Poisson regression

[komal Prakash]

Hi everyone!
this platform has been a great help for me. I would like to ask questions on Poison regression

Background- i want to estimate the elasticity of income on number of children in the household and income elasticity of years of education. log-log model could give me the required answers but, there are 0's in income variable and other two dependent variable which years of schooling and number of children.
So, I thought of using Poisson GLM.

Questions;
1) Does Poisson mean log-linear form of the regression and If I take log of dependent variable ( number of children/years of education) which has zeros in it, Would that be fine?
2) As independent variable I am using income which has huge values. To lower the magnitude I use log of income. Can I take log of income (independent variable) in Poisson?
3) Can I log transform the income knowing that it has 0 values in few observations.
4) will the coefficient of log-log model and Poisson regression with log_income as independent same?
5) How to take into account the assumption of "mean and variance equal" in Poisson distribution while using Poison regression?
I really look forward for the help.
Thank you in advance

[Nick Cox]

Poisson regression for some writers implies a discrete outcome variable, but a more generous form allows the outcome variable to be continuous too. See e.g. https://blog.stata.com/2011/08/22/us...tell-a-friend/ With this generosity y = exp(Xb) is the main idea.

Even on a narrow definition,

1. Zeros are allowed outcome values. Indeed, occasional negative values are allowed as the bigger idea is that the mean function is positive, which doesn't rule out some values being zero or negative. On the other hand, standard definitions of the Poisson distribution don't allow negative values.

2. You cannot take logarithms of zero. Log zero is utterly undefined. Trying to do in Stata, or any other software, will at best yield missing values.

3. You should not try to take logarithms even if all outcomes are positive. Poisson regression comes with an expectation that the variance around mean outcomes will not be constant, so that you need an estimation routine designed for that situation.