[Sebastian Kripfganz]

If ln_GDPc is nonstationary in levels, then the system GMM estimator in Case 1 is likely inconsistent. The additional Blundell-Bond instruments for the level model require a stationarity assumption that is unlikely to be satisfied in this case. A difference GMM estimator could still be applicable, although the usual instruments might be weak. The resulting identification problems could possibly be avoided if the Ahn-Schmidt nonlinear moment conditions are added to the difference GMM estimator.

If you use the the first difference of ln_GDPc as the dependent variable, estimation should be fine (assuming that the instruments are strong enough), but you cannot get any long-run effects on the level of ln_GDPc. The two cases are only equivalent if no such level effects exist, because the latter are ignored in Case 2.

[Sarah Magd]

Thanks very much, Prof. Sebastian Kripfganz for your reply,
I just figured out that I explained my model in the wrong way. However, I think your answer in #406 could guide us if our dependent variable is nonstationary in level.

I am regressing renewable energy consumption (REN) on the log of GDP per capita (ln_GDPc), using the two-step sys GMM model for a sample of 26 countries over the period 2005 to 2017. The panel unit root analysis shows that ln_GDPc is stationary in first difference, whereas REN is stationary in level. (So the independent variable in the model is ln_GDPc and it is non-stationary at level).

This leads to two cases:
- Case 1: If I regress REN on ln_GDPc in level: the coefficient of ln_GDPc is 0.3 and statistically significant.
- Case 2: If I regress REN on ln_GDPc in the first difference (Δln_GDPc) on REN: the coefficient of Δln_GDPc becomes 0.9 and statistically significant.

In this context, are the two cases equivalent to each other? Should I care about the stationarity given the small-time period I have?

[Sebastian Kripfganz]

There is an issue at the conceptual level: It does not really make sense to model the effect of a nonstationary variable on a stationary variable (if you believe the stationarity tests). Such a model would be unbalanced in terms of the integration orders. I recommend to do the following:

Case 3: Regress REN on ln_GDPc and L.ln_GDPc. By adding the lag of ln_GDPc, the model in Case 2 becomes nested in Case 3. Case 2 is essentially the same model as Case 3 but with the restriction that the coefficients of ln_GDPc and L.ln_GDPc are identical with opposite signs. If you estimate Case 3 and indeed find that the estimates support this restriction, you can go for the simplified Case 2. Otherwise, I recommend to stick to Case 3.

Case 4: Equivalently, regress REN on ln_GDPc and Δln_GDPc. If the coefficient of ln_GDPc is zero, then again Cases 2, 3, and 4 are equivalent.

With a small time horizon, you do not need to worry about stationarity for the reason it can cause trouble in time series models (nonstandard distributions etc.). However, as mentioned in my previous post, nonstationarity can lead to invalidity of instruments for the level model and weak instruments in general.

Sometimes, it helps to include a deterministic time trend in the regression (and, importantly, in the unit-root tests). Otherwise, the unit-root tests might mistake the deterministic time trend for a unit root.

[Sarah Magd]

Thanks, Prof Sebastian Kripfganz,

- Yes, when I run two different panel unit root tests, they give contradicting results for the stationarity of ln_GDPc.

- I want to make sure that when I use "predict residual, ue" after estimating the model with xtdpdgmm, this gives the residuals. Am I right?

[Sebastian Kripfganz]

Yes, option ue of predict gives you the usual residuals.

[Neyati Ahuja]

Hello Prof. Sebastian

I have a query on variables that can be included in the dynamic panel data model.
In my study I am working on firm level data (N=1400 companies, T=11 years). Besides the firm level variable as independent variables, I am also interested in analysing the impact of macroeconomic variable ((country specific) which are cross section invariant and time variant ) on my dependent variable which is firm performance.
1. Can these time variant cross section invariant variable eg GDP incorportated in the dynamic paniel model along with my time varying cross section varying variable.
2. If ans to 1. is no, is there a way to analyse the impact of these cross section invariant variable.
3. Can we incorporate cross section variant and time invariant variable in the dynamic panel mode eg industry to which the firm belong using a dummy variable. If yes, please elaborate.

