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Poisson regression using Panel data requires the dependent variable ($y_{it}$) to be a non-negative count variable. I need to take the first difference of the dependent variable to deal with reverse causality of the regressors. Cameron and Trivedi's Microeconometrics using Stata outline a transformation

$\frac{\lambda_{i,t-1}}{\lambda_{i,t}}*y_{i,t}-y_{i,t-1}$

where

$\lambda_{i,t}=exp(x_{it}'\beta)$

which, they state, can be used to eliminate the FE and as the basis for estimation of dynamic panel count models. My question is how can one actually estimate $\lambda_{it}$ (ideally in Stata) so that it can then be used to create first differences?

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It seems that I need to use the gmm program evaluator to take the differencing as required in the question. I wrote the following program but it's not running and giving the error

could not calculate numerical derivatives -- flat or discontinuous region encountered

the program that I wrote is as follows

capture program drop gmm_poifd
program gmm_poifd
    version 11
    syntax varlist if, at(name) myrhs(varlist) mylhs(varlist) myidvar(varlist)
    quietly {
    tempvar mu muLag yLag
    gen double `mu' = 0 `if'
    local j = 1
    foreach var of varlist `myrhs' {
        replace `mu' = `mu' + `var'*`at'[1,`j'] `if'
        local j = `j' + 1
    }
    replace `mu' = exp(`mu')
    gen double `muLag' = L.`mu' `if'
    gen double `yLag' = L.`mylhs' `if'  
    replace `varlist' = `muLag'/`mu' * `mylhs' - `yLag' `if'
}
end

gmm gmm_poifd, mylhs(PostMile1) myrhs(Traffic $Messages $Time $Weather) /// 
myidvar(MessageBoardID) nequations(1) parameters(Traffic $Messages $Time $Weather) ///
instruments(Traffic $Messages $Time $Weather) onestep

Can somebody please help me identify the problem and fix this answer?

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