Robust Inference with Quantile Regression in Stochastic Volatility Models with application to Value at Risk calculation

Abstract

Stochastic Volatility (SV) models play an integral role in modeling time varying volatility, with widespread application in finance. Due to the absence of a closed form likelihood function, estimation is a challenging problem. In the presence of outliers, and the high kurtosis prevalent in financial data, robust estimation techniques are desirable. Also, in the context of risk assessment when the underlying model is SV, computing the one step ahead predictive return densities for Value at Risk (VaR) calculation entails a numerically indirect procedure. The Quantile Regression (QR) estimation is an increasingly important tool for analysis, which helps in fitting parsimonious models in lieu of full conditional distributions. We propose two methods (i) Regression Quantile Method of Moments (RQMM) and (ii) Regression Quantile - Kalman Filtering method (RQ-KF) based on the QR approach that can be used to obtain robust SV model parameter estimates as well as VaR estimates. The RQMM is a simulation based indirect inference procedure where auxiliary recursive quantile models are used, with gradients of the RQ objective function providing the moment conditions. This was motivated by the Efficient Method of Moments (EMM) approach used in SV model estimation and the Conditional Autoregressive Value at Risk (CAViaR) method. An optimal linear quantile model based on the underlying SV assumption is derived. This is used along with other CAViaR specifications for the auxiliary models. The RQ-KF is a computationally simplified procedure combining the QML and QR methodologies. Based on a recursive model under the SV framework, quantile estimates are produced by the Kalman filtering scheme and are further refined using the RQ objective function, yielding robust estimates. For illustration purposes, comparison of the RQMM method with EMM under different data scenarios show that RQMM is stable under model misspecification, presence of outliers and heavy-tailedness. Comparison of the RQ-KF method with the existing QML method provide competitive results in terms of model estimation. Also, risk evaluation test results show desirable statistical properties of the quantile estimates obtained from these methods. Applications to real data and simulation studies on different parameter settings of the SV model provide empirical support in favor of the quantile model specifications. We also propose an algorithm, based on a Gram Charlier density approximation for the conditional predictive volatility density given past returns, to compute the one step ahead predictive return densities in the existing Nonlinear Filtering (NF) scheme. This approach is used in likelihood and VaR computations. This algorithm provides an alternative approximation in the reduction of the infinite-dimensional state vector and is based on fewer computational steps compared to the existing methods. Results based on the algorithm are comparable to existing methods.