this paper we exploit Gibbs sampling to provide a likelihood framework for the analysis of stochastic volatility models, demonstrating how to perform either maximum likelihood or Bayesian estimation. The paper includes an extensive Monte Carlo experiment which compares the efficiency of the maximum likelihood estimator with that of quasi-likelihood and Bayesian estimators proposed in the literature. We also compare the fit of the stochastic volatility model to that of ARCH models using the likelihood criterion to illustrate the flexibility of the framework presented. Some key words: ARCH, Bayes estimation, Gibbs sampler, Heteroscedasticity, Maximum likelihood, Quasi-maximum likelihood, Simulation, Stochastic EM algorithm, Stochastic volatility, Stock returns. 1 INTRODUCTION

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Non-Gaussian processes of Ornstein-Uhlenbeck type, or OU processes for short, offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory. Keywords: Background driving L'evy process; Econometrics; L'evy density; L'evy process; Option pricing; OU process; Particle filter; Stochastic volatility; Subordination; Superposition. Authors' note: This paper supersedes our previously circulated but unpublished papers "Aggregation and model ...

In this paper we will rigourously study some of the properties of continuous time stochastic volatility models. We have five main results: (i) the stochastic volatility class can be linked to Cox process based models of tick-by-tick financial data; (ii) we characterise the moments, autocorrelation function and spectrum of squared returns; (iii) based only on discrete time returns, we give a simple consistent and asymptotically normally distributed estimator of continuous time volatility models without any simulation or discretisation error. Furthermore, we review a new class of Ornstein-Uhlenbeck processes of volatility, introduced in a companion paper, which allows (iv) the discrete time returns to be simulated without any form of discretisation error, (v) explicit modelling of correlation structures and allow analytic calculations of the properties of returns. 1 Contents 1

We develop a class of ARCH models for series sampled at unequal time intervals set by trade or quote arrivals. Our approach combines insights from the temporal aggregation for GARCH models discussed by Drost and Nijman (1993) and Drost and Werker (1994), and the autoregressive conditional duration model of Engle and Russell (1996) proposed to model the spacing between consecutive nancial transactions. The class of models introduced here will be called ACD-GARCH. It can be described as a random coefficient GARCH, or doubly stochastic GARCH, where the durations between transactions determine the parameter dynamics. The ACD-GARCH model becomes genuinely bivariate when past asset return volatilities are allowed to affect transaction durations and vice versa. Otherwise the spacings between trades are considered exogenous to the volatility dynamics. This assumption is required in a two-step estimation procedure. The bivariate setup enables us to test for Granger causality between volatility a...

The empirical application of Stochastic Volatility (SV) models has been limited due to the difficulties involved in the evaluation of the likelihood function. However, recently there has been fundamental progress in this area due to the proposal of several new estimation methods that try to overcome this problem, being at the same time, empirically feasible. As a consequence, several extensions of the SV models have been proposed and their empirical implementation is increasing. In this paper, we review the main estimators of the parameters and the volatility of univariate SV models proposed in the literature. We describe the main advantages and limitations of each of the methods both from the theoretical and empirical point of view. We complete the survey with an application of the most important procedures to the S&P 500 stock price index.

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In order to assess the e#ect of jumps on realised variance calculations, we study some of the econometric properties of time-changed Levy processes. We show that in general realised variance is an inconsistent estimator of the time-change, however we can derive the second order properties of realised variances and use these to estimate the parameters of such models. Our analytic results give a first indication of the degrees of inconsistency of realised variance as an estimator of the time-change in the non-Brownian case. Further, our results suggest volatility is even more predictable than has been shown by the recent econometric work on realised variance.

The efficient method of moments (EMM) estimation bases its inference on the versatile auxiliary model estimated by a seminonparametric (SNP) method. When the auxiliary model is appropriately chosen, the EMM estimates are near fully efficient and the EMM criterion can be used as a specification test on the structural model. In this paper, we propose a new specification test which allows us to choose an appropriate auxiliary model. Using the auxiliary model selected by our new test statistic, we test the stochastic volatility models fitted to daily S&P 500 stock price index. Our results indicate that while the standard stochastic volatility models are rejected overwhelmingly their extensions with thick tail errors cannot be rejected. 1 Introduction The idea of using the scores of an auxiliary model (called a score generator) to summarize systematically the characteristics of the data, and then confronting a structural model with this score generator is proposed in Bansal, Gallant, Hus...