Bauwens-Lubrano (1998) Bayesian Inference on GARCH Models Using the Gibbs Sampler

Summary

Bauwens and Lubrano show how the Gibbs sampler can perform Bayesian inference on GARCH models even though the full conditional posterior densities are not of any known analytical form. Their solution combines Gibbs sampling with a one-dimensional deterministic numerical-integration rule applied to each coordinate: each univariate full conditional is evaluated on a grid, numerically integrated to obtain its distribution function, and inverted to produce a draw (the "griddy-Gibbs" sampler of Ritter and Tanner). The method is automatic — it needs only pointwise evaluation of the conditional density, not a tuned global approximation — and is shown to be feasible and competitive with importance sampling and the Metropolis–Hastings algorithm. It is applied to an asymmetric Student-t GARCH model for a stock-index return and used to compute predictive option prices; the paper also proves that a flat prior on the degrees-of-freedom parameter produces an improper posterior.

Key Claims

• GARCH breaks conjugate Gibbs. In a regression model with GARCH errors, the conditional posterior of one parameter block given the others is not a standard distribution, so the usual Gibbs sampler (which draws from known conditionals) cannot be applied directly.
• Griddy-Gibbs solution. Apply a univariate deterministic integration rule to each coordinate: evaluate the (unnormalized) full conditional on a grid of points, numerically integrate to get the cumulative distribution function, then invert it at a uniform draw. Cycling over coordinates yields a Markov chain converging to the joint posterior. Only pointwise density evaluation is required.
• Automatic and competitive. Unlike importance sampling or Metropolis–Hastings, griddy-Gibbs needs no carefully tuned proposal/importance density; the paper shows it is feasible and competitive in accuracy and effort for GARCH posteriors, and yields smooth marginal density estimates.
• Asymmetric Student-t GARCH application. Estimated on a stock-exchange index return with fat tails and an asymmetry (leverage) term, and used to compute Bayesian predictive option prices on the index from the posterior predictive density of returns.
• Improper-posterior warning. A flat (improper) prior on the Student-t degrees-of-freedom parameter $\nu$ leads to an improper posterior; $\nu$ must be given a prior that ensures integrability.

Concepts Introduced or Extended

• Griddy-Gibbs Sampler
• GARCH and BEKK-GARCH
• Gibbs Sampler

Entities Mentioned

• Luc Bauwens
• Michel Lubrano

Quotes

"We show that the Gibbs sampler can be combined with a unidimensional deterministic integration rule applied to each coordinate of the posterior density."

"We prove, moreover, that a flat prior on the degrees of freedom parameter leads to an improper posterior density."

My Take

This is one of the papers that made Bayesian GARCH practical before Metropolis-within-Gibbs and adaptive samplers became routine. The insight is pragmatic: for GARCH you can always evaluate a full conditional up to a constant, so rather than search for a good global proposal, just grid the univariate conditional and invert its numerical CDF — no tuning, no acceptance rate to babysit. The cost is a grid per coordinate per sweep (which scales poorly in dimension), which is why later work moved to Metropolis-within-Gibbs and reparameterized block samplers, but the recipe is still the cleanest way to explain Bayesian GARCH estimation. The lasting theoretical residue is the improper-posterior result on the Student-t degrees of freedom — a standard citation whenever someone reaches for a "flat" prior on $\nu$. It anchors the Bayesian-GARCH line that runs through Bauwens–Lubrano–Richard (1999) to the modern GARCH and stochastic-volatility MCMC literature.