Koop-Poirier-Tobias (2003) Bayesian Semiparametric Inference in Multiple Equation Models

Summary

This paper generalizes the single-equation Bayesian partial linear model of Koop-Poirier to multiple-equation systems — seemingly unrelated regressions (SUR) and simultaneous-equations models with nonparametric components. Each nonparametric regression line's points are treated as unknown parameters, and a smoothness prior on each line delivers valid posterior inference despite there being more parameters than observations. The key structural move is that the semiparametric system can be written as a standard SUR model with an independent Normal-Wishart prior, so textbook results for estimation, model comparison, prediction, and computation apply directly; the smoothing hyperparameters are chosen by empirical Bayes.

Key Claims

• Semiparametric SUR. The system $y_{ij}=z_{ij}'\beta_j + f_j(x_{ij}) + \varepsilon_{ij}$ ($j=1,\dots,m$ equations), with each $f_j$ nonparametric, is recast — by treating the function values $\gamma_j=(f_j(x_{1j}),\dots)$ as parameters — as a standard SUR in $y=W\delta+\varepsilon$. As a (possibly restricted) reduced form, this also covers semiparametric simultaneous / triangular simultaneous-equations models, enabling nonparametric instrumental variables and endogeneity handling.
• Independent Normal-Wishart prior, not natural-conjugate. Unlike the single-equation case, in the SUR system the natural-conjugate prior imposes undesirable restrictions (it forces the same explanatory variables across equations / cross-equation covariance restrictions), so an independent Normal-Wishart prior is used — at the cost of requiring posterior simulation (Gibbs), but keeping everything within a textbook model.
• Smoothness prior supplies identification. As in the single-equation work, only prior information about each line's smoothness (a single hyperparameter $\eta_j$ per equation) is needed to make the over-parameterised regression's posterior proper.
• Empirical Bayes for the smoothing parameters. $\eta_j$ are estimated from the data by empirical Bayes (marginal-likelihood maximisation), avoiding subjective choice; the analytically-tractable single-equation building block keeps the $\eta$-search cheap.
• Application — returns to schooling. A two-equation structural model: a wage equation (nonparametric in TENURE) and a schooling equation (nonparametric in ABILITY), with schooling endogenous. Empirical Bayes selects $\eta_1=5\times10^{-6}$, $\eta_2=10^{-11}$; the error correlation (endogeneity) has posterior mean 0.102 (s.d. 0.142). A log Bayes factor of 4.645 favours the semiparametric model over linear-in-schooling.
• Substantive nonlinearity. The ability→schooling relation is strongly nonlinear: nearly flat for below-mean ability, steep above the mean — i.e. schooling choices of high-ability individuals are most responsive to ability. The estimated return to schooling (posterior mean 0.058) is slightly lower than the parametric estimate.

Concepts Introduced or Extended

• Bayesian Semiparametric Regression

Entities Mentioned

• Gary Koop
• Dale J. Poirier
• Justin L. Tobias

Quotes

"An advantage of our semiparametric model is that it is written as a seemingly unrelated regressions model with independent Normal-Wishart prior. Since this model is a common one, textbook results for posterior inference, model comparison, prediction and posterior computation are immediately available."

"Prior information about the smoothness of the nonparametric regression line was all that was required to ensure valid posterior inference."

My Take

The elegance carried over from Koop-Poirier (2004) is the same trick — nonparametric flexibility reduced to a familiar linear model plus a smoothness prior — but the multi-equation setting is where it earns its keep: once you have a system, you unlock nonparametric IV / endogeneity correction, which is the setting applied microeconometricians actually care about (the returns-to-schooling application is exactly the canonical endogenous-regressor problem). The honest tradeoff, clearly stated, is that the single-equation natural-conjugate magic (closed-form, no MCMC) does not survive intact: the SUR natural-conjugate prior is too restrictive, so they switch to an independent Normal-Wishart and pay for it with Gibbs sampling — though the analytic single-equation block still makes the empirical-Bayes $\eta$-search cheap. The substantive finding — that ability matters for schooling mainly in the upper tail — is a nice demonstration that the flexibility isn't just decorative: a quadratic-in-ability parametric model would have missed the flat-then-steep shape. It's the natural systems companion to the single-equation concept and belongs alongside it under Bayesian Semiparametric Regression.