A Special Case of Independence
Dudley and Montmarquette (1976) analyzed whether or not the United States gives foreign aid to a particular country and, if it does, how much foreign aid it gives using a special case of the model (10.7.1), where the independence of uu and u2t is assumed. In their model the sign of y* determines whether aid is given to the rth country, and yf,- determines the actual amount of aid. They used the probit MLE to estimate (assuming ax = 1) and the least squares regression of y2i on x2i to estimate f)2. The LS estimator of ft2 is consistent in their model because of the assumed independence between uu and u2i. This makes their model computationally advantageous. However, it seems unrealistic to assume that the potential amount of aid, yf, is independent of the variable that determines whether or not aid is given, yf. This model is the opposite extreme of the T obit model, which can be regarded as a special case of Type 2 model where there is total dependence between у f and yf, in the whole spectrum of models (with correlation between yf and yf varying from — 1 to +1) contained in Type 2.
Because of the computational advantage mentioned earlier this “independence” model and its variations were frequently used in econometric applications in the 1960s and early 1970s. In many of these studies, authors made the additional linear probability assumption: P(y* > 0) = х’и^, which enabled them to estimate Д (as well asfi2) consistently by the least squares method. For examples of these studies, see the articles by Huang (1964) and Wu (1965).