The generic form of a bivariate limited dependent variable model is
where the disturbances, and , have joint normal distribution with zero mean, standard deviations and , and correlation of . and are latent variables. The dependent variables and are observed if the latent variables and fall in certain ranges:
is a transformation from to . For example, if and are censored variables with lower bound 0, then
There are three cases for the log likelihood of . The first case is that and . That is, this observation is mapped to one point in the space of latent variables. The log likelihood is computed from a bivariate normal density,
where is the density function for standardized bivariate normal distribution with correlation ,
The second case is that one observed dependent variable is mapped to a point of its latent variable and the other dependent variable is mapped to a segment in the space of its latent variable. For example, in the bivariate censored model specified, if observed and , then and . In general, the log likelihood for one observation can be written as follows (the subscript i is dropped for simplicity): If one set is a single point and the other set is a range, without loss of generality, let and ,
where and are the density function and the cumulative probability function for standardized univariate normal distribution.
The third case is that both dependent variables are mapped to segments in the space of latent variables. For example, in the bivariate censored model specified, if observed and , then and . In general, if and , the log likelihood is