ESSAYS ON MONTE CARLO METHODS FOR STATE SPACE MODELS

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1 VRIJE UNIVERSITEIT ESSAYS ON MONTE CARLO METHODS FOR STATE SPACE MODELS ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus prof.dr. L.M. Bouter, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de Faculteit der Economische Wetenschappen en Bedrijfskunde op maandag.17 december 2012 om uur in de aula van de universiteit, De Boelelaan 1105 door Marcel Scharth Figueiredo Pinto geboren te Rio de Janeiro, Brazilie

2 Contents 1 Introduction State space models Why Monte Carlo methods are necessary State space models and financial econometrics Thesis overview 10 2 Numerically accelerated importance sampling for nonlinear non-gaussian state space models Importance sampling for state space models Nonlinear and non-gaussian state space model Likelihood evaluation via importance sampling The Gaussian importance density Selecting the importance sampling parameters Numerically Accelerated Importance Sampling The construction of the importance sampler via numerical integration...., Importance sampling and control variables First new control variable Second new control variable Monte Carlo and empirical evidence Likelihood estimation Three models Simulation results Parameter estimation for a higher dimensional model 40

3 2.3.5 Empirical application Conclusion 45 2.A Appendix 48 2.A.I Specifications with analytical solutions 48 2.A.2 The high-dimensional EIS method 50 2.A.3 The bias in the EIS method."..-.-." 52 3 The analysis of stochastic volatility in the presence of daily realised measures Stochastic Volatility and Realised Measures A general discrete time stochastic volatility model Stochastic volatility with leverage Stochastic volatility with long range dependence Realised Stochastic Volatility Conditional return distribution The dependence between daily returns and measurement errors Overnight returns Maximum likelihood estimation The likelihood function based on full information A likelihood function based on selected information Two-step estimation for the model without leverage Two-step estimation for the model with leverage Simulation study Empirical Analysis Data and measurement Model specification Estimation and filtering, Forecasting The distribution of return innovations 90 3.A Appendix 96 3.A.I Deletion Smoothing 96

4 4 Predicting time-varying parameters with parameter-driven and observationdriven models Modelling time-varying parameters Dynamic model specifications Observation densities Parameter-driven versus observation-driven models Observation-driven continuous mixture models Weibull-gamma and exponential-gamma mixture models Double gamma mixture models Maximum Likelihood Estimation Observation-driven models: maximum likelihood Parameter-driven models: simulated maximum likelihood Predictive analysis: a Monte Carlo study Design of the Monte Carlo study Results Analysis based on model confidence sets Multi-step forecasting Conclusion A Appendix A.I Numerically accelerated importance sampling A.2 Forecasting for the parameter-driven models Efficient likelihood evaluation for non-gaussian measurement state space models with multiple time-varying parameters Basics ( State Space model Importance sampling Approximating linear state space model Efficient likelihood evaluation : Importance parameters ' An efficient algorithm ' Diagonalising the approximating linear state space model Illustrations 147

5 5.3.1 Models Likelihood evaluation Accuracy for parameter estimation Empirical application Conclusion Conclusion Contributions Importance sampling for state space models Realised stochastic volatility Parameter-driven and observation-driven models Extensions 172 Bibliography 175 Samenvatting (Summary in Dutch) 187

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