1. GMM Estimation- class notes
ECO 311
Professor Erdinç

2. Dynamic Panel Estimation
Suppose we estimate a dynamic model:
In Stata: xtabond Y X1 X2 X3, lags(1)
where we assume X1, X2 and X3 as exogenous and they enter as instruments in the first-difference equation (diff-GMM) for themselves (no lags). If you want to use also lags of these variables as instruments, then you have to use either the pre or the endogenous option after the comma..

3. xtabond with endogenous variables
xtabond Y X1, lags(1) pre(X2) endogenous (X3)
where we assume X2 as predetermined and X3 as endogenous. Predetermined variables use instruments in levels from t-p-1, where p is the maximum lag of the dependent variable (# in lags option)used as explanatory variable (lagged dependent) whereas endogenous variables enter in the instrument matrix in levels from t-p-2 unless you limit the number of lags for both cases.

4. xtabond2 command
Another alternative is to use xtabond2 command (ssc install xtabond2), a stata routine to implement also the Blundell and Bond System- GMM estimation for dynamic panel (1998).
xtabond2 does diff-GMM (with nolevel option) as well as system GMM (default)

5. System GMM
The problem with diff-GMM is that the lags of predetermined lagged dependent variables used as instruments are uncorrelated (orthogonal) with the to differenced error term (validity!) but are not strictly correlated with the variable instrumented (lack of relevance, or weak instrument problem).
The idea is to improve the Arellano and Bond estimators with a linear GMM model where the variable l.y in level is instrumented with t-p-1 lags of y and predetermined variable in first difference and the exogenous variable in level. Hence, the system-GMM improves the correlation of instruments with the instrumented variable.

6. xtabond2 command
xtabond2 replicates diff-GMM xtabond in the following way:
xtabond y X1 X2 X3, lags(1) noconst
is equivalent to
xtabond2 y l.y X1 X2 X3, noleveleq gmm(l.y) iv(X1 X2 X3)
Note: you have to drop the constant in xtabond to replicate the xtabond results (diff-GMM) with xtabond2.
Also, for xtabond2 to apply diff-GMM, use noleveleq or just nolevel option, otherwise you get system-GMM results.

7. xtabond2 command
In xtabond2, the iv option displays the variables that are strictly exogenous (they enter in first difference (fd) equation in first difference (fd) form and the command gmm specifies the endogenous variable (l.y!) and the predetermined variables (that enter in the fd equation in levels from t-p-1).
xtabond y, lags(1) pre( X1 X2 X3) noconst
is equivalent to
xtabond2 y l.y X1 X2 X3, gmm(l.y X1 X2 X3) noleveleq
Note: There is repetition in variable list in xtabond2 as compared to xtabond

8. xtabond2 command
Differently from xtabond, in xtabond2, one can exactly set the number of instruments.
For example: if you want to use only t-2 and t-3 level of y, X1, X2 and X3 as additional instruments then USE
xtabond2 y l.y X1 X2 X3, gmm (l.y X1 X2 X3, lag (2 3)) noleveleq
If you want to use all the levels of y and only t-2 and t-3 of Xs, then
xtabond2 y l.y X1 X2 X3, gmm (l.y) gmm(X1 X2 X3, lag (2 3)) noleveleq

9. xtabond2 command
if you want to use all the level of Xs but only t-2 and t-3 level of y, then
xtabond2 y l.y X1 X2 X3, gmm (l.y, lag(2 3)) gmm(X1 X2 X3) noleveleq
In addition, you can use in xtabond command, twostep robust option for twostep estimation (default is onestep)
Similarly, in xtabond2, you can specify twostep robust.
As can be seen, xtabond2 does everything xtabond does with many additional features. But you have to type the options yourselves (good for flexibility).

10. xtabond2 command
If you want to limit the number of instruments in an efficient way, in xtabond2 y l.y X1 X2 X3, gmm (l.y, lag(2 3) collapse) gmm(X1 X2 X3) nolevel small
Small for small sample size and t stats rather than z
With robust option, no Sargan but its equivalent Hansen test is generated with the same null: “instruments are valid or exogenous: cov(z,u)=0”: model is well-specified. Additionally use AR(1) and AR(2) for model diagnostics. Also, . Estat overid for Hansen’s test.
If time dummies are used in the model, include them in iv(..) option as they are exogenous.
As always, with large T but small N, you do not need to do GMM (as xtivreg with gmm option will work just as well)
With large T, need to check stationarity of your panel variables: use xtunitroot fisher Y, lags(aic 10) to determine the optimal number of lags with aic minimization.
If you estimate system-GMM without nolevel option, add nocons to the options to estimate the level equation without the constant.

11. xtabond2 command
If you want to limit the number of instruments in an efficient way, in xtabond2 y l.y X1 X2 X3, gmm (l.y, lag(2 .) collapse) gmm(X1 X2 X3, lag(2 3)) small robust
Note that with nolevel option, we have one-step robust system-GMM. With lag specifications as above, for l.y, we use t-2 and earlier dated lags of y as instruments for the fd equations, and correspondingly, the lagged first-differences (t-1 and earlier as instruments for the equations in levels.
For X1-X3 (endogenous, we use t-2 and t-3 lags (only) of these variables for the fd model, and t-1 and t-2 lags of first-differences for equations in levels. Recall system GMM combines these in its instrument set to gain in efficiency.