![]() In the maximum-likelihood estimator for Poisson, what does depend on the assumption that E( y j) = Var( y j) are the estimated standard errors of the coefficients b 0, b 1, …, b k. It turns out that the estimated coefficients of the maximum-likelihood Poisson estimator in no way depend on the assumption that E( y j) = Var( y j), so even if the assumption is violated, the estimates of the coefficients b 0, b 1, …, b k are unaffected. Nonetheless, I suggest you fit this model using Poisson regression rather than linear regression. Indeed, it would be absurd to think one could predict income so accurately based solely on years of schooling and job experience. If a person has an expected income of $45,000, there is no reason to think that the variance around that mean is 45,000, which is to say, the standard deviation is $212.13. There is simply no reason to think that the the variance of the log of income is equal to its mean. Ln(income j) = b 0 + b 1*education j + b 2*experience j + b 3*experience j 2 + ε j If your goal is to fit something like a Mincer earnings model, In any case, in a Poisson process, the mean is equal to the variance. Actually regression does assume the variance is constant but since we are working the logs, that amounts to assuming that Var( y j) is proportional to y j, which is reasonable in many cases and can be relaxed if you specify vce(robust). Whereas linear regression merely assumes E(ln( y j)) = b 0 + b 1 x 1 j + b 2 x 2 j + … + b k x kj and places no constraint on the variance. Poisson regression assumes the variance is equal to the mean,Į( y j) = Var( y j) = exp( b 0 + b 1 x 1 j + b 2 x 2 j + … + b k x kj) Y j = exp( b 0 + b 1 x 1 j + b 2 x 2 j + … + b k x kj + ε j) Which is to say, fit instead a model of the form The next time you need to fit such a model, rather than fitting a regression on ln( y), consider typing The above is just an ordinary linear regression except that ln( y) appears on the left-hand side in place of y. Ln( y j) = b 0 + b 1 x 1 j + b 2 x 2 j + … + b k x kj + ε j
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