# Efficient Bayesian estimation for GARCH-type models via Sequential Monte Carlo

## TL;DR

This paper introduces an efficient Bayesian estimation method for GARCH models using Sequential Monte Carlo, enabling better uncertainty quantification, model selection, and exact inference for complex models.

## Contribution

It develops novel SMC-based algorithms for GARCH parameter estimation, model selection, and likelihood estimation, addressing limitations of classical methods.

## Key findings

- Posterior distributions are non-normal, justifying Bayesian methods.
- Proposed methods outperform classical inference in long time series.
- Unbiased likelihood estimator enables exact Bayesian inference for complex models.

## Abstract

The advantages of sequential Monte Carlo (SMC) are exploited to develop parameter estimation and model selection methods for GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) style models. It provides an alternative method for quantifying estimation uncertainty relative to classical inference. Even with long time series, it is demonstrated that the posterior distribution of model parameters are non-normal, highlighting the need for a Bayesian approach and an efficient posterior sampling method. Efficient approaches for both constructing the sequence of distributions in SMC, and leave-one-out cross-validation, for long time series data are also proposed. Finally, an unbiased estimator of the likelihood is developed for the Bad Environment-Good Environment model, a complex GARCH-type model, which permits exact Bayesian inference not previously available in the literature.