9
Under the assumption that u
i
follows a normal distribution this equation corresponds to a
standard probit model. To generate data that correspond to a non-random selection
process, we need to introduce correlation between the stochastic components of these two
models. We accomplish this by taking draws from the bivariate normal with correlation
and transform one of the terms to follow an exponential distribution. Thus we start with
draws from the following bivariate normal distribution:
1
1
,
0
0
~
i
i
e
u
and then convert e
i
first to a uniform through the normal c.d.f., denoted by , and then to
a standard exponential by applying the inverse exponential c.d.f.:
))
(
-
log(1
-
i
i
e
=
.
9
We start with a total of 5000 observations on the independent variables, X and W,
which are jointly distributed according to a standard bivariate normal distribution with
correlation 0.5.
10
These variables are drawn once and then held fixed throughout the rest
of the simulations. We then generate 5000 draws from the joint error distribution and
determine the corresponding values of c
i
and y
i
. After applying the selection criterion,
about one third of the original 5000 observations remain. We then estimate our duration
models on the uncensored observations and save the estimated parameters. The process is
then repeated 1000 times for a specific value of . Once this is completed, the value of
is changed and 1000 simulations are performed for this new value.
In practice, most studies that use duration models do not rely on the exponential
model for a variety of reasons. Foremost among these is that this model assumes that the
9
We ran a series of simulated draws of the error terms alone and found that the correlation between the
resulting exponential and normal errors is about the same as the true correlation.
10
The distribution of the independent variables and the parameter values were selected to generate roughly
equal variation in the systematic and stochastic components of the models.
Convention