Two-Step Estimator of Heckman
Heckman (1976a) proposed the two-step estimator of the reduced-form parameters (which we discussed in Section 10.8.1); but he also reestimated the labor supply model of Heckman (1974) using the structural equations version. Because (10.8.6) is a reduced-form as well as a structural equation, the estimation of fi2 is done in the same way as discussed in Section 10.8.1, namely, by applying least squares to the regression equation for E( W,Hf> 0) after estimating the argument of X (the hazard rate) by probit MLE. So we shall discuss only the estimation of (10.8.7), which we can rewrite as
Hi = y~lWt – г’,ау~1 ~ rlvt – (Ю.8.9)
By subtracting E(v,Hf > 0) from vt and adding the same, we can rewrite (10.8.9) further as
щ = ri Wi – Z’ay~‘ – olvo^rlK*uPJox)- Г’е„ (10.8.10)
where aXv = Cov (uu, vt), a = Vuu, and €t = vt — E(vtH* > 0). Then the consistent estimates of jr1, ay-1, and аХваї 1У~1 are obtained by the least squares regression applied to (10.8.10) after replacing fix /сг, by its probit MLE and Wf by Wt, the least squares predictor of Wt obtained by applying Heckman’s two-step estimator to (10.8.6). The asymptotic variance-covariaiice matrix of this estimator can be deduced from the results in the article by Heckman (1978), who considered the estimation of a more general model (which we shall discuss in the section on Type 5 Tobit models).
Actually, there is no apparent reason why we must first solve (10.8.7) for H, and proceed as indicated earlier. Heckman could justas easily have subtracted and added E(vtHf > 0) to (10.8.7) itself and proceeded similarly. This method would yield alternative consistent estimates. Inferring from the well – known fact that the two-stage least squares estimates of the standard simultaneous equations model yield asymptotically equivalent estimates regardless of which normalization is chosen, the Heckman two-step method applied to
(10.8.7) and (10.8.9) should also yield asymptotically equivalent estimates of у and a.
Lee, Maddala, and Trost (1980) extended Heckman’s simultaneous equations two-step estimator and its WLS version (taking account of the hetero – scedasticity) to more general simultaneous equations Tobit models and obtained their asymptotic variance-covariance matrices.