An example: Poisson Regression

A Poisson regression is estimated explaining the number of Olympic Games medals won by various countries as a function of the logarithms of population and gross domestic product (in 1995 dollars). The estimated coefficents are as follows:

1. Based on the output in the above table, are the signs of explanatory variables consistent with your expectation? Explain.
   Answer: Both the size of the country and wealth of the country have a positive and significant impact on the number of medals won. This is reasonable, since we expect that a country with larger pupulation, there are more people being talent for sports. And also a wealthier country can devote larger amount of financial funds to sports training. Therefore, the more number of medals expect to be won.
2. In 1988 Canada had GDP=5.19E+11 and pupulation of 26.9 million, predict the number of medals that Canada would win.
   Answer: $Ln NUM=c+\beta_{POP}ln(POP)+\beta_{GDP}ln(GDP) \\\hat{NUM}=\exp(c+\beta_{POP}ln(POP)+\beta_{GDP}ln(GDP))=15.59$LnNUM=c+βPOP​ln(POP)+βGDP​ln(GDP)NUM^=exp(c+βPOP​ln(POP)+βGDP​ln(GDP))=15.59
3. calculate the probability that they would win 5 medals or less.
   Answer: As we know, for Poisson distribution,
   $P(Y=y)=\frac{e^{-\lambda}\lambda^y}{y!}$P(Y=y)=y!e−λλy​
   so that
   $P(Y\leq 5)=[P(Y=0)+...+P(Y=5)]\\\quad\\=15.59^0 e^{-15.59}{0!}+...+15.59^5 e^{-15.59}{5!}=0.0018492$P(Y≤5)=[P(Y=0)+...+P(Y=5)]=15.590e−15.590!+...+15.595e−15.595!=0.0018492
4. What is the marginal effect of a change in population on the numbers of medals won? How do you interpret it?
   $\frac{\partial \hat{NUM}}{\partial POP}=\exp(c+\beta_{POP}ln(POP)+\beta_{GDP}ln(GDP))*\beta_{POP}=15.59*0.18=2.8062$∂POP∂NUM^​=exp(c+βPOP​ln(POP)+βGDP​ln(GDP))∗βPOP​=15.59∗0.18=2.8062
5. In addition to pupulation and GDP, the file Olympics contains a dummy variable PLANNED, which includes nonmarket, typically conmmunist countries and the dummy variable HOST, which indicates the country hosting the Olympic Games. The Poisson regression model that adds these two varia les to the specification is estimated as in the following Table:
   In 2000, the GDP (in 1995 US $) of Canada was 6.41256E + 11. The Canadian population in 2000 was 30.689 million. Using these figures, predict the number of medals won by Canada based on the estimates. Note that the 2000 games wre held in Sydney, Australia.
   Answer: $\hat Y=\exp(c+\beta_{POP}Ln(POP)+...+\beta_{PLANNED}PLANNED)=17.495$Y^=exp(c+βPOP​Ln(POP)+...+βPLANNED​PLANNED)=17.495
   The calculated mean is consistent with that computed from the EViews output. Sinece, in 2000, Canada actually won 14 medals, the predicted number of medals won by Australia is relatively close to the acutal value.

Microeconometrics course, University of LIVERPOOL