The COUNTREG Procedure

In the zero-inflated Poisson (ZIP) regression model, the data generation process referred to earlier as Process 2 is

where . Thus the ZIP model is defined as

The conditional expectation and conditional variance of are given by

Note that the ZIP model (as well as the ZINB model) exhibits overdispersion since .

In general, the log-likelihood function of the ZIP model is

Once a specific link function (either logistic or standard normal) for the probability is chosen, it is possible to write the exact expressions for the log-likelihood function and the gradient.

### ZIP Model with Logistic Link Function

First, consider the ZIP model in which the probability is expressed with a logistic link function, namely

The log-likelihood function is

The gradient for this model is given by

### ZIP Model with Standard Normal Link Function

Next, consider the ZIP model in which the probability is expressed with a standard normal link function: . The log-likelihood function is

The gradient for this model is given by

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