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So I am estimating a production function based on Wooldridge (2009) GMM adaptation of preexisting semi-parametric, 2-stage techniques. One of the upsides of GMM is simultaneous instead of sequential estimation, which alleviates correlation problems of residuals across equations.

In effect, I am estimating simultaneously the following equations:
(Eq1) : $y_{it}=\beta_0+\beta_Ll_{it}+\beta_Kk_{it}+poly(k_{it},m_{it})\lambda+e_{it}$
(Eq2) : $y_{it}=\alpha_0+\beta_Ll_{it}+\beta_Kk_{it}+poly(k_{it-1},m_{it-1})\lambda+u_{it}$
Where $l$ denotes labor, $k$ denotes capital, $m$ is material, and $poly(...)$ is a degree-3 polynomial (without the the constant term and the term of degree 1 in $k$ since those already appear in the equation).

Instruments are as follows:
(Z1) : $(k_{it},l_{it},poly(k_{it},m_{it}))$
(Z2) : $(k_{it},l_{it-1},poly(k_{it-1},m_{it-1}))$

Note that coefficients are constrained to be identical across equations.

Looking at the variance-covariance matrix output by the routine I used, common coefficients, though identical, have a different variance in each equation (which isn't particularly surprising). The covariance of a given coefficient across equations is also always positive.

Meanwhile, "canned" packages for productivity estimation return just one standard error per coefficient. One of such packages simply returned the standard errors from (Eq 1). This equation does make the most sense economically, since (Eq 2) is there only for identification purposes. However, I have my doubts about the package. As for the other package I tried... I don't know what it did to find this single standard error.

Thus, I'm wondering: when simultaneously estimating equations by GMM, is there a way to compute some sort of "common" standard errors for common coefficients?

Thanks in advance for your answers.

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