Many microeconometric applications (including binary, discrete choice, tobit, and generalized tobit analyses) involve the use of latent data. These latent data are unobserved by the econometrician, but the observed choices economic agents make typically impose some type of truncation or ordering among the latent variables.
In this chapter we show how the Gibbs sampler can be used to fit a variety of common microeconomic models involving the use of latent data. In particular, we review how data augmentation [see, e.g., Tanner and Wong (1987), Chib (1992), and Albert and Chib (1993)] can be used to simplify the computations in these models. Importantly, we recognize that many popular models in econometrics are essentially linear regression models on suitably defined latent data. Thus, conditioned on the latent data's values, we can apply all of the known techniques discussed in previous chapters (especially Chapter 10) for inference in the linear regression model. The idea behind data augmentation is to add (or augment) the joint posterior distribution with the latent data itself. In the Gibbs sampler, then, we can typically apply standard techniques to draw from the model parameters given the latent data, and then add an additional step to draw values of the latent data given the model parameters.
To review the idea behind data augmentation in general terms, suppose that we are primarily interested in characterizing the joint posterior p(θ|y) of a k-dimensional parameter vector θ.