Estimation of Seemingly Unrelated Tobit Regressions via the EM Algorithm

In this article we consider the estimation of two seemingly unrelated Tobit regressions in which the dependent variables are truncated normal. The model is useful, since it can be viewed as the reduced form of a simultaneous-equations Tobit model. The proposed estimation method and algorithm are interesting in themselves for the following reasons. In the estimation of a simultaneous equations model, for example, Nelson and Olson (1978) proposed a procedure analogous to the two-stage least squares method. In the first stage of estimating the reduced form, however, the disturbances are assumed uncorrelated; hence some iterative algorithms of maximum likelihood or instrumental-variable techniques apply to each of the reduced-form equations separately. A more efficient first-stage estimate of the truncated dependent variables can be obtained by taking into account the nonzero covariance between the disturbances, even though the regressors are identical in the different Tobit regressions. In this article, the expectation-maximization (EM) algorithm of Dempster, Laird, and Rubin (1977) is applied to compute the maximum likelihood estimates in the case of nonzero covariance. We then provide an illustrative example on the determination of life-health insurance and pension benefits.