Regards

[Sebastian Kripfganz]

1. You can include "global" variables (invariant over cross sections) as long as you do not also include time dummies for each year. If those global variables do not account sufficiently for common time effects, then this might result in biased estimates of all coefficients.

2. There is no simple way of analyzing the impact of global variables in the presence of time effects. (In theory, the same two-stage approach as in 3. below could be applied. In practice, due to the necessary correction of the standard errors, this is more complicated because the respective Stata command was not designed to do this for time effects.)

3. In principle, the same issues as above apply to time-invariant variables in the presence of firm-fixed effects. We discuss potential solutions in the following paper:

• Kripfganz, S., and C. Schwarz (2019). Estimation of linear dynamic panel data models with time-invariant regressors. Journal of Applied Econometrics 34 (4), 526-546.

You can add appropriate instruments which are assumed to be uncorrelated with the firm-fixed effects, e.g. by assuming that the time-invariant variables themselves are uncorrelated with the fixed effects. This can be a quite strong assumption. If violated, it could bias all of the coefficient estimates. (With industry dummies, this should be fine. The firm-fixed effects would then need to be interpreted as firm-specific differences which are not due to being in a specific industry. You can then simply add the industry dummy as a regressor and a standard instrument for the level model.) To gain some robustness for the time-varying regressors, you could apply a two-stage procedure (also discussed in the above paper). The latter can be implemented with my xtseqreg command.

[Neyati Ahuja]

Thanks a lot
I ll refer the above paper and will try implement xtseqreg command.

Thank You

[Sebastian Kripfganz]

Addendum: For a simple example with industry dummies (not using the two-stage approach), see slide 86 of my 2019 London Stata Conference presentation:

• Kripfganz, S. (2019). Generalized method of moments estimation of linear dynamic panel data models. Proceedings of the 2019 London Stata Conference.

[Sarah Magd]

Sebastian Kripfganz
Dear Prof. Kripfganz,
I have a question regarding the "estat overid" after running the two-step and one-step system GMM. I run the same specification (i.e., with the same number of lags) and get different results for the overidentification test. For the one-step estimation, I never obtain an insignificant P-value for the 1-step moment functions and 1-step weighting matrix. Even if I change the specification of the model. Please find below the details of the output. My sample has 27 cross-section units and 13 years.
What does this conflicting result mean?

##########################
#estat overid after the one step
##########################
Sargan-Hansen test of the overidentifying restrictions
H0: overidentifying restrictions are valid

1-step moment functions, 1-step weighting matrix chi2(21) = 122.9740
note: * Prob > chi2 = 0.0000

1-step moment functions, 2-step weighting matrix chi2(21) = 27.9882
note: * Prob > chi2 = 0.1405

* asymptotically invalid if the one-step weighting matrix is not optimal
####################################
#estat overid after the two step
####################################
Sargan-Hansen test of the overidentifying restrictions
H0: overidentifying restrictions are valid

2-step moment functions, 2-step weighting matrix chi2(21) = 27.3389
Prob > chi2 = 0.1599

2-step moment functions, 3-step weighting matrix chi2(21) = 27.9939
Prob > chi2 = 0.1403

[Sebastian Kripfganz]

As the note says, both 1-step tests are asymptotically invalid because the one-step weighting matrix of the system GMM estimator is not optimal. That is especially true for the first of the two reported tests (1-step moment functions, 1-step weighting matrix). You should just ignore that test.

[Sarah Magd]

Thanks, Prof. Sebastian Kripfganz
In this case, do you mean that I should ignore the 1-step weighting matrix chi2 test and report the 2-step weighting matrix chi2 only?

[Sebastian Kripfganz]

In the case of system GMM, yes.

[Sarah Magd]

############################################
#Interaction variables with xtdpdgmm
#############################################
Can you please write a post about how we should instrument the interaction variable X3 (X1*X2) in the following two cases?
Case 1: X1 and X2 are both endogenous
Case 2: X1 is endogenous whereas X2 is predeterminant

[Sebastian Kripfganz]

If either X1 or X2 is endogenous, then it usually makes sense to assume that their interaction X3 is endogenous as well. You can then just treat it the same way as any other endogenous variable.