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# Poisson Regression

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1. Poisson Regression A presentation by Jeffry A. Jacob Fall 2002 Eco 6375

2. Poisson Distribution • A Poisson distribution is given by: Where, is the average number of occurrences in a specified interval • Assumptions: • Independence • Prob. of occurrence In a short interval is proportional to the length of the interval • Prob. of another occurrence in such a short interval is zero

3. Poisson Model • The dependent variable is a count variable taking small values (less than 100). • It has been proposed that the count dependent variable follows a Poisson process whose parameters are determined by the exogenous variables and the coefficients • Justified when the variable considered describes the number of occurrences of an event in a give time span eg. # of job-related accidents=f(factory charact.), ship damage=f(type, yr.con., pd.op.)

4. Specification of the Model • The primary equation of the model is • The most common formulation of this model is the log-linear specification: • The expected number of events per period is given by

5. Specification…. • The major assumption of the Poisson model is : • Thus: • Later on when we do diagnostic testing, we will test this assumption. It is called testing for over-dispersion (if Var[y]>E[y]) or under- • dispersion (if Var[y]<E[y])

6. Estimation • We estimate the model using MLE. The Likelihood function is non linear: • The parameters of this equation can be estimated using maximum likelihood method • Note that the log-likelihood function is concave in and has a unique maxima. (Gourieroux)

7. Estimation…. • The Hessian of this function is: • From this, we can get the asymptotic variance- covariance matrix of the ML estimator: • Finally, we use the Newton-Raphson iteration to find the parameter estimates:

8. Interpretation of the coefficients • Once we obtain the parameter estimates, i.e. estimates , we can calculate the conditional mean: Which gives us the expected number of eventsper period. • Further, if xik is the log of an economic variable, i.e. xik = logXki, can be interpreted as an elasticity

9. Diagnostic Testing • As we had mentioned before, a major assumption of the Poisson model is: • Here the diagnostic tests are concerned with checking for this assumption • Cameron and Trivedi (1990) test H0 : Var (yi) = H1 : Var (yi) = + g( ), usually g( )= or • Test for over or under dispersion is =0 in We check the t-ratio for

10. Diagnostic Testing… • An alternative approach is by Wooldridge(1996) which involves regressing the square of standardized residuals-1 on the forecasted value and testing alpha = 0 in the following test equation • In case of miss-specification, we can compute QMLestimators, which are robust – they are consistent estimates as long as the conditional mean in correctlyspecified, even if the distribution is incorrectlyspecified.

11. Diagnostic Testing… • With miss-specification, the std errors will not be consistent. We can compute robust std errors using Huber/White (QML) option or GLM , which corrects the std errors for miss-specification. • For Poisson, MLE are also QMLE • The respective std errors are: And, Where,

12. Empirical Examples Done in Eviews