poisson regression (ppmlhdfe) with multiplicative error

[Ramin Forouzandeh]

Is there a way to force ppmlhdfe to use multiplicative error term instead of additive?

I wan to run a poisson regression with instruments and as this thread suggests the only way it makes sense is if the error term is multiplicative:
https://www.statalist.org/forums/for...-fixed-effects

I know that the ivpoisson command can do that but I need a command that can deal with high dimentional fixed effects as well.

I appreciate any help.

[Fei Wang]

You may use a similar control function method to the one proposed by Jeff in the cited thread, given that you have lots of FEs to control for. I run a couple of simulations before posting here and they are fairly good.

reghdfe y2 x1 ... xJ z1 ... zK, a(FE1 FE2 ...) res(v)
ppmlhdfe y x1 ... xJ y2 v, a(FE1 FE2 ...)

where y is the DV, y2 is the EEV, x1-xJ are exogenous covariates, z1-zK are excluded IVs, and FE1... are FEs.

[Ramin Forouzandeh]

Dear Fei,

Thanks for your answer. I wanted to do the exact procedure you outlined here, but my concern was the fact that ppmlhdfe error term is additive and the command does not have an option for changing it to a multiplicative error (like ivpoisson does). Following this thread:
https://www.statalist.org/forums/for...ive-iv-poisson
it seems that the error should be multiplicative when we use instruments. I couldn't find any theoretical basis for an IV poisson estimation when the second stage errors are additive.
(sorry I referred to a wrong link in the main post)

[Fei Wang]

In my simulations, no matter the true model is additive or multiplicative, -ppmlhdfe- will always give consistent estimates for slope coefficients. I think there must be some underlying theories, but it's beyond my scope of knowledge.

[Fei Wang]

Dear Fei,

Thanks for your answer. I wanted to do the exact procedure you outlined here, but my concern was the fact that ppmlhdfe error term is additive and the command does not have an option for changing it to a multiplicative error (like ivpoisson does). Following this thread:
https://www.statalist.org/forums/for...ive-iv-poisson
it seems that the error should be multiplicative when we use instruments. I couldn't find any theoretical basis for an IV poisson estimation when the second stage errors are additive.
(sorry I referred to a wrong link in the main post)

I figured it out a little bit by going through the cited thread. When estimating a poisson model, no matter which command, poisson, xtpoisson, ppmlhdfe, etc. is used, the crucial part is to specify the conditional mean function of y. Once the mean function is determined, one would estimate coefficients with a routine ML approach. Additive or multiplicative errors don't matter when all regressors are exogenous, as shown below. In the multiplicative case, the key is that E(exp(u)|x) is not a function of x as x are exogenous. No matter what "a" is, it's no more than a constant term within an exponential function and won't impact the shape of the mean function and the slope coefficients estimated from ML.

Additive: y = exp(xb) + u --> E(y|x) = exp(xb), if E(u|x) = 0
Multiplicative: y = exp(xb + u) --> E(y|x) = a*exp(xb), if E(exp(u)|x) = a

When a relevant variable, r, is omitted to error terms, the form of errors do matter. Just like what Jeff showed in the thread,

Additive: y = exp(xb) + u --> E(y|x, r) = exp(xb) + c*r
Multiplicative: y = exp(xb + u) --> E(y|x, r) = exp(xb + d*r)

The mean functions are essentially different and we will obtain different sets of coefficients if we observe r. When we plug in the first-stage residual, v, to the second stage using a control function method, we have implicitly assume multiplicative errors -- Otherwise, we won't be able to include the residual as a usual regressor into the exponential function, as in #2.

E(y|x, v) = exp(xb + m*v)

So yes, when control function approaches are used in this way, we are actually "forcing" ppmlhdfe and other commands to use multiplicative errors.

ADD: In ivpoisson, additive or multiplicative errors matter for gmm estimation as moment conditions differ if error forms differ. But for -ivpoisson cfunction-, multiplicative error is assumed (and there is no option for additive errors), probably because the idea of control function, controlling for the first-stage residual together with other covariates, is naturally a reflection of multiplicative error structure in poisson.

[Ramin Forouzandeh]

Thank you very much. I couldn't hope for a better explanation.