How to treat time-invariant variables in system GMM?

[Laura Zutphen]

Dear Statalist-users,

I am using the system GMM method for dynamic panel data. I was wondering how to deal with time-invariant regressors in this method. Roodman (2009) says:
"In system GMM, one can include time-invariant regressors, which would disappear in difference GMM. Asymptotically, this does not affect the coefficient estimates for other regressors because all instruments for the levels equation are assumed to be orthogonal to fixed effects, indeed to all time-invariant variables."

However, I do not understand how one can use the lagged first-difference as instrument in the level equation and the lagged level as instrument in the first-difference equation, as the first-difference is by definition equal to zero. The only option, in my opinion, is to treat these time-invariant regressors as exogenous and only include them in the level equations, as this will make Stata only use the lagged levels as instruments for the levels (no first-difference is used).

What do you think? Is there a way to treat time-invariant variables as endogenous in the system GMM method? If so, how?

To be complete, this is the command in Stata I use, in which BF*_c is time-invariant:

xi: xtabond2 logsavings L.logsavings BF*_c cc_hh ni incunc age agesq edu i.year, gmm(L.logsavings, lag(2 .)) gmm(cc_hh ni incunc age agesq  edu, lag(2 .)) iv(BF*_c i.year, equation(level)) two r nocons

Thank you in advance!

[Sebastian Kripfganz]

Your doubts are absolutely valid. Indeed, first differences of time-invariant variables cannot be used as instruments because they disappear, and the included first differences of the other (time-varying) variables generally do not qualify as valid instruments. There is hardly any situation where such differences can be reasonably believed to be uncorrelated with the unobserved time-invariant effects ("fixed effects") but at the same time be sufficiently correlated with the observed time-invariant regressors. The dangerous trap is that those first differences of the time-varying variables are not perfectly orthogonal to the time-invariant regressors in any given sample such that Stata will still report estimates for the coefficients of the latter (which, however, are not reliable at all).

One solution would indeed be to assume that the time-invariant regressors are exogenous with respect to the unobserved effects (which, strictly speaking, would then no longer be "fixed effects"). As you have done in your example, you can then use those time-invariant variables as instruments for themselves in the level equation.

If you instead want to treat them as endogenous, you need to find some other suitable instrumental variables. These could be some external instruments or internal instruments following a Hausman-Taylor type strategy (see help xthtaylor for this strategy in a static panel context).

I have provided some further thoughts on this topic in earlier Statalist posts, for example:

• why xtabond and xtdpdsys produce the same results (both the coefficient and standard error) for certain variables?
• Appropriate econometric technique for estimating invariant regressors in panel data analysis
• Fixed Effects and time-invariant variables

A longer discussion can be found in the following working paper:

• Kripfganz, S. and C. Schwarz (2015). Estimation of linear dynamic panel data models with time-invariant regressors. ECB Working Paper 1838, European Central Bank.

[Laura Zutphen]

Thanks for the replay Sebastian! That helps a lot.

However, if I want to use external instruments for the time-invariant endogenous regressors, these instruments obviously have to be relevant (besides the fact that they have to be exogenous). In other words, they have to be sufficiently correlated with the endogenous regressor. How can I test if an instrument is relevant in the system GMM framework?

[swati saini]

Hi,
I have a similar doubt.I am running a regression of income inequality on iprs and other control variables.Since I have a dynamic panel specification, I am using xtabond2 command.My regression eq looks like:
Income Ineq = b1 +b2. L.Income inequality +b3.IPRs + b4.Controls
My variable IPRs is an IPR index ( on a scale of 0 to 5) which does not vary much over time.So my data looks like

country	year	gini_net	ipr
Australia	1	28.1872	2.49
Australia	2	29.0448	2.49
Australia	3	28.7599	3.28
Australia	4	29.9817	4.33
Australia	5	31.1922	4.33
Australia	6	31.7795	4.33

I want to treat IPR as an endogenous variable but I am not very clear about how to instrument it using its own lags?Suppose if I consider one period lag as instruments, this means my observation for 6th year : IPR = 4.33 will be instrumented by 0 .( levels are instrumented by differences of lags in system-gmm) which seems illogical.How should I proceed ?Please correct me if i am wrong.