Appendix E
Type-1 Tobit Model
(Source: Franses and Paap [1])
For a type-1 Tobit model, the censored variable 
is 0 if the unobserved latent variable 
is less than or equal to 0 and 
if 
is positive,
(E.1) 
(E.2) 
with 
. For observations 
that are 0, we know only that
Maximum likelihood estimation is used to estimate the Tobit model. The likelihood function consists of two parts: the probability that an observation is censored is given by Equation E.3; and the density of the non-censored observations is a standard normal density. The likelihood function is
(E.4) 
where 
. It is more convenient to reparameterize the model according to 
and 
. The log-likelihood function in terms of 
reads
(E.5) 
The first-order derivatives of the log-likelihood function with respect to 
and 
are
(E.6) 
(E.7) 
and the second-order derivatives are
(E.8) 
(E.9) 
(E.10) 