Advanced Econometrics Takeshi Amemiya
Model of Nelson and Olson
The empirical model actually estimated by Nelson and Olson (1978) is more general than Type 4 and is a general simultaneous equations Tobit model (10.9.4). The Nelson-Olson empirical model involves four elements of the vector y*:
yf Time spent on vocational school training, completely observed if yf > 0, and otherwise observed to lie in the interval (—°°, 0] yf Time spent on college education, observed to lie in one of the three intervals (—<», 0], (0, 1], and (1, °°) yf Wage, always completely observed yf Hours worked, always completely observed
These variables are related to each other by simultaneous equations. However, they merely estimate each reduced-form equation seperately by various appropriate methods and obtain the estimates of the structural parameters from the estimates of the reduced-form parameters in an arbitrary way.
The model that Nelson and Olson analyzed theoretically in more detail is the two-equation model:
У и = УіУн + хн«і + vu
and
Ун = УіУи + *2І<*2 + v2i,
where y2l is always observed and y* is observed to be yu if У и > 0. This model may be used, for example, if we are interested in explaining only yf and y* in the Nelson-Olson empirical model. The likelihood function of this model may be characterized by P(y{ < 0, y2) • Р(Уі, у2), and therefore, the model is a special case of Type 4.
Nelson and Olson proposed estimating the structural parameters of this model by the following sequential method:
Step 1. Estimate the parameters of the reduced-form equation foryfby the Tobit MLE and those of the reduced-form equation for y2 by LS.
Step 2. Replace y2i in the right-hand side of (10.9.5) by its LS predictor obtained in step 1 and estimate the parameters of (10.9.5) by the Tobit MLE.
Step 3. Replace yf, in the right-hand side of (10.9.6) by its predictor obtained in step 1 and estimate the parameters of (10.9.6) by LS.
Amemiya (1979) obtained the asymptotic variance-covariance matrix of the Nelson-Olson estimator and showed that the Amemiya GLS (see Section 10.8.4) based on the same reduced-form estimates is asymptotically more efficient.
