Periodic autoregressive stochastic volatility

This paper proposes a stochastic volatility model (PAR-SV ) in which the log-volatility follows a …rst-order periodic autoregression. This model aims at representing time series with volatility displaying a stochastic periodic dynamic structure, and may then be seen as an alternative to the familiar periodic GARCH process. The probabilistic structure of the proposed PAR-SV model such as periodic stationarity and autocovariance structure are …rst studied. Then, parameter estimation is examined through the quasi-maximum likelihood (QML) method where the likelihood is evaluated using the prediction error decomposition approach and Kalman …ltering. In addition, a Bayesian MCMC method is also considered, where the posteriors are given from conjugate priors using the Gibbs sampler in which the augmented volatilities are sampled from the Griddy Gibbs technique in a single-move way. As a-by-product, period selection for the PAR-SV is carried out using the (conditional) Deviance Information Criterion (DIC). A simulation study is undertaken to assess the performances of the QML and Bayesian Griddy Gibbs estimates in …nite samples while applications of Bayesian PAR-SV modeling to daily, quarterly and monthly S&P 500 returns are considered. Keywords and phrases: Periodic stochastic volatility, periodic autoregression, QML via prediction error decomposition and Kalman …ltering, Bayesian Griddy Gibbs sampler, single-move approach, DIC. Mathematics Subject Classi…cation: AMS 2000 Primary 62M10; Secondary 60F99 Proposed running head: Periodic AR Stochastic volatility. 1. Introduction Over the past three decades, stochastic volatility (SV ) models introduced by Taylor (1982) have played an important role in modelling …nancial time series which are characterized by a time-varying volatility feature. Faculty of Mathematics, University of Science and Technology Houari Boumediene, Algiers, Algeria, e-mail: [email protected]. yThis is a long version of the paper Aknouche (2017) [Periodic autoregressive stochastic volatility. Statistical Inference for Stochastic Processes, 20, 139-177)] which contains many materials submitted in earlier versions but do not appear in the published paper.