Mark J. Jensen

Bayesian Semiparametric Stochastic Volatility Modeling

This paper extends the existing fully parametric Bayesian literature on stochastic volatility to allow for more general return distributions. Instead of specifying a particular distribution for the return innovation, nonparametric Bayesian methods are used to flexibly model the skewness and kurtosis of the distribution while the dynamics of volatility continue to be modeled with a parametric structure. Our semiparametric Bayesian approach provides a full characterization of parametric and distributional uncertainty. A Markov chain Monte Carlo sampling approach to estimation is presented with theoretical and computational issues for simulation from the posterior predictive distributions. An empirical example compares the new model to standard parametric stochastic volatility models.

Bayesian semiparametric multivariate GARCH modeling

This paper proposes a Bayesian nonparametric modeling approach for the return distribution in multivariate GARCH models. In contrast to the parametric literature, the return distribution can display general forms of asymmetry and thick tails. An infinite mixture of multivariate normals is given a flexible Dirichlet process prior. The GARCH functional form enters into each of the components of this mixture. We discuss conjugate methods that allow for scale mixtures and nonconjugate methods, which provide mixing over both the location and scale of the normal components. MCMC methods are introduced for posterior simulation and computation of the predictive density. Bayes factors and density forecasts with comparisons to GARCH models with Student-t innovations demonstrate the gains from our flexible modeling approach.

Estimating a Semiparametric Asymmetric Stochastic Volatility Model with a Dirichlet Process Mixture

In this paper, we extend the parametric, asymmetric, stochastic volatility model (ASV), where returns are correlated with volatility, by flexibly modeling the bivariate distribution of the return and volatility innovations nonparametrically. Its novelty is in modeling the joint, conditional, return-volatility distribution with an infinite mixture of bivariate Normal distributions with mean zero vectors, but having unknown mixture weights and covariance matrices. This semiparametric ASV model nests stochastic volatility models whose innovations are distributed as either Normal or Student-t distributions, plus the response in volatility to unexpected return shocks is more general than the fixed asymmetric response with the ASV model. The unknown mixture parameters are modeled with a Dirichlet process prior. This prior ensures a parsimonious, finite, posterior mixture that best represents the distribution of the innovations and a straightforward sampler of the conditional posteriors. We develop a Bayesian Markov chain Monte Carlo sampler to fully characterize the parametric and distributional uncertainty. Nested model comparisons and out-of-sample predictions with the cumulative marginal-likelihoods, and one-day-ahead, predictive log-Bayes factors between the semiparametric and parametric versions of the ASV model shows the semiparametric model projecting more accurate empirical market returns. A major reason is how volatility responds to an unexpected market movement. When the market is tranquil, expected volatility reacts to a negative (positive) price shock by rising (initially declining, but then rising when the positive shock is large). However, when the market is volatile, the degree of asymmetry and the size of the response in expected volatility is muted. In other words, when times are good, no news is good news, but when times are bad, neither good nor bad news matters with regards to volatility.

Bayesian Inference of Long-Memory Stochastic Volatility via Wavelets

In this paper we are concerned with estimating the fractional order of integration associated with a long-memory stochastic volatility model. We develop a new Bayesian estimator based on the Markov chain Monte Carlo sampler and the wavelet representation of the log-squared returns to draw values of the fractional order of integration and latent volatilities from their joint posterior distribution. Unlike short-memory stochastic volatility models, long-memory stochastic volatility models do not have a state-space representation, and thus their sampler cannot employ the Kalman filters simulation smoother to update the chain of latent volatilities. Instead, we design a simulator where the latent long-memory volatilities are drawn quickly and efficiently from the near independent multivariate distribution of the long-memory volatilitys wavelet coefficients. We find that sampling volatility in the wavelet domain, rather than in the time domain, leads to a fast and simulation-efficient sampler of the posterior distribution for the volatilitys long-memory parameter and serves as a promising alternative estimator to the existing frequentist based estimators of long-memory volatility.

Robust estimation of nonstationary, fractionally integrated, autoregressive, stochastic volatility

Abstract Empirical volatility studies have discovered nonstationary, long-memory dynamics in the volatility of the stock market and foreign exchange rates. This highly persistent, infinite variance, but still mean reverting, behavior is commonly found with nonparametric estimates of the fractional differencing parameter, d, for financial volatility. In this paper, a fully parametric Bayesian estimator, robust to nonstationarity, is designed for the fractionally integrated, autoregressive, stochastic volatility (SV-FIAR) model. Joint estimates of the autoregressive and fractional differencing parameters of volatility are found via a Bayesian, Markov chain Monte Carlo (MCMC) sampler. Like [Jensen, M. J. 2004. “Semiparametric Bayesian Inference of Long-memory Stochastic Volatility.” Journal of Time Series Analysis 25: 895–922.], this MCMC algorithm relies on the wavelet representation of the log-squared return series. Unlike the Fourier transform, where a time series must be a stationary process to have a spectral density function, wavelets can represent both stationary and nonstationary processes. As long as the wavelet has a sufficient number of vanishing moments, this paper’s MCMC sampler will be robust to nonstationary volatility and capable of generating the posterior distribution of the autoregressive and long-memory parameters of the SV-FIAR model regardless of the value of d. Using simulated and empirical stock market return data, we find our Bayesian estimator producing reliable point estimates of the autoregressive and fractional differencing parameters with reasonable Bayesian confidence intervals for either stationary or nonstationary SV-FIAR models.

Measuring the Impact Intradaily Events Have on the Persistent Nature of Volatility

In this chapter we measure the effect a scheduled event, like the opening or closing of a regional foreign exchange market, or a unscheduled act, such as a market crash, a political upheaval, or a surprise news announcement, has on the foreign exchange rate’s level of volatility and its well documented long-memory behavior. Volatility in the foreign exchange rate is modeled as a non-stationary, long-memory, stochastic volatility process whose fractional differencing parameter is allowed to vary over time. This non-stationary model of volatility reveals that long-memory is not a spurious property associated with infrequent structural changes, but is a integral part of the volatility process. Over most of the sample period, volatility exhibits the strong persistence of a long-memory process. It is only after a market surprise or unanticipated economic news announcement that volatility briefly sheds its strong persistence.