A Gibbs Sampling Approach to Cointegration

Summary

Bauwens and Giot (1997) demonstrate the feasibility of Griddy-Gibbs sampling for Bayesian estimation of the cointegrated vector autoregression (VAR) model of Bauwens and Lubrano (1994a), replacing importance sampling with a Markov chain Monte Carlo (MCMC) approach. The paper also makes a methodological contribution on Gibbs convergence analysis using spectral diagnostics, and shows how blocking correlated parameters into a joint draw can reduce estimator variance 5–7x at roughly the same cost per unit of accuracy.

Key Claims

• The posterior density of the cointegrating matrix $\beta$ from Bauwens-Lubrano (1994a) has no closed form for $r > 1$; Gibbs sampling is feasible and converges well in their Belgium imports application.
• Griddy-Gibbs (evaluate full conditional on a grid, integrate numerically to get the cumulative distribution function (CDF), invert by interpolation) is a practical fallback when full conditionals have no analytical form.
• The Rao-Blackwell estimator $\hat\mu_i^c = k^{-1}\sum_j E(X_i \mid x_{-i}^{(j)})$ consistently dominates the direct estimator $\hat\mu_i = k^{-1}\sum_j x_i^{(j)}$ in variance; with Griddy-Gibbs it comes nearly for free.
• Sampler variance equals $S(0)/k$ where $S(0)$ is the spectral density at zero frequency of the sampled series; this can be estimated via spectral windows (truncated periodogram, Daniell, Tukey-Hanning) to give confidence intervals on posterior mean estimates.
• High posterior correlation between parameters ($\delta_f$, $\delta_d \approx -0.8$) causes high chain autocorrelation and slow convergence; drawing them jointly from the bivariate conditional reduces variance 5–7x.

Concepts Introduced or Extended

• Cointegration (Bauwens-Lubrano vector error-correction model (VECM) framework; posterior of $\beta$; identification)
• Gibbs Sampler (Griddy-Gibbs; Rao-Blackwell; blocked sampling; spectral convergence diagnostics)

Entities Mentioned

• Luc Bauwens

Quotes

"Gibbs sampling is relatively easy to implement for the researcher, although it may be more demanding in computing time than importance sampling (once a good importance function has been found)."

"The quality of these results must be examined by appropriate techniques."

My Take

This is primarily a computational methods paper from the early days of MCMC econometrics. The cointegration model itself is Bauwens-Lubrano (1994a); the contribution here is showing Gibbs is workable and investigating convergence rigorously via spectral analysis. The spectral diagnostic framework — estimating $S(0)/k$ to get quantitative confidence intervals on posterior means — is underused in later applied work, where CUSUM (cumulative sum) plots or simple autocorrelogram inspection dominate. The blocked sampler idea anticipates the more systematic treatment in Liu-Wong-Kong (1994), which Villani (2008) later uses explicitly.