Implications for Empirical Estimation:

Before proceeding further let us pause to look at some distinguishing characteristics of  this model. First suppose the variance of E,  is multiplied by some positive constant k².  The latent regressioi will  now be,  Y_i* = x_i,β +kε_r. But,  Y*/k = x_i,β + ε_r is the same model with the same data. The observed data will remain as it is. Y,  is still 0 or 1, depending only on  the sign of Y_i*  not on  its scale. In other words the data have no information about k and so it cannot be estimated. Secondly, if the model contains a constant term, say a, then,  

Since  α  is  unknown,  the difference  (a - c)  remains  an unknown  parameter. Therefore, we may write

The problem becomes one of choosing  an appropriate probability distribution  for ε_i. Successful estimation  of  the model ,outlined  above is  dependent on  making  the correct choice  of  functional  fdnh  for  G(x_i,β)  or  the appropriate  probability distribution for ε_r. Choices made myst  be  such that the predicted  probabilities are consistent with  the underlying theory. Choosing G(x_i,β)  to be any proper, continuous  probability distribution, say F(x_i,β),  defined over the real line will ensure logical consistency.