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Poisson Regression

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Poisson Regression

A presentation by Jeffry A. Jacob

Fall 2002

Eco 6375

- 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

- 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.)

- 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

- 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])

- 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[1991])

- 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:

- 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

- 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

- 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.

- 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,

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