## Info

Obs with Dep=0 451 Total obs 601

Obs with Dep = 1 150

assigning Y = 0 for individuals who did not have any affairs and Y = 1 for those who had such affairs, giving the results shown in Table 15.17. With the knowledge of probit modeling, readers should be able to interpret the probit results given in this table on their own.

620 PART THREE: TOPICS IN ECONOMETRICS

15.12 MODELING COUNT DATA: THE POISSON REGRESSION MODEL

There are many phenomena where the regressand is of the count type, such as the number of vacations taken by a family per year, the number of patents received by a firm per year, the number of visits to a dentist or a doctor per year, the number of visits to a grocery store per week, the number of parking or speeding tickets received per year, the number of days stayed in a hospital in a given period, the number of cars passing through a toll booth in a span of, say, 5 minutes, and so on. The underlying variable in each case is discrete, taking only a finite number of values. Sometimes count data can also refer to rare, or infrequent, occurrences such as getting hit by lightning in a span of a week, winning more than one lottery within 2 weeks, or having two or more heart attacks in a span of 4 weeks. How do we model such phenomena?

Just as the Bernoulli distribution was chosen to model the yes/no decision in the linear probability model, the probability distribution that is specifically suited for count data is the Poisson probability distribution. The pdf of the Poisson distribution is given by41:

where f (Y) denotes the probability that the variable Y takes non-negative integer values, and where Y! (read Y factorial) stands for Y! = Y x (Y — 1) x (Y — 2) x 2 x 1. It can be proved that

Notice an interesting feature of the Poisson distribution: Its variance is the same as its mean value.

The Poisson regression model may be written as:

where the Y's are independently distributed as Poisson random variables with mean ¡i for each individual expressed as

¡¡i = E(Yi) = ft + ft *2i + ft X3i +■ ■ ■ + PkXki (15.12.5)

where the X's are some of the variables that might affect the mean value. For example, if our count variable is the number of visits to the Metropolitan Museum of Art in New York in a given year, this number will depend on

41See any standard book on statistics for the details of this distribution.

CHAPTER FIFTEEN: QUALITATIVE RESPONSE REGRESSION MODELS 621

variables such as income of the consumer, admission price, distance from the museum, and parking fees.

For estimation purposes, we write the model as:

with i replaced by (5.12.5). As you can readily see, the resulting regression model will be nonlinear in the parameters, necessitating nonlinear regression estimation discussed in the previous chapter. Let us consider a concrete example to see how all this works out.

AN ILLUSTRATIVE EXAMPLE: GERIATRIC STUDY OF FREQUENCY OF FALLS

The data used here were collected by Netere et al.42 The data relate to 100 individuals 65 years of age and older. The objective of the study was to record the number of falls ( = Y) suffered by these individuals in relation to gender (X2 = 0 female and 1 for male), a balance index (X3), and a strength index (X4). The higher the balance index, the more stable is the subject, and the higher the strength index, the stronger is the subject. To find out if education or education plus aerobic exercise has any effect on the number of falls, the authors introduced an additional variable (X1), called the intervention variable, such that X1 = 0 if only education and X1 = 1 if education plus aerobic exercise training. The subjects were randomly assigned to the two intervention methods.

Using Eviews 4, we obtained the output in Table 15.18.