Geweke-McCausland (2001) Bayesian Specification Analysis in Econometrics

Summary

Introduces Bayesian predictive specification analysis as a formal framework for econometric model checking. Rather than testing a null hypothesis, the analyst specifies a vector of diagnostically relevant statistics $\psi(y)$, computes their predictive (or posterior predictive) distribution under the model, and asks whether the observed $\psi(y^o)$ is plausible under that distribution. Illustrated by applying a Gaussian i.i.d. model and a t-GARCH model to stock returns; the former fails spectacularly while the latter misfits on kurtosis.

Key Claims

• Predictive vs. postpredictive. Predictive specification: compute $p(\psi | A) = \int p(\psi | \theta, A)\, p(\theta | A)\, d\theta$ (prior predictive); check whether $\psi(y^o)$ falls in a high-density region before seeing data. Postpredictive specification (posterior predictive checking): use $p(\tilde\psi | y^o, A) = \int p(\psi | \theta, A)\, p(\theta | y^o, A)\, d\theta$; checks whether the fitted model could regenerate the observed statistics. The postpredictive approach uses the data twice, potentially being too conservative.
• Vector of interest $\psi$. Rather than checking the full likelihood, the analyst selects a low-dimensional vector of statistics capturing the aspects of the data-generating process (DGP) the model claims to represent — here: first- and 20th-order volatility autocorrelations, volatility decay ratio, excess kurtosis, quantile ratio, skewness, and leverage coefficients (lead and lag). The choice of $\psi$ is an explicit modelling decision.
• Gaussian i.i.d. fails completely. All nine statistics of daily stock returns fall outside the 1%–99% predictive interval of the Gaussian i.i.d. model: volatility clustering (first-order volatility autocorrelation = 0.218, predictive median = 0.000) and fat tails (excess kurtosis = 22.4, predictive interval $(-0.083, 0.090)$) are missed by orders of magnitude.
• t-GARCH misfits on tail behaviour. The t-GARCH model fits first-order volatility autocorrelation well (data: 0.218, postpredictive median: 0.474, interval covers data). But the excess kurtosis postpredictive distribution has median 1312 and 25th–75th percentile (712, 2388), while the data kurtosis is 22.4 — the model dramatically over-predicts tail thickness when the degrees-of-freedom parameter is integrated out. The GARCH model effectively forces all kurtosis into the $t$ errors, producing an over-fat-tailed predictive.
• Model predictive density. Integrating over parameters produces the predictive density $p(y|A) = \int p(y|\theta,A)\,p(\theta|A)\,d\theta$; this is the natural Bayesian model evaluation criterion. The posterior predictive $p(\tilde y | y^o, A)$ is the postpredictive analogue. Both are evaluated via Markov chain Monte Carlo (MCMC) simulation.

Concepts Introduced or Extended

• Posterior Predictive Check
• GARCH and BEKK-GARCH

Entities Mentioned

• John Geweke
• William McCausland

Quotes

"All econometric models are wrong, but some are useful. They mirror only certain aspects of reality, and these imperfectly."

My Take

A clear exposition of Bayesian predictive specification analysis, but the paper is short (7 pages) and the contribution is mainly pedagogical — the framework was already present in Gelman-Meng-Stern (1996) and related work. The key value is the concrete application showing that t-GARCH over-generates kurtosis when degrees of freedom are integrated out; this is a practically important finding that helps explain why t-GARCH in-sample fits can look good on kurtosis statistics that are computed at the maximum likelihood estimate (MLE), where df is fixed, but fail when parameter uncertainty is accounted for. The "vector of interest" framing is useful for structuring model checks in practice.