Periodic autoregressive stochastic volatility

This paper proposes a stochastic volatility model (PAR-SV) in which the log-volatility follows a first-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 first 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 filtering. 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. Applications of Bayesian PAR-SV modeling to daily, quarterly and monthly S&P 500 returns are considered

Unified quasi-maximum likelihood estimation theory for stable and unstable Markov bilinear processes

A unified quasi-maximum likelihood (QML) estimation theory for stationary and nonstationary simple Markov bilinear (SMBL) models is proposed. Such models may be seen as generalized random coefficient autoregressions (GRCA) in which the innovation and the random coefficient processes are fully correlated. It is shown that the QML estimate (QMLE) for the SMBL model is always asymptotically Gaussian without assuming strict stationarity, meaning that there is no knife edge effect. The asymptotic variance of the QMLE is different in the stationary and nonstationary cases but is consistently estimated using the same estimator. A perhaps surprising result is that in the nonstationary domain, all SMBL parameters are consistently estimated in contrast with unstable GARCH and GRCA models where the QMLE of the conditional variance intercept is inconsistent. As a result, strict stationarity testing for the SMBL is studied. Simulation experiments and a real application to strict stationarity testing for some financial stock returns illustrate the theory in finite samples

Unified quasi-maximum likelihood estimation theory for stable and unstable Markov bilinear processes

A unified quasi-maximum likelihood (QML) estimation theory for stationary and nonstationary simple Markov bilinear (SMBL) models is proposed. Such models may be seen as generalized random coefficient autoregressions (GRCA) in which the innovation and the random coefficient processes are fully correlated. It is shown that the QML estimate (QMLE) for the SMBL model is always asymptotically Gaussian without assuming strict stationarity, meaning that there is no knife edge effect. The asymptotic variance of the QMLE is different in the stationary and nonstationary cases but is consistently estimated using the same estimator. A perhaps surprising result is that in the nonstationary domain, all SMBL parameters are consistently estimated in contrast with unstable GARCH and GRCA models where the QMLE of the conditional variance intercept is inconsistent. As a result, strict stationarity testing for the SMBL is studied. Simulation experiments and a real application to strict stationarity testing for some financial stock returns illustrate the theory in finite samples

Periodic autoregressive stochastic volatility

This paper proposes a stochastic volatility model (PAR-SV) in which the log-volatility follows a first-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 first 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 filtering. 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. Applications of Bayesian PAR-SV modeling to daily, quarterly and monthly S&P 500 returns are considered

Periodic autoregressive stochastic volatility

This paper proposes a stochastic volatility model (PAR-SV) in which the log-volatility follows a first-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 first 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 filtering. 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. Applications of Bayesian PAR-SV modeling to daily, quarterly and monthly S&P 500 returns are considered

Two-stage weighted least squares estimator of the conditional mean of observation-driven time series models

General parametric forms are assumed for the conditional mean λ_{t}(θ₀) and variance υ_{t}(ξ₀) of a time series. These conditional moments can for instance be derived from count time series, Autoregressive Conditional Duration (ACD) or Generalized Autoregressive Score (GAS) models. In this paper, our aim is to estimate the conditional mean parameter θ₀, trying to be as agnostic as possible about the conditional distribution of the observations. Quasi-Maximum Likelihood Estimators (QMLEs) based on the linear exponential family fulfill this goal, but they may be inefficient and have complicated asymptotic distributions when θ₀ contains zero coefficients. We thus study alternative weighted least square estimators (WLSEs), which enjoy the same consistency property as the QMLEs when the conditional distribution is misspecified, but have simpler asymptotic distributions when components of θ₀ are null and gain in efficiency when υ_{t} is well specified. We compare the asymptotic properties of the QMLEs and WLSEs, and determine a data driven strategy for finding an asymptotically optimal WLSE. Simulation experiments and illustrations on realized volatility forecasting are presented