I emphasize to the students that, first and foremost, the reason we use the probit and logit models
is to obtain more reasonable functional forms for the response probability.
Once we move to a
nonlinear model with a fully specified conditional distribution, it makes sense to use the efficient
estimation procedure, maximum likelihood.
It is important to spend some time on interpreting
probit and logit estimates.
In particular, the students should know the rules-of-thumb for
comparing probit, logit, and LPM estimates. Beginners sometimes mistakenly think that,
because the probit and especially the logit estimates are much larger than the LPM estimates, the
explanatory variables now have larger estimated effects on the response probabilities than in the
This may or may not be true, and can only be determined by computing partial
effects for the probit and logit models. The two most common ways of compute the partial
effects are the so-called “average partial effects” (APE), where the partial effects for each unit
are averaged across all observations (or interesting subsets of observations), and the “partial
effects at the average” (PAE). The PAEs are routinely computed by many econometrics
packages, but they seem to be less useful than the APEs. The APEs have a more straightforward
meaning in most cases and are more comparable to linear probability model estimates.
I view the Tobit model, when properly applied, as improving functional form for corner solution
(I believe this motivated Tobin’s original work, too.) In most cases, it is wrong to
view a Tobit application as a data-censoring problem (unless there is true data censoring in
collecting the data or because of institutional constraints).
For example, in using survey data to
estimate the demand for a new product, say a safer pesticide to be used in farming, some farmers
will demand zero at the going price, while some will demand positive pounds per acre.
no data censoring here: some farmers simply find it optimal to use none of the new pesticide.
The Tobit model provides more realistic functional forms for E(
) and E(
) than a linear
With the Tobit model, students may be tempted to compare the Tobit estimates
with those from the linear model and conclude that the Tobit estimates imply larger effects for
the independent variables.
But, as with probit and logit, the Tobit estimates must be scaled down
to be comparable with OLS estimates in a linear model.
[See Equation (17.27); for examples,
see Computer Exercise C17.3 and C17.12 The latter goes through calculation an average partial
effect for a (roughly) continuous explanatory variable.]
Poisson regression with an exponential conditional mean is used primarily to improve over a
linear functional form for E(
) for count data.
The parameters are easy to interpret as semi-
elasticities or elasticities.
If the Poisson distributional assumption is correct, we can use the
Poisson distribution to compute probabilities, too.