Appendix M
Estimation of Tobit Model with Selection
(Source: Thomas [1])
The basic assumptions are that the error terms in Equations 5.4 and 5.5 are bivariate normal – that is, 
, 
– and are not correlated with the regressors. Therefore, a truncated bivariate normal distribution is used to derive the likelihood function for the Tobit model with selection. Specifically, an individual's conditional likelihood function is as follows:
Given the assumption of normality of the error terms, Equation (M.1) can be respecified as follows:
where
(M.3) 
and 
is the standard deviation of 
for segment 
, 
is the covariance between 
and 
for segment 
, and SBVN is the standard bivariate normal distribution.
Given the conditional likelihood function in Equation (M.2), Kamakura and Russell's [2] approach asserts that the unconditional likelihood can be determined by the following:
(M.4) 
such that
(M.5) 
where 
is the probability that a consumer is in segment 
, and 
is a segment-specific parameter. Based on this specification, the sample likelihood is
(M.6) 
where N is the total number of observations.