Fitting complex stochastic volatility models using Laplace approximation

The paper proposes the use of Laplace approximation (LA) to estimate complex univariate symmetric and asymmetric stochastic volatility (SV) models with flexible distributions for standardized returns. LA is a method for approximating integrals, especially in Bayesian statistics, and is often used to approximate the posterior distribution of the model parameters. This method simplifies complex problems by focusing on the most critical areas and using a well-understood approximation. We show how easily complex SV models can be estimated and analyzed using LA, with changes to specifications, priors, and sampling error distributions requiring only minor changes to the code. The simulation study shows that the LA estimates of the model parameters are close to the true values in finite samples and that the proposed estimator is computationally efficient and fast. It is an effective alternative to existing estimation methods for SV models. Finally, we evaluate the in-sample and out-of-sample performance of the models by forecasting one-day-ahead volatility. We use four well-known energy index series: two for clean energy and two for conventional (brown) energy. In the out-of-sample analysis, we also examine the impact of climate policy uncertainty and energy prices on the volatility forecasts. The results support the use of asymmetric SV models for clean energy series and symmetric SV models for brown energy indices conditional on these state variables.