the model parameters and obtaining a set of likelihood equations very similar in form to the Yule—Walker equations. The main contribution of this paper is a very simple expression for the derivatives and the resulting likelihood equations in terms of the components of a ( p + 1) x ( p + 1) function of the data, the model parameters ( s̀ 2 , φ ) and the autocovariances at lags 0 through p. We propose an iterative algorithm for solving the likelihood equations by alternately solving two linear systems, first for ( s̀ 2 , φ ) given current estimates of the autocovariances, then for updated estimates of the autocovariances given current estimates of ( s̀ 2 , φ ). The number of operations per iteration is independent of the series length since the algorithm uses the data only through the value of the ( p + 1) x ( p + 1) sufficient statistic.