Estimation and Inference for Higher-Order Stochastic Volatility Models With Leverage

Abstract

Statistical inference—estimation and testing—for stochastic volatility models is challenging and computationally expensive. This problem is compounded when leverage effects are allowed. We propose efficient, simple estimators for higher-order stochastic volatility models with leverage [SVL$\left(p\right)$], based on a small number of moment equations derived from ARMA representations associated with SVL models, along with the possibility of using “winsorization” to improve stability and efficiency (W-ARMA estimators). The asymptotic distributional theory of the estimators is derived. The computationally simple estimators proposed allow one to easily perform simulation-based (possibly exact) tests, such as Monte Carlo tests (MCTS) or bootstrap procedures. In simulation experiments, we show that: (1) the proposed W-ARMA estimators dominate alternative estimators (including Bayesian estimators) in terms of bias, root-mean-square error, and computation time; (2) local and maximized Monte Carlo tests based on W-ARMA estimators yield good control of the size and power of LR-type tests; (3) taking into account leverage improves volatility forecasting. The methods developed are applied to daily returns for three major stock indices (S&P 500, Dow Jones, Nasdaq), confirming the superiority of SVL$\left(p\right)$ models over competing conditional volatility models in terms of forecast accuracy.