The COUNTREG Procedure |

Poisson Regression |

The most widely used model for count data analysis is Poisson regression. This assumes that , given the vector of covariates , is independently Poisson distributed with

and the mean parameter — that is, the mean number of events per period — is given by

where is a parameter vector. (The intercept is ; the coefficients for the regressors are .) Taking the exponential of ensures that the mean parameter is nonnegative. It can be shown that the conditional mean is given by

The name *log-linear model* is also used for the Poisson regression model since the logarithm of the conditional mean is linear in the parameters:

Note that the conditional variance of the count random variable is equal to the conditional mean in the Poisson regression model:

The equality of the conditional mean and variance of is known as *equidispersion*.

The marginal effect of a regressor is given by

Thus, a one-unit change in the th regressor leads to a *proportional* change in the conditional mean of .

The standard estimator for the Poisson model is the maximum likelihood estimator (MLE). Since the observations are independent, the log-likelihood function is written as

The gradient and the Hessian are, respectively,

The Poisson model has been criticized for its restrictive property that the conditional variance equals the conditional mean. Real-life data are often characterized by *overdispersion* — that is, the variance exceeds the mean. Allowing for overdispersion can improve model predictions since the Poisson restriction of equal mean and variance results in the underprediction of zeros when overdispersion exists. The most commonly used model that accounts for overdispersion is the negative binomial model.

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