Likelihood Functions for State Space Models with Diffuse Initial Conditions

Tinbergen Institute Discussion Paper No. TI 2008-040/4

26 Pages Posted: 16 Apr 2008

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

Multiple version iconThere are 2 versions of this paper

Date Written: April 2006

Abstract

State space models with nonstationary processes and 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 paper we consider profile, diffuse and marginal likelihood functions. The marginal likelihood 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 specific data transformation that may depend on parameters. Therefore, the diffuse likelihood can not be used generally for parameter estimation. Our newly proposed marginal likelihood function is based on an orthonormal transformation that does not depend on parameters. Likelihood functions for state space models are evaluated using the Kalman filter. The diffuse Kalman filter is specifically designed for computing the diffuse likelihood function. We show that a modification of the diffuse Kalman filter is needed for the evaluation of our proposed marginal likelihood function. Diffuse and marginal likelihood functions have better small sample properties compared to the profile likelihood function for the estimation of parameters in linear time series models. The results in our paper confirm the earlier findings and show that the diffuse likelihood function is not appropriate for a range of state space model specifications.

Keywords: Diffuse likelihood, Kalman filter, Marginal likelihood, Multivariate time series models, Profile likelihood

JEL Classification: C13, C22, C32

Suggested Citation

Francke, Marc and Koopman, Siem Jan and de Vos, Aart F., Likelihood Functions for State Space Models with Diffuse Initial Conditions (April 2006). Available at SSRN: https://ssrn.com/abstract=1120262 or http://dx.doi.org/10.2139/ssrn.1120262

Marc Francke (Contact Author)

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
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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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