Omitted variable bias in Poisson regression

[Gabor Mugge]

Hi,

This is more of a conceptual/econometrics question.

I have a Poisson model where the true relationship is:

E(y|x,z)=exp(b₁+b₂*x+b₃*z)

but z is not observable and so it is omitted from the estimated regression.

I read here that when z is independent from x, the estimate of b₂ should not be biased. Is it possible to derive the sign or magnitude of the bias when x and z are not independent? (The paper discusses this case briefly but I was not able to follow the argument there.)

The specific case to which I would like to apply the answer is where the omitted variable is z=x*u with some unobservable u that follows N(0,1). I have tried to understand this case with simulations by I wonder if it possible to say something more definite. Thank you.

[Jeff Wooldridge]

Set b₃ = 1 since it can't be identified, anyway. This means z is defined to have a positive effect on y. Depending on the nature of z, it might make sense to assume E(bz|x) = exp(a₁ + a₂*z), and then E(y|x) = exp(b₁+b₂*x)*exp(a₁ + a₂*z) = exp(a₁ + b₁ + (a₂ + b₂ )*x). So the inconsistency (not really bias) in the estimator is determined by a₂ -- very much like in the linear case.

[Gabor Mugge]

Thank you very much. That is very helpful.

If you don't mind an extension question: I am looking at a case where E(y|x,z)=exp(b*x+c*x*z+f(x)); f() is some non-linear function (specifically, the log of the cumulative normal distribution function), and x and z are unrelated variables. My intuition is that omitting f(x) should not bias the interaction term in the linear case, but simulations indicate that the interaction is inconsistent in the Poisson model. Do you perhaps have any advice on why this could be the case?