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Exact Maximum Likelihood Estimation In Autoregressive Processes

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  • James W. Miller

Abstract

. 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 (s̀2, 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 (s̀2, φ) given current estimates of the autocovariances, then for updated estimates of the autocovariances given current estimates of (s̀2, 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.

Suggested Citation

  • James W. Miller, 1995. "Exact Maximum Likelihood Estimation In Autoregressive Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 16(6), pages 607-615, November.
    • Handle: RePEc:bla:jtsera:v:16:y:1995:i:6:p:607-615
    DOI: 10.1111/j.1467-9892.1995.tb00258.x
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    References listed on IDEAS

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    1. T. W. Anderson & Raúl P. Mentz, 1980. "On The Structure Of The Likelihood Function Of Autoregressive And Moving Average Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(2), pages 83-94, March.
    2. M. Minozzo & A. Azzalini, 1993. "On The Unimodality Of The Exact Likelihood Function For Normal Ar(2) Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 14(5), pages 497-509, September.
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