Journal of Business & Economic Statistics,July 1987, Vol.5, No. 3 Estimation of Seemingly Unrelated Tobit Regressions via the EM Algorithm Cliff J. Huang and Frank A. Sloan Department of Economics, Vanderbilt University, Nashville, TN 37235 KillardW. Adamache Department of Health Economics, Virginia Commonwealth University, Richmond, VA 23298 An expectation-maximum (EM) likelihood algorithm is used to estimatetwo seemingly unrelated Tobit regressions inwhich the dependent variables are truncated normal. Anillustrative example on the determination of the life-health insurance and pension benefits is also given. KEYWORDS:Truncated normal; Conditional expectation; Maximum likelihood estimation; Suf- ficient statistics; Latent variables; Censored models. 1. INTRODUCTION In this article we consider the estimation of two seem- ingly unrelatedTobit regressions in which the depen- dent variables are truncated normal.The model is use- ful, since it can be viewed as the reduced form of a simultaneous-equations Tobit model. The proposed es- timation method and algorithm are interesting in them- selves for the following reasons. In the estimation of a simultaneous equations model, for example, Nelson and Olson (1978) proposed a procedureanalogous to the two-stage least squares method. In the first stage of estimating the reduced form, however, the disturb- ances are assumed uncorrelated; hence some iterative algorithms of maximum likelihood or instrumental-vari- able techniques apply to eachof the reduced-form equa- tions separately. A more efficient first-stage estimate of the truncated dependent variablescan be obtained by taking into account the nonzerocovariance between the disturbances, even though the regressors are iden- tical in the differentTobit regressions. In this article, the expectation-maximization (EM) algorithm of Dempster, Laird, and Rubin (1977) is ap- plied to compute the maximum likelihoodestimatesin the case of nonzero covariance. We then provide an illustrative example on the determination of life-health insurance and pension benefits. 2. THE MODEL AND THE EM ALGORITHM Considerthe estimationof two seemingly unrelated Tobit regressions, Z1, _ Xit 0 B1 + Ul Z2t 0 X2t B2 U2t or Z, = X, B + U,, (1) where Z, is a vector of random latent variables and the matrix of exogenous variables Xt is assumed to be non- stochastic.The observeddata Yit are related to the la- tent variable Zi, by the observation function Y(Zit), Yi = Y(Zi,) = Z, iff Z, > 0 = 0iff Zi, 0. (2) The vector of disturbances, U' = (U1t U2t), is assumed to be distributed according to N(O, L), where (T11 O 12 (021 Cr22 and E(U, Uj,') = 0 for s : s'. Naturally, the model (1), along with the observation function (2), can be considered as the reducedform of a simultaneous-equations model, such as the model of Amemiya (1974) or the model of Nelson and Olson (1978). The model of seemingly unrelated Tobit regres- sion (SUTR) considered here conforms to the reduced form of the latter model, but not to Amemiya's. The reduced form of the Amemiya model is different in the sense that the observationfunction (2) is replacedby Yit = Y(Zit) = Zit if all observed dependent variables are positive and Yit = 0 otherwise. Let 0 = (B, E) be the unknown parameter set. The likelihood function of the latent variables Z = (Z1, ..., Zn) is L(Z; 0) = (270)-n 1l'-n/2 x exp - Yij Zit Zt 2 _ Z i j t=l - 2 j 'i ( SitZ.i)Bi i j t=l + .i B ( XtXjt)Bj }, i 1 t=l (3) 425 ? 1987 American Statistical Association