Appendix K

Poisson Regression Model

(Source: Greene [1])

The Poisson regression model specifies that each is drawn from a Possion distribution with parameter , which is related to the regressors . The primary equation of the model is

(K.1)

The most common formulation for is the log-linear model,

(K.2)

It is easily shown that the expected number of events per period is given by

(K.3)

so

(K.4)

With the parameter estimates in hand, this vector can be computed using any data vector desired.

In practice, the Poisson model is simply a nonlinear regression. But it is far easier to estimate the parameters with maximum likelihood techniques. The log-likelihood function is

(K.5)

The likelihood equations are

(K.6)

The Hessian is

(K.7)

The Hessian is negative definite for all x and . Newton's method is a simple algorithm for this model and will usually converge rapidly. At convergence,

provides an estimator of the asymptotic covariance matrix for the parameter estimates. Given the estimates, the prediction for observation is . A standard error for the prediction interval can be formed by using a linear Taylor series approximation. The estimated variance of the prediction will be , where is the estimated asymptotic covariance matrix for .