System GMM estimation

[Taz Raihan]

Suppose I want to estimate the following model using System GMM:

\[ \Delta logy_t \equiv logy_t-logy_{t-1} = \alpha logy_{t-1} + \beta x_t + e \]

Given that the estimation of System GMM involves taking orthogonal deviations (or first differences), I am confused as to which of the following ways should I actually write my Stata commands?

Command 1:

 xtabond2  Δlogyt logyt-1  xt , ///
          gmm(logyt-1, collapse) ivstyle( xt , equation(diff)) ///
          ivstyle(xt , equation(level)) ///
          twostep robust orthogonal

Command 2:

 xtabond2 logyt logyt-1  xt, ///
          gmm(logyt-1, collapse) ivstyle( xt, equation(diff)) ///
          ivstyle(xt, equation(level)) ///
          twostep robust orthogonal

I am asking this question because in the paper of Roodman titled "How to do xtabond2: An introduction to difference and system GMM in Stata", it is mentioned on page 17 that the equations implied by Command 1 and 2 are equivalent. But when estimating, he seems to estimate using the equation implied by Command 2. Given my objective model, which command should I use here? Thank you for your help.

[Sebastian Kripfganz]

The models are equivalent up to the fact that in the second model the coefficient of the lagged dependent variable is \(\alpha + 1\) (given your notation). You should be able to observe this relationship when you compare the estimates from both versions.

[Taz Raihan]

Sebastian Kripfganz yes, I observed this, it's mentioned in that paper too. My question is about the implementation of this. Is my first command appropriate given the model I want to estimate?

[Taz Raihan]

The reason I ask this question is when I estimate both models (with more controls, but same controls in both models), I get different coefficients for all of the controls, not just for the lagged dependent variable.

[Sebastian Kripfganz]

That should not happen, but if you want to be on the safe side I would recommend version 2.

[Taz Raihan]

Sebastian Kripfganz Thank you for your input.