Likelihood specification in simultaneous equation models for discrete data

Abstract

In this article, we examine the foundations of a rich and diverse literature in economics and derive the likelihood function of simultaneous equation models for discrete data as the invariant distribution of a suitably specified Markov process. This formulation offers a well-defined reduced form of the model and dispenses with the need for controversial recursivity requirements and ad hoc indeterminacy rules. The derivation resolves puzzling paradoxes highlighted in earlier work and shows that the likelihood is unique, proper, coherent, complete, and theoretically grounded in conditional distribution modeling – a framework that has yet to be popularized in economics. We note possible extensions and relevant links with other models, comment on computational issues, and implement the methodology in three empirical applications involving female labor force participation, the interactions between health and wealth, and the interplay between banks’ lending practices and their reliance on assistance from the lender of last resort during the Great Depression.