Econometrica: Jul 2004, Volume 72, Issue 4

Efficient Semiparametric Estimation of Censored and Truncated Regressions via a Smoothed Self‐Consistency Equation
p. 1277-1293

Stephen R. Cosslett

An asymptotically efficient likelihood‐based semiparametric estimator is derived for the censored regression (tobit) model, based on a new approach for estimating the density function of the residuals in a partially observed regression. Smoothing the self‐consistency equation for the nonparametric maximum likelihood estimator of the distribution of the residuals yields an integral equation, which in some cases can be solved explicitly. The resulting estimated density is smooth enough to be used in a practical implementation of the profile likelihood estimator, but is sufficiently close to the nonparametric maximum likelihood estimator to allow estimation of the semiparametric efficient score. The parameter estimates obtained by solving the estimated score equations are then asymptotically efficient. A summary of analogous results for truncated regression is also given.

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