Joanna J.J. Wang

Stochastic volatility models with leverage and heavy-tailed distributions: A Bayesian approach using scale mixtures

This paper studies a heavy-tailed stochastic volatility (SV) model with leverage effect, where a bivariate Student-t distribution is used to model the error innovations of the return and volatility equations. Choy et al. (2008) studied this model by expressing the bivariate Student-t distribution as a scale mixture of bivariate normal distributions. We propose an alternative formulation by first deriving a conditional Student-t distribution for the return and a marginal Student-t distribution for the log-volatility and then express these two Student-t distributions as a scale mixture of normal (SMN) distributions. Our approach separates the sources of outliers and allows for distinguishing between outliers generated by the return process or by the volatility process, and hence is an improvement over the approach of Choy et al. (2008). In addition, it allows an efficient model implementation using the WinBUGS software. A simulation study is conducted to assess the performance of the proposed approach and its comparison with the approach by Choy et al. (2008). In the empirical study, daily exchange rate returns of the Australian dollar to various currencies and daily stock market index returns of various international stock markets are analysed. Model comparison relies on the Deviance Information Criterion and convergence diagnostic is monitored by Gewekes convergence test.

Modelling stochastic volatility using generalized t distribution

In modelling financial return time series and time-varying volatility, the Gaussian and the Student-t distributions are widely used in stochastic volatility (SV) models. However, other distributions such as the Laplace distribution and generalized error distribution (GED) are also common in SV modelling. Therefore, this paper proposes the use of the generalized t (GT) distribution whose special cases are the Gaussian distribution, Student-t distribution, Laplace distribution and GED. Since the GT distribution is a member of the scale mixture of uniform (SMU) family of distribution, we handle the GT distribution via its SMU representation. We show this SMU form can substantially simplify the Gibbs sampler for Bayesian simulation-based computation and can provide a mean of identifying outliers. In an empirical study, we adopt a GT–SV model to fit the daily return of the exchange rate of Australian dollar to three other currencies and use the exchange rate to US dollar as a covariate. Model implementation relies on Bayesian Markov chain Monte Carlo algorithms using the WinBUGS package.

Non-ignorable missingness in logistic regression

Nonresponses and missing data are common in observational studies. Ignoring or inadequately handling missing data may lead to biased parameter estimation, incorrect standard errors and, as a consequence, incorrect statistical inference and conclusions. We present a strategy for modelling non-ignorable missingness where the probability of nonresponse depends on the outcome. Using a simple case of logistic regression, we quantify the bias in regression estimates and show the observed likelihood is non-identifiable under non-ignorable missing data mechanism. We then adopt a selection model factorisation of the joint distribution as the basis for a sensitivity analysis to study changes in estimated parameters and the robustness of study conclusions against different assumptions. A Bayesian framework for model estimation is used as it provides a flexible approach for incorporating different missing data assumptions and conducting sensitivity analysis. Using simulated data, we explore the performance of the Bayesian selection model in correcting for bias in a logistic regression. We then implement our strategy using survey data from the 45 and Up Study to investigate factors associated with worsening health from the baseline to follow-up survey. Our findings have practical implications for the use of the 45 and Up Study data to answer important research questions relating to health and quality-of-life. Copyright

On the impact of nonresponse in logistic regression: application to the 45 and Up study

BackgroundIn longitudinal studies, nonresponse to follow-up surveys poses a major threat to validity, interpretability and generalisation of results. The problem of nonresponse is further complicated by the possibility that nonresponse may depend on the outcome of interest. We identified sociodemographic, general health and wellbeing characteristics associated with nonresponse to the follow-up questionnaire and assessed the extent and effect of nonresponse on statistical inference in a large-scale population cohort study.MethodsWe obtained the data from the baseline and first wave of the follow-up survey of the 45 and Up Study. Of those who were invited to participate in the follow-up survey, 65.2% responded. Logistic regression model was used to identify baseline characteristics associated with follow-up response. A Bayesian selection model approach with sensitivity analysis was implemented to model nonignorable nonresponse.ResultsCharacteristics associated with a higher likelihood of responding to the follow-up survey include female gender, age categories 55–74, high educational qualification, married/de facto, worked part or partially or fully retired and higher household income. Parameter estimates and conclusions are generally consistent across different assumptions on the missing data mechanism. However, we observed some sensitivity for variables that are strong predictors for both the outcome and nonresponse.ConclusionsResults indicated in the context of the binary outcome under study, nonresponse did not result in substantial bias and did not alter the interpretation of results in general. Conclusions were still largely robust under nonignorable missing data mechanism. Use of a Bayesian selection model is recommended as a useful strategy for assessing potential sensitivity of results to missing data.

Contaminated Variance-Mean mixing model

The Generalised Normal Variance-Mean (GNVM) model in which the mixing random variable is Gamma distributed is considered. This model generalises the popular Variance-Gamma (VG) distribution. This GNVM model can be interpreted as the addition of noise to a (skew) VG base. The discussion is based on goodness of fit criteria and on parameter estimation. The conclusion is that the shape of the VG distribution can be adjusted in a favourable way by adding noise.