Appropriate econometric technique for estimating invariant regressors in panel data analysis

[Omer Javed]

I am trying to estimate the impact of institutional determinants on macroeconomic instability and also on economic growth separately, using a panel of 129 IMF member countries (using the GMM and System GMM econometric techniques). Some of the variables like investment freedom, monetary freedom and property rights are invariant variables, and the impact is not been captured correctly it seems. It would be helpful to know as to which econometric technique might be appropriate here?

[Stephen Jenkins]

the impact is not been captured correctly it seems

Please explain in more detail what you mean by this, e.g. showing example output. Are the models that you cite fixed effects models? So, is the problem that the effects of time-invariant regressors cannot be identified in principle let alone practice?

NB please don't assume all list members know what the models are that you are referring to. So, please elaborate and given explicit and full citations to relevant literature and/or the Stata manual(s) and commands.

[Omer Javed]

Thanks Stephens. I am using ~~ GMM approach [ suggested by Arellano and Bover (1995) and Blundell and Bond (1998)]. Also, the problem with results is wrong sign for investment freedom. In literature it has a positive impact on macroeconomic stability but in my output it is negative. While I respect results but I think it might be also because the variable is very slow moving, so thought it might be a case that I need to regress on the following lines, "We propose a two-stage estimation procedure to identify the effects of time-invariant regressors. We first estimate the coecients of the time-varying regressors and subsequently regress the first-stage residuals on the time-invariant regressors to recover the coecients of the latter" (Kripfganz and Schwartz, 2013)?

[Sebastian Kripfganz]

Essentially, you have two alternatives. (i) You may estimate all coefficients simultaneously with a system GMM estimator. (ii) You may apply the two-stage approach that we are discussing in our paper. However, you should be aware that the fundamental identification problem is present with both approaches. You either have to assume (!) that your time-invariant regressors are uncorrelated with the fixed effects, or you will need to use appropriate instruments. The latter can be of the Hausman/Taylor type, for example. The advantage of the two-stage approach is that wrong assumptions on the exogeneity of time-invariant regressors do not affect the consistent estimation of the coefficients of the time-varying regressors. Moreover, you can combine the two-stage approach with alternative first-stage estimators that cannot be used to identify effects of time-invariant regressors, for example the first-difference (Arellano/Bond) GMM estimator, or a first-difference QML estimator. On the other hand side, you will have to correct the standard errors because the second-stage regression depends on first-stage estimates.

Regarding your empirical results it would be helpful if you post the Stata syntax for your estimation to see if you have specified the GMM estimator appropriately. Let me give you an example a system GMM estimation with Hausman/Taylor instruments for a dynamic panel data model. Suppose you want to regress a dependent variable y on its own lag L.y, two time-varying variables, x1 and x2, and two time-invariant variables, z1 and z2. It is assumed that the variables x1 and z1 are uncorrelated with the fixed effects, while x2 and z2 are correlated with them. You may then use x1 as an instrument for z2, provided that both variables are sufficiently correlated. A basic Stata estimation syntax would then read as follows:

xtdpd y L.y x1 x2 z1 z2, div(x1 x2) liv(x1 z1) dgmmiv(y, lag(2 .)) lgmm(y, lag(1)) hascons

Importantly, you have to specify the "hascons" option. Otherwise, your time-invariant regressors will be dropped. The liv() option essentially specifies the instruments for the time-invariant regressors. You may also use the "twostep" option or specify different lag orders for the instruments. (Do not confuse the "twostep" option with our two-stage estimator. The "twostep" option computes the efficient GMM estimator with the optimal weighting matrix but still computes all coefficients simultaneously, while our two-stage procedure would estimate the coefficients sequentially.)

[Sebastian Kripfganz]

As an important remark to my above stylized example: Stata will also report results for all coefficients including those of the time-invariant regressors z1 and z2 if you do not specify the liv() option. However, in that case only the first differences of the variable y as specified with the lgmmiv() option are used to "identify" these coefficients because the options div() and dgmmiv() refer to the first-differenced equation that does not contain the time-invariant variables. However, the first differences of y do not qualify as instruments for the time-invariant variables. (Ask yourself: Why should first differences of y be uncorrelated with unobserved time-invariant fixed effects but correlated with observed time-invariant variables?) The results of such a regression without the liv() option are therefore pure garbage. I am emphasizing this because I have heard the argument quite often that system GMM was able to "identify" coefficients of time-invariant variables because Stata reports estimates for them. However, actual identification requires that you specify for each time-invariant regressor at least one valid (!) instrument with the liv() option. In the above example, this is z1 as an instrument for itself, and x1 as and instrument for z2. (See also the Stata documentation to xthtaylor.)