GMM estimation: when one of dependent variables minus a constant

[haiyan lin]

Dear everyone,

I had noticed a very interesting phenomenon in GMM estimation, which I can't figure out why. Hope you can give me some hints. Great thanks.

Take the following equation (the one listed in the help file of the code xtdpdgmm) as an example:

************************************************** ******************************************
webuse abdata

//gen new variable by minus a constant
gen w_new=w-2

//OLS
reg n l.n w k
reg n l.n w_new k

//GMM
xtdpdgmm L(0/1).n w k, gmm(L.n w k, l(1 4) c m(d)) iv(L.n w k, d) two vce(r)
xtdpdgmm L(0/1).n w_new k, gmm(L.n w_new k, l(1 4) c m(d)) iv(L.n w_new k, d) two vce(r)

************************************************** ***********************************************

Logically, when the independent variable minus a constant, we should expect the coefficients won't change, that is the case of OLS estimation. However, the coefficients vary for all independent variables in the GMM estimation (both in code xtdpdgmm and xtabond2). That's really hard for me to understand. Could anyone enlighten me on what leads to this outcome?

Best,
Haiyan

[Sebastian Kripfganz]

That is an interesting observation. In a nutshell, there are two reasons for the observed behavior:

1. In a linear regression model estimated by OLS, the intercept recenters the regressors. This is why regress yields identical results (besides the constant term). When estimating the model with an IV/GMM estimator, the intercept also recenters the regressors and the instruments. In general, we should thus expect again that using w_new instead of w does not matter. However, recentering happens for the model in levels. The intercept has no effect on the instruments for the model in first differences. That is comparable to running a 2SLS regression in first differences without an intercept:

ivregress 2sls d.n dl.n (d.w = l.w) d.k, nocons
ivregress 2sls d.n dl.n (d.w_new = l.w_new) d.k, nocons

Once you include an intercept in the first-differenced model, the results coincide again. We can tweak the system GMM estimator to also recenter the instruments for the first-differenced model by adding a vector of ones as an instrument for the first-differenced model:

gen byte ones = 1
xtdpdgmm L(0/1).n w k, gmm(L.n w k, l(1 2) c m(d)) iv(ones, m(d)) iv(L.n w k, d) two vce(r)
xtdpdgmm L(0/1).n w_new k, gmm(L.n w_new k, l(1 2) c m(d)) iv(ones, m(d)) iv(L.n w_new k, d) two vce(r)

The two specifications are now identical, but they differ from the specifications without the additional instrument. It is an interesting question whether it is beneficial to recenter the moment conditions for the first-differenced model. I vaguely remember that I may have seen a paper on this issue somewhere but I do not recall which paper it was.

2. The above comes with a caveat. You may have noticed that I restricted the maximum lag order in the gmm() option to 2 for the example to work. The reason is that for higher lag orders, there will be missing values in the instrument matrix which are replaced by zeros. Adding the vector of ones essentially recenters the moment conditions after replacing those missing values by zeros, while equivalence of the models with w and w_new is only achieved if the moments are recentered before replacement of the missing values.

[haiyan lin]

Great thanks, Sebastian, for such a detailed and clear explanation. I find minus a constant in one dependent variable greatly changes dif-GMM estimation, while a small difference happens in sys-GMM estimation. Your explanation that recentering happens for the model in levels makes sense for this phenomenon.

No recentering in the first-differenced model is really a pitfall, however, for people who use demeaned data with GMM estimation, especially the dif-GMM model. You don't know what would happen to coefficients of other variables when you use the demeaned data of one dependent variable.