Predicting Panel Data Binary Choice with the Gibbs Posterior

Abstract

This letter considers Bayesian binary classification where data are assumed to consist of multiple time series (panel data) with binary class labels (binary choice). The observed data can be represented as { y it , x it } T, t=1 i = 1, . . . , n. Here y it ∈ {0, 1} represents binary choices, and x it represents the exogenous variables. We consider prediction of y it by its own lags, as well as by the exogenous components. The prediction will be based on a Bayesian treatment using a Gibbs posterior that is constructed directly from the empirical error of classification. Therefore, this approach is less sensitive to the misspecification of the probability model compared to the usual likelihood-based posterior, which is confirmed by Monte Carlo simulations. We also study the effects of various choices of n and T both numerically (by simulations) and theoretically (by considering two alternative asymptotic situations: large n and large T ). We find that increasing T helps to reduce the prediction error more effectively compared to increasing n. We also illustrate the method in a real data application on the brand choice of yogurt purchases.

General Inequalities for Gibbs Posterior With Nonadditive Empirical Risk

“…2. Although we have assumed iid data for both GMM and ranking examples, in principle our general inequalities can be applied to dependent data or panel data and improve the convergence results of, e.g., Jiang and Tanner (2010) and Yao et al (2011). 3.…”

“…Compared to the posterior distribution derived from a likelihood-based procedure, the Gibbs posterior may no longer have the usual interpretation of conditional probability given observed data unless λR n (θ) is exactly the negative log-likelihood. However, it can achieve better risk performance under model misspecification compared to the likelihood-based Bayesian method, since the Gibbs posterior is directly associated with the risk function of interest (Jiang and Tanner, 2008;Yao, Jiang, and Tanner, 2011).…”

“…6) have established a nonasymptotic and assumption-free relation between the risk performance of Gibbs posterior and a probability of uniform large deviation, which allows for a theoretical study of the risk performance of Gibbs posterior in very general situations, including dependent data (e.g., Jiang and Tanner, 2010) and panel data (e.g., Yao et al, 2011). In principle this general relation is applicable to accommodate nonadditive empirical risks such as the GMM criterion function.…”