Likelihood Functions for State Space Models with Diffuse Initial Conditions

8 Pages Posted: 12 Oct 2010

See all articles by Marc Francke

Marc Francke

University of Amsterdam - Faculty of Economics and Business (FEB); Ortec Finance

Siem Jan Koopman

Vrije Universiteit Amsterdam - School of Business and Economics; Tinbergen Institute; Aarhus University - CREATES

Aart F. de Vos

Vrije Universiteit Amsterdam, School of Business and Economics

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Abstract

State space models with non-stationary processes and/or fixed regression effects require a state vector with diffuse initial conditions. Different likelihood functions can be adopted for the estimation of parameters in time-series models with diffuse initial conditions. In this article, we consider profile, diffuse and marginal likelihood functions. The marginal likelihood function is defined as the likelihood function of a transformation of the data vector. The transformation is not unique. The diffuse likelihood is a marginal likelihood for a data transformation that may depend on parameters. Therefore, the diffuse likelihood cannot be used generally for parameter estimation. The marginal likelihood function is based on an orthonormal data transformation that does not depend on parameters. Here we develop a marginal likelihood function for state space models that can be evaluated by the Kalman filter. The so-called diffuse Kalman filter is designed for computing the diffuse likelihood function. We show that a minor modification of the diffuse Kalman filter is needed for the evaluation of our marginal likelihood function. Diffuse and marginal likelihood functions have better small sample properties compared with the profile likelihood function for the estimation of parameters in linear time series models. The results in our article confirm the earlier findings and show that the diffuse likelihood function is not appropriate for a range of state space model specifications.

Suggested Citation

Francke, Marc and Koopman, Siem Jan and de Vos, Aart F., Likelihood Functions for State Space Models with Diffuse Initial Conditions. Journal of Time Series Analysis, Vol. 31, Issue 6, pp. 407-414, November 2010, Available at SSRN: https://ssrn.com/abstract=1688651 or http://dx.doi.org/10.1111/j.1467-9892.2010.00673.x

Marc Francke

University of Amsterdam - Faculty of Economics and Business (FEB) ( email )

Plantage Muidergracht 12
Amsterdam, 1018 TV
Netherlands

HOME PAGE: http://www.uva.nl/en/contact/staff/item/m.k.francke.html?f=francke

Ortec Finance ( email )

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Amsterdam, 1043 CP
Netherlands
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+ 31 20 7009 701 (Fax)

HOME PAGE: http://www.ortec-finance.com

Siem Jan Koopman

Vrije Universiteit Amsterdam - School of Business and Economics ( email )

De Boelelaan 1105
Amsterdam, 1081 HV
Netherlands
+31205986019 (Phone)

HOME PAGE: http://sjkoopman.net

Tinbergen Institute ( email )

Gustav Mahlerplein 117
1082 MS Amsterdam
Netherlands

HOME PAGE: http://personal.vu.nl/s.j.koopman

Aarhus University - CREATES ( email )

School of Economics and Management
Building 1322, Bartholins Alle 10
DK-8000 Aarhus C
Denmark

Aart F. De Vos

Vrije Universiteit Amsterdam, School of Business and Economics ( email )

De Boelelaan 1105
Amsterdam, 1081HV
Netherlands

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