Parameter inference for multivariate stochastic processes with jumps
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This dissertation addresses various aspects of estimation and inference for multivariate stochastic processes with jumps. The first chapter develops an unbiased Monte Carlo estimator of the transition density of a multivariate jump-diffusion process. The drift, volatility, jump intensity, and jump magnitude are allowed to be state-dependent and non-affine. The density estimator proposed enables efficient parametric estimation of multivariate jump-diffusion models based on discretely observed data. Under mild conditions, the resulting parameter estimates have the same asymptotic behavior as maximum likelihood estimators as the number of data points grows, even when the sampling frequency of the data is fixed. In a numerical case study of practical relevance, the density and parameter estimators are shown to be highly accurate and computationally efficient. In the second chapter, I examine continuous-time stochastic volatility models with jumps in returns and volatility in which the parameters governing the jumps are allowed to switch according to a Markov chain. I estimate the parameters and the latent processes using the S&P 500 and Nasdaq indices from 1990 to 2014. The Markov-switching parameters characterize well the periods of market stress, such as those in 1997-1998, 2001 and 2007-2010. Several statistical tests favor the model with Markov-switching jump parameters. These results provide empirical evidence about the state-dependent and time-varying nature of asset price jumps, a feature of asset prices that has recently been documented using high-frequency data. The third chapter considers applying Markov-switching affine stochastic volatility models with jumps in returns and volatility, where the jump parameters are not regime-switching. The estimation is performed via Markov Chain Monte Carlo methods, allowing to obtain the latent processes induced by the structure of the models. Furthermore, I propose some misspecification tests and develop a Markov-switching test based on the odds ratios. The parameters and the latent processes are estimated using the S&P 500 index from 1970 to 2014. I show that the S&P 500 stochastic volatility exhibits a Markov-switching behavior, and that most of the high volatility regimes coincide with the recessions identified ex-post by the National Bureau of Economic Research.