6 Multivariate tobit models in econometrics

Abstract

This chapter provides a brief survey on multivariate tobit models. The term “tobit models” here refers solely to models with either censored or truncated dependent variables. Multivariate tobit models are models which generalize univariate tobit models to systems of equations. The chapter emphasizes on models formulated with neoclassical microeconomic theories. Consumer demand models are important members. The chapter provides some familiar multivariate and simultaneous equation models with multivariate tobit variables, which are generalizations of the classical multivariate regression model and the classical linear simultaneous equation model, and discusses the formulation of consumer demand systems and production systems, which are compatible with microeconomic theories and incorporate features of microeconomic data with zero expenditures or kink points. The derived equation systems are essentially nonlinear simultaneous equations with complicated cross-equation constraints. Some of the simultaneous equation tobit models are nonlinear in variables. Such models may not be well-defined stochastic models without appropriate restrictions on some of the structural parameters. This is an issue on model coherency. The estimation of multivariate tobit models by the classical maximum likelihood method is known to be computationally inefficient, as the computation of multivariate normal probabilities of high dimensions is difficult. The chapter reviews some recent developments on estimation methods that are computationally tractable, and model specification strategies that provide computationally tractable and reasonably flexible structures. Instrumental variable estimation methods and simulated methods of moments are two of such methods. Lagrange multiplier tests are computationally simple diagnostic tests for such models. Some of such tests are presented.