[Sebastian Kripfganz]

For the level model, once usually wants to difference the instruments. A key system GMM assumption is that those first-differenced instruments are valid instruments for the level model (i.e. uncorrelated with the unobserved group-specific effects, which are still present in the level error term). For the differenced model, you may or may not difference the instruments as well; both is possible. Often, no differencing of the instruments is applied for the differenced model.

The one-step and two-step (system) GMM estimators differ in the way that the two-step estimator uses an optimal estimate of the weighting matrix, while the one-step estimator uses a pre-specified weighting matrix which is typically not optimal (in an asymptotic variance minimizing sense). Estimating this weighting matrix can be very inaccurate when the sample size is small and/or the number of instruments is large.

In dynamic panel models, the main idea is to use lagged variables as instruments (rather than other variables excluded from the model), assuming that those lags do not have a direct effect on Y but through their serial correlation over time affect the unlagged X regressors. The particular way of transforming the model and/or instruments then ensures that those instruments are uncorrelated with the error term, especially the group-specific component in the error term.

GMM-type instruments are typically used for endogenous variables, but technically IV-type instruments are just collapsed versions of GMM-type instruments. What really matters is to use the right lags and transformations to ensure the endogeneity of those instruments, irrespective of whether they are in GMM-type or IV-type format.

[Erlend Salhus]

Thank you for your response. I understand that I must collapse the instrument set since 103 instruments become far too many. Furthermore, looking at your Stata presentation I understand that the diff/level option inside gmmiv and iv only accounts for the transformation of the instruments. How do you decide whether you want to difference- or level transform the instruments?

What is the main difference between the one-step sys-gmm and the two-step sys-gmm? I have tried to read the common articles but cannot find any good explanations.

From my understanding, the gmm-type instruments can only be endogenous instruments, while iv-style instruments should be exogenous. In traditional econometrics instrumental variables only affect Y through the Xs. What if I do not have variables that impact the X's standalone? I am mainly using GMM to account for the unobserved heterogeneity and the dynamic characteristics of the underlying data.

Thank you for your brilliant work and insights.

[Sebastian Kripfganz]

103 instruments are definitely way too many for just 23 groups. You should normally remain clearly below the number of groups, which already becomes difficult given your small N. Aside from that, neither of your codes really performs a system GMM estimation, at least not in the conventional way. For l.investments fundraising, you have only specified instruments for the level model. For realrate, there is only an instrument for the differenced model. And for loans EV_EBITDA GDPGrowth, you did not specify any instruments. Please have a look at my 2019 London Stata Conference presentation for examples of how to specify a system GMM estimator. (The presentation is for my xtdpdgmm command, but the syntax is very similar.)

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

[Erlend Salhus]

Dear Professor Kripfganz. I have a research design where T=14 and N=29. My main explanatory variable is a dummy variable, which is the reason why I want to run a sequential regression.
With some guidance, I have decided to pursue a one-step system GMM, in addition to trying to limit the instruments used.

I have two questions. The first is regarding the specification in STATA - do these commands replicate the one-step system GMM?
The second question: If I do not collapse the instrument count, the number of instruments becomes 103. This seems a bit too much given the underlying data, however, the estimates fit my research hypotheses. Collapsing the lag order produces results that do not match my initial hypotheses at all.

First sequence without collapsing:

 xtseqreg investments l.investments fundraising loans realrate EV_EBITDA GDPGrowth, gmmiv(l.investments fundraising, difference lag(1 4) model(level)) iv(realrate, difference model(diff)) teffects vce(robust)

Group variable: Country                      Number of obs         =       322
Time variable: Year                          Number of groups      =        23

                                             Obs per group:    min =        14
                                                               avg =        14
                                                               max =        14

                                             Number of instruments =       103

                               (Std. Err. adjusted for 23 clusters in Country)
------------------------------------------------------------------------------
             |               Robust
 investments |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
 investments |
         L1. |    .490159   .1416168     3.46   0.001     .2125952    .7677229
             |
 fundraising |   .3290689   .0940575     3.50   0.000     .1447197    .5134182
       loans |   .0734566   .0649467     1.13   0.258    -.0538366    .2007498
    realrate |  -.2176656     .09787    -2.22   0.026    -.4094874   -.0258439
   EV_EBITDA |   .3007255   .1406747     2.14   0.033     .0250082    .5764428
   GDPGrowth |   .3428593   .2096171     1.64   0.102    -.0679827    .7537013
             |
        Year |
       2009  |   1.088307   .8770698     1.24   0.215     -.630718    2.807332
       2010  |   .1624718   .8496158     0.19   0.848    -1.502745    1.827688
       2011  |    1.01128    .908088     1.11   0.265      -.76854    2.791099
       2012  |   .1774802   .5373513     0.33   0.741     -.875709    1.230669
       2013  |  -.1022629   .7235758    -0.14   0.888    -1.520445     1.31592
       2014  |  -.1601438    .734644    -0.22   0.827     -1.60002    1.279732
       2015  |  -.4042953    .871267    -0.46   0.643    -2.111947    1.303357
       2016  |  -2.037866   .9979397    -2.04   0.041    -3.993791   -.0819398
       2017  |  -1.562841    1.63792    -0.95   0.340    -4.773105    1.647423
       2018  |  -3.113656   .8736665    -3.56   0.000    -4.826011   -1.401301
       2019  |  -1.528436   1.269343    -1.20   0.229    -4.016302    .9594295
       2020  |  -2.863026   1.310453    -2.18   0.029    -5.431466   -.2945848
       2021  |  -2.433613   1.797741    -1.35   0.176     -5.95712    1.089894
             |
       _cons |          0  (omitted)
------------------------------------------------------------------------------

