1 / 12

# Poisson Regression - PowerPoint PPT Presentation

Poisson Regression. A presentation by Jeffry A. Jacob Fall 2002 Eco 6375. Poisson Distribution. A Poisson distribution is given by:. Where, is the average number of occurrences in a specified interval. Assumptions: Independence

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Poisson Regression' - kitra-mayer

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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,

Done in Eviews