Poisson Regression

Model Specification

The model specification for a Poisson regression is based on assuming that a count-dependent variable yihas, conditional on a vector of independent variables xi, a Poisson distribution independent across other observations i with mean μi and the following probability distribution function:

where β is a parameter vector (with the same dimension as xi) to be estimated. Note that the exponential link function between μi and xi ensures that μi is strictly positive; this is essential because the mean of the Poisson distribution cannot be nonpositive. Based on a set of n observations (yi, xi), equations (1) and (2) together generate a log-likelihood function of

where log (·) denotes natural logarithm. The first-order conditions from this log-likelihood function—there is one such condition per each element of β—are nonlinear in β and cannot be solved in closed form. Therefore, numerical methods must be used to solve for the value of β^ that maximizes equation (3).

Interpretation of Poisson regression output is similar to interpretation of other nonlinear regression models. If a given element of β^ is positive, then it follows that increases in the corresponding element of xi are associated with relatively large count values in yi. However, comparative statics in a Poisson regression model are complicated by the fact that the derivative of μi with respect to an element of xi in equation (2) depends on all the elements of xi (cross-partial derivatives of μiwith respect to different elements of xi are not zero, as they are in an ordinary least squares regression). Hence, with a vector estimate β^, one can vary an element of xi and, substituting β^ for β in equation (2), generate a sequence of estimated μi values based on [Page 830]a modified xi vector. It is then straightforward to generate a probability distribution over the nonnegative integers—an estimated distribution for yi—using the link function in equation (2). This distribution, it is important to point out, will be conditional on the elements of xi that are not varied.

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