This note is concerned with specification for modelling financial leverage effect in the context of stochastic volatility (SV) models. Two alternative specifications coexist in the literature. One is the Euler approximation to the well known continuous time SV model with leverage effect and the other is the discrete time SV model of Jacquier, Polson and Rossi (2004, Journal of Econometrics, forthcoming). Using a Gaussian nonlinear state space form with uncorrelated measurement and transition errors, I show that it is easy to interpret the leverage e#ect in the conventional model whereas it is not clear how to obtain the leverage effect in the model of Jacquier et al. Empirical comparisons of these two models via Bayesian Markov chain Monte Carlo (MCMC) methods reveal that the specification of Jacquier et al is inferior. Simulation experiments are conducted to study the sampling properties of the Bayes MCMC for the conventional model.

The literature on multivariate stochastic volatility (MSV) models has developed significantly over the last few years. This paper reviews the substantial literature on specification, estimation, and evaluation of MSV models. A wide range of MSV models is presented according to various categories, namely, (i) asymmetric models, (ii) factor models, (iii) time-varying correlation models, and (iv) alternative MSV specifications, including models based on the matrix exponential transformation, the Cholesky decomposition, and the Wishart autoregressive process. Alternative methods of estimation, including quasi-maximum likelihood, simulated maximum likelihood, and Markov chain Monte Carlo methods, are discussed and compared. Various methods of diagnostic checking and model comparison are also reviewed.

Bayesian methods have been efficient in estimating parameters of stochastic volatility models for analyzing financial time series. Recent advances made it possible to fit stochastic volatility models of increasing complexity, including covariates, leverage effects, jump components and heavy-tailed distributions. However, a formal model comparison via Bayes factors remains difficult. The main objective of this paper is to demonstrate that model selection is more easily performed using the deviance information criterion (DIC). It combines a Bayesian measure-of-fit with a measure of model complexity. We illustrate the performance of DIC in discriminating between various different stochastic volatility models using simulated data and daily returns data on the S&P100 index.

Abstract. Although stochastic volatility (SV) models have an intuitive appeal, their empirical application has been limited mainly due to difficulties involved in their estimation. The main problem is that the likelihood function is hard to evaluate. However, recently, several new estimation methods have been introduced and the literature on SV models has grown substantially. In this article, we review this literature. We describe the main estimators of the parameters and the underlying volatilities focusing on their advantages and limitations 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.

In this paper, we propose a new stochastic volatility model, called A-LMSV, to cope simultaneously with the leverage effect and long-memory. We derive its statistical properties and compare them with the properties of the FIEGARCH model. We show that the dependence of the autocorrelations of squares on the parameters measuring the asymmetry and the persistence is different in both models. The kurtosis and autocorrelations of squares do not depend on the asymmetry in the A-LMSV model while they increase with the asymmetry in the FIEGARCH model. Furthermore, the autocorrelations of squares increase with the persistence in the A-LMSV model and decrease in the FIEGARCH model. On the other hand, the autocorrelations of absolute returns increase with the magnitude of the asymmetry in the FIEGARCH model while they can increase or decrease depending on the sign of the asymmetry in the L-MSV model. Finally, the cross-correlations between squares and original observations are, in general, larger in the FIEGARCH model than in the A-LMSV model. The results are illustrated by fitting both models to represent the dynamic evolution of volatilities of daily returns of the S&P500 and DAX indexes.

We show that the fitting of nonlinear hierarchical random effects models by max-imum likelihood can be made automatic to the same extent that Bayesian model fitting can be automated by the program BUGS. The word ‘automatic ’ here means that the technical details of computation are made transparent to the user. We achieve this by combining a technique from computer science known as ‘automatic differentiation ’ with the Laplace approximation for calculating the marginal like-lihood. Automatic differentiation, which should not be confused with symbolic differentiation, is mostly unknown to statisticians, and hence we review basic ideas and results. A software prototype that implements our approach has been devel-oped, and the computational performance is reported for a selection of models from the mixed-model literature. In general, our approach performs well in comparison with existing software.

This document is copyrighted by the American Psychological Association or one of its allied publishers. This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. Bayesian estimation for 2 groups provides complete distributions of credible values for the effect size, group means and their difference, standard deviations and their difference, and the normality of the data. The method handles outliers. The decision rule can accept the null value (unlike traditional t tests) when certainty in the estimate is high (unlike Bayesian model comparison using Bayes factors). The method also yields precise estimates of statistical power for various research goals. The software and programs are free and run on Macintosh, Windows, and Linux platforms.

This study investigates a probabilistic approach for the inverse problem associated with blending time-dependent dynamic models of marine ecosystems with observations. The goal is to combine prior information, in the form of model dynamics and substantive knowledge about uncertain parameters, with available measurements in order to produce posterior estimates of the time-varying ecological state variables, along with their uncertainty. Ecological models of interacting populations are considered in the context of the nonlinear, non-Gaussian state space model. This comprises a nonlinear stochastic difference equation for the model dynamics, and an observation equation relating the model state to the measurements. Complex error processes are readily incorporated. The posterior probability density function provides a complete solution to the inverse problem. Bayes' theorem allows one to obtain this posterior density through synthesis of the prior information and the observations. To illustrate this Bayesian inverse method, these ideas are applied to a simple ecosystem box model concerned with predicting the seasonal co-evolution of a population of grazing shellfish and its two food sources, plankton and detritus. Observations of shellfish growth over time are available. Lognormal system noise was incorporated into the ecosystem equations at all time steps. Ingestion and respiration parameters for shellfish growth are considered as uncertain quantities described by beta distributions. Stochastic simulation was carried out and provided predictions of the model state with uncertainty estimates. The Bayesian inverse method was then used to assimilate the additional information contained in the observations. Posterior probability density functions for the parameters and time-var...

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Paper No. 30 – 2012 smu economics & statistics working paper series