 xtseqreg investments (l.investments fundraising loans realrate EV_EBITDA GDPGrowth) qe_D, iv(qe_D, model(level)) vce(robust)

Group variable: Country                      Number of obs         =       322
Time variable: Year                          Number of groups      =        23

------------------------------------------------------------------------------
Equation _first                              Equation _second
Number of obs         =       322            Number of obs         =       322
Number of groups      =        23            Number of groups      =        23

Obs per group:    min =        14            Obs per group:    min =        14
                  avg =        14                              avg =        14
                  max =        14                              max =        14

Number of instruments =       103            Number of instruments =         2

                                (Std. Err. adjusted for clustering on Country)
------------------------------------------------------------------------------
             |               Robust
 investments |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
_first       |
 investments |
         L1. |    .490159   .1416168     3.46   0.001     .2125952    .7677229
             |
 fundraising |   .3290689   .0940575     3.50   0.000     .1447197    .5134182
       loans |   .0734566   .0649467     1.13   0.258    -.0538366    .2007498
    realrate |  -.2176656     .09787    -2.22   0.026    -.4094874   -.0258439
   EV_EBITDA |   .3007255   .1406747     2.14   0.033     .0250082    .5764428
   GDPGrowth |   .3428593   .2096171     1.64   0.102    -.0679827    .7537013
       _cons |          0  (omitted)
-------------+----------------------------------------------------------------
_second      |
        qe_D |  -1.777723    .862589    -2.06   0.039    -3.468366   -.0870796
       _cons |  -.1945286   .5189867    -0.37   0.708    -1.211724    .8226667

Second sequence with the collapse option:

 xtseqreg investments l.investments fundraising loans realrate EV_EBITDA GDPGrowth, gmmiv(l.investments fundraising, model(level) lagrange(1 4) collapse) iv(realrate, difference model(diff)) teffects vce(robust)

Group variable: Country                      Number of obs         =       322
Time variable: Year                          Number of groups      =        23

                                             Obs per group:    min =        14
                                                               avg =        14
                                                               max =        14

                                             Number of instruments =        23

                               (Std. Err. adjusted for 23 clusters in Country)
------------------------------------------------------------------------------
             |               Robust
 investments |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
 investments |
         L1. |   .4429917    .249676     1.77   0.076    -.0463643    .9323478
             |
 fundraising |   .2205416   .1397837     1.58   0.115    -.0534294    .4945127
       loans |   .2482312   .2097539     1.18   0.237     -.162879    .6593414
    realrate |  -.5433681   .3479326    -1.56   0.118    -1.225303    .1385673
   EV_EBITDA |   .5111535   .2761733     1.85   0.064    -.0301363    1.052443
   GDPGrowth |   1.544522   1.776925     0.87   0.385    -1.938187    5.027232
             |
        Year |
       2009  |    2.61303   1.932554     1.35   0.176    -1.174707    6.400766
       2010  |  -1.396378   2.079974    -0.67   0.502    -5.473051    2.680295
       2011  |  -.6476817   2.146449    -0.30   0.763    -4.854645    3.559282
       2012  |   -.413738   1.139913    -0.36   0.717    -2.647927    1.820451
       2013  |  -1.412886   2.266768    -0.62   0.533    -5.855669    3.029897
       2014  |   -1.71271   2.292105    -0.75   0.455    -6.205153    2.779734
       2015  |  -3.245102   3.497507    -0.93   0.353    -10.10009    3.609886
       2016  |  -6.038229   4.439052    -1.36   0.174    -14.73861    2.662154
       2017  |  -6.911909   5.704652    -1.21   0.226    -18.09282    4.269002
       2018  |  -7.788161    5.47485    -1.42   0.155    -18.51867    2.942347
       2019  |  -6.247621   4.997203    -1.25   0.211    -16.04196    3.546718
       2020  |  -5.574032   4.096693    -1.36   0.174     -13.6034    2.455339
       2021  |  -9.078443   7.165498    -1.27   0.205    -23.12256    4.965675
             |
       _cons |          0  (omitted)
------------------------------------------------------------------------------

 xtseqreg investments (l.investments fundraising loans realrate EV_EBITDA GDPGrowth) qe_D, iv(qe_D, model(level)) vce(robust)

Group variable: Country                      Number of obs         =       322
Time variable: Year                          Number of groups      =        23

