Notes on Poisson Regression and Some Extensions

Pr(Y=y)0 .1 .2 .30 2 4 6 8yFigure 1: Comparison of Poisson Model µ = 2 and Observed Data. poisson childs cons, noconsPoisson regression Number of obs = 1501Wald chi2(1) = 1277.14Log likelihood = -2764.5247 Prob > chi2 = 0.0000------------------------------------------------------------------------------childs | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------cons | .6623651 .0185344 35.74 0.000 .6260383 .6986919------------------------------------------------------------------------------. poisson, irrchilds | IRR Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------cons | 1.939374 .0359452 35.74 0.000 1.870187 2.01112------------------------------------------------------------------------------We find that the average number of kids is roughly 2. Suppose we take this rate as the theoreticalmean rate in the population. We can generate a Poisson distribution under this assumption andcompare to the observed distribution as shown in Figure 1. Below are the Stata commands to dothis.gen y = childsmatrix bpois = e(b)gen lampois = exp(bpois[1,1])scalar lampois = 2* generate the predicted distribution for this value of lambda* and plot against empirical distributiongen PrY_pois = exp(-lampois)*lampois^y/exp(lngamma(y+1))twoway (histogram y, discrete fraction blcolor(black) bfcolor(none) ///legend(off) ytitle(Pr(Y=y)) ) (connected PrY_pois y, sort legend(off))A listing of the distribution might also useful for comparison to the theoretical distributionµ = 2.4