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I want to estimate a panel data model (not yet decided if FE, RE or an alternative that is not yet known to me). The equation is:

$ {y_{it}} = \alpha_{i} + \lambda_{t} + \beta_{1} X_{{it}} + \beta_{2} Z_{{it}} + \epsilon_{{it}}$

I am mainly interested in X, but I suspect that might be endogenous. I investigate some options to deal with this problem, and I came up with the FDIV solution that involves: 1) Take the first differences of all variables in the model; 2) Instrument the "troublesome" $ {\Delta}X_i$ with $ {X_{{it-1}}} $ and use the $ \hat{{\Delta}X_i}$ in order to estimate the first-differences FD model:

$ {\Delta y_{i}} = \lambda_{t} + \alpha_{1} \hat{{\Delta}X_i} + \alpha_{2} \Delta Z_{{i}} + \Delta \epsilon_{{i}}$

My question is: is this procedure valid, provided that I count only with two waves of the panel (so t=2)? I have more or less 300 observations per wave, so i=600 (roughly). My panel is balanced.

Your orientation and hints would be greatly appreciated

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    $\begingroup$ You should check out the dynamic panel literature, which in some cases instruments for levels with lagged first differences, and in other cases instruments for first differences with sufficiently lagged first differences. Some of the main references would be Arellano and Bond 1991(people.stern.nyu.edu/wgreene/Econometrics/Arellano-Bond.pdf) and Blundell and Bond (ucl.ac.uk/~uctp39a/Blundell-Bond-1998.pdf). One of the biggest challenges with doing this is the weak instruments problem: sufficiently lagged levels may not have strong correlation with first differences. $\endgroup$ – Hessian Dec 13 '16 at 14:30
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It depends on why you think Delta x is endogenous. It is a very special kind of endogeneity if you think that x_it-1 is exogenous. Because in a sense, you already have x_it-1 in your equation (namely inside Delta x_it). I have a hard time thinking of any situation like that.

But sometimes you may be willing to assume that x_it-2 is exogenous as an instrument for Delta x_it.

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