------------------------------------------------------------------------------
Equation _first                              Equation _second
Number of obs         =       322            Number of obs         =       322
Number of groups      =        23            Number of groups      =        23

Obs per group:    min =        14            Obs per group:    min =        14
                  avg =        14                              avg =        14
                  max =        14                              max =        14

Number of instruments =        23            Number of instruments =         2

                                (Std. Err. adjusted for clustering on Country)
------------------------------------------------------------------------------
             |               Robust
 investments |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
_first       |
 investments |
         L1. |   .4429917    .249676     1.77   0.076    -.0463643    .9323478
             |
 fundraising |   .2205416   .1397837     1.58   0.115    -.0534294    .4945127
       loans |   .2482312   .2097539     1.18   0.237     -.162879    .6593414
    realrate |  -.5433681   .3479326    -1.56   0.118    -1.225303    .1385673
   EV_EBITDA |   .5111535   .2761733     1.85   0.064    -.0301363    1.052443
   GDPGrowth |   1.544522   1.776925     0.87   0.385    -1.938187    5.027232
       _cons |          0  (omitted)
-------------+----------------------------------------------------------------
_second      |
        qe_D |  -4.515909   3.353083    -1.35   0.178    -11.08783    2.056013
       _cons |  -1.777259   1.884741    -0.94   0.346    -5.471283    1.916765
------------------------------------------------------------------------------

Any help is highly appreciated!

[Alexandre Sonderegger]

Thank you very much for your help.

[Sebastian Kripfganz]

If your model is overidentified - that is, you have more exogenous time-varying regressors than endogenous time-invariant regressors - then you can use overidentification tests. However, these tests can never test the joint validity of all instruments, and are therefore only of limited help. Prior information is still required. Out of the top of my head, I do not have a better suggestion.

[Alexandre Sonderegger]

Dear Professor Kripfganz,

According to my current understanding, FEF-IV(static) and sGMM2(dynamic) although less sensible to instruments misspecification with respect to Hausman Taylor (HT), do require prior knowledge/assumption of which time-variant regressors can be included as instruments (which time varying regressors are exogenous to the individual effects).
For the static case, Chatelain & Ralf (2021) propose an instrument selection method based on t-test statistic of the time varying regressors estimated with GLS (random effects) on Krishnakumar's (2006) augmented model. The original model can then be estimated with HT using the previously selected instruments.

Do you have any recommendations for similar "automatic" selection of instruments for FEF-IV and sGMM2 respectively ?

References:

- Krishnakumar, J. (2006). Chapter 5 Time Invariant Variables and Panel Data Models: A Generalised Frisch–Waugh Theorem and its Implications. Contributions to Economic Analysis, 274. https://doi.org/10.1016/S0573-8555(06)74005-6
- Chatelain, J.-B., & Ralf, K. (2021). Inference on time-invariant variables using panel data: A pretest estimator. Economic Modelling, 97, 157–166. https://doi.org/10.1016/j.econmod.2021.01.014

[Alexandre Sonderegger]

Thank you for your quick reply and your precious help, I somehow did not see your reply before today.

Best,
Alexandre

[Sebastian Kripfganz]

You can have multiple fixed effects, as long as you are not trying to estimate the model in more than 2 stages. You would have occupation-fixed effects in the second stage, and all other fixed effects in the first stage.

[Alexandre Sonderegger]

Hi Professor Kripfganz,

Thank you for your package and contributions to time-invariant regressors literature.

I am wondering whether your xtseqreg package can be used for models with multiple fixed effects.

For example, can it be used on a model with occupations, industry, and time fixed effects (unit of observation being an occupation-industry-time aggregate cell). The dimension of time-invariant regressors being the occupation level, and the relation between industries and occupations being Many-to-Many (occupation-in-industry).

Thank you,
Best regards,

Alexandre

[Sebastian Kripfganz]

The instrumenting variable is allowed to vary at the country level even though the instrumented variable does not. The county-level variation is just not useful to identify the effects. I am afraid that out of the top of my head I do not have a useful reference for you. However, I do not think there is anything special about a regional dummy. Yes, it might make sense to treat it as endogenous (i.e. correlated with the unobserved country-specific effects). As with any other endogenous time-invariant variable, you would then need to find an appropriate instrument.

[Ana Ortiz Salazar]

Hi Dr. Kripfganz,

Do you have any recommendations about instrumenting a regional dummy in the second-stage? Can the instrumenting variable vary at the country-level even though the regional dummy does not? My colleages and I do not think we can justify this regional dummy as exogenous..

If you have any material/guides you can reference for applications where the time-invariant variable is a similar “group” variable like region, I would appreciate it.

Thank you!

Ana

[Debajyoti Chakrabarty]

Hi Sebastian,
Okay, thank you for your quick response.
Regards,
Debajyoti

[Sebastian Kripfganz]

It depends how you use it. As long as you make sure that the number of instruments does not become too large, it should not be a problem.

[Debajyoti Chakrabarty]

Hi Sebastian,
I wanted to ask if xtseqreg can be used for large N and large T panels.
Regards,
Debajyoti