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Time Series Analysis by State Space MethodsSecond Edition$
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James Durbin and Siem Jan Koopman

Print publication date: 2012

Print ISBN-13: 9780199641178

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199641178.001.0001

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Maximum likelihood estimation of parameters

Maximum likelihood estimation of parameters

(p.170) 7 Maximum likelihood estimation of parameters
Time Series Analysis by State Space Methods

J. Durbin

S.J. Koopman

Oxford University Press

This chapter discusses maximum likelihood estimation of parameters both for the case where the distribution of the initial state vector is known and for the case where at least some elements of the vector are diffuse or are treated as fixed and unknown. For the linear Gaussian model it shows that the likelihood can be calculated by a routine application of the Kalman filter, even when the initial state vector is fully or partially diffuse. It details the computation of the likelihood when the univariate treatment of multivariate observations is adopted. It considers how the loglikelihood can be maximised by means of iterative numerical procedures. An important part in this process is played by the score vector. The chapter shows how this is calculated, both for the case where the initial state vector has a known distribution and for the diffuse case. A useful device for maximisation of the loglikelihood in some cases, particularly in the early stages of maximisation, is the EM algorithm; details are provided for the linear Gaussian model. The chapter also considers biases in estimates due to errors in parameter estimation and ends with a discussion of some questions of goodness-of-fit and diagnostic checks.

Keywords:   state vector, linear Gaussian model, kalman filter, loglikelihood, em algorithm, score vector

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