Truncated normal hurdle model


In econometrics, the truncated normal hurdle model is a variant of the Tobit model and was first proposed by Cragg in 1971.
In a standard Tobit model, represented as, where This model construction implicitly imposes two first order assumptions:
  1. Since: and, the partial effect of on the probability and the conditional expectation: has the same sign:
  2. The relative effects of and on and are identical, i.e.:
However, these two implicit assumptions are too strong and inconsistent with many contexts in economics. For instance, when we need to decide whether to invest and build a factory, the construction cost might be more influential than the product price; but once we have already built the factory, the product price is definitely more influential to the revenue. Hence, the implicit assumption doesn't match this context. The essence of this issue is that the standard Tobit implicitly models a very strong link between the participation decision or and the amount decision. If a corner solution model is represented in a general form:, where is the participate decision and is the amount decision, standard Tobit model assumes:
To make the model compatible with more contexts, a natural improvement is to assume:
where the error term is distributed as a truncated normal distribution with a density as
and are independent conditional on.
This is called Truncated Normal Hurdle Model, which is proposed in Cragg. By adding one more parameter and detach the amount decision with the participation decision, the model can fit more contexts. Under this model setup, the density of the given can be written as:
From this density representation, it is obvious that it will degenerate to the standard Tobit model when This also shows that Truncated Normal Hurdle Model is more general than the standard Tobit model.
The Truncated Normal Hurdle Model is usually estimated through MLE. The log-likelihood function can be written as:
From the log-likelihood function, can be estimated by a probit model and can be estimated by a truncated normal regression model. Based on the estimates, consistent estimates for the Average Partial Effect can be estimated correspondingly.