Stationarity in difference GMM

[Tobias Wendler]

Hey everyone,

I am rather confused by the question on whether in difference GMM we do need stationary variables in levels or first-differences, or at all ?

To my understanding as we are using first differences in the model these being stationary should be sufficient, or am I wrong with this?

If somebody could shed some light upon this issue it would really help.

Thanks and best
Tobi

[Sebastian Kripfganz]

If a variable follows a unit-root process, then its lagged levels will be uncorrelated with the first-differenced regressor. The consequence will be a weak-instruments problem.

[Tobias Wendler]

Thanks for the quick and straight reply, didn`t find a clear statement such as this in most threads.
How would one then proceed,
1) Should I look for external instruments for the unit root variables ?
2) Can I include the first differences directly in the model (regressor would then be first differences of first differences), so that the lagged levels are first differences ?
3) Should I switch to system GMM or another estimator ?

If you have some practical tip on how to proceed I would be very grateful as I am getting a little lost in the wealth of possibilities.

Thanks and best wishes
Tobias

[Sebastian Kripfganz]

First of all, I wonder how you are establishing that your variable is nonstationary. Most unit-root tests require a large time horizon but the panel data GMM framework is designed for short-T panels.

1) If you have reasonable external instruments that you can justify, go for them. But usually this is a difficult task.
2) You would usually start with an econometric model that is motivated by some economic theory. You do not want to transform some of your variables into first differences just for econometric reasons because the resulting model would no longer be in line with your economic theory.
3) Strictly speaking, my earlier statement in comment #2 above was a bit abridged. It refered to a situation where a variable just follows a random walk. A variable can be nonstationary if it is caused by another nonstationary variable. When there exists an error-correction mechanism for the first variable, its first difference will still be correlated with the lagged levels despite the nonstationarity. Implicitly, the existence of such a mechanism is what you are assuming when your dependent variable and at least one of your independent variables are nonstationary. If your dependent variable is stationary but your independent variables are not, then you might question your initial economic model unless there is an error-correction mechanism between these variables themselves.

In short, unless your variables follow a simple random walk, nonstationarity per se is not necessarily a problem. Yet, you might want to double check if you have an economic theory that justifies the postulated relationship between your nonstationary variables in the first place. With a difference GMM estimator, you would get into trouble if the true autoregressive coefficient approaches unity (which implies that there is no error-correction mechanism). A system GMM estimator could help, although the required initial-observations condition might be hard to justify. An alternative might be the nonlinear Ahn-Schmidt GMM estimator.

[Tobias Wendler]

Dear Sebastian,

thank you for all these extremely helpful hints, I will think about them closely, if it is fine with you I would like to show you more closely my results below for eventual comments and hints whether I made severe mistakes in my implementation and interpretation.

Considering your first point, I in fact estimate a macro-panel with T and N both equal to 21. I considered the difference GMM estimator in reaction to the paper by Judson and Owen (1999).

Your second point is very helpful, according to economic theory I should stay in first-differences (or in levels) as my context is the IPAT-hypothesis, that environmental impact is influenced by the economic scale and affluence as well as technology employed.

According to unit root test, only the Inno-Variable is stationary in levels (though plotting the data for individual countries questions these test results), while the others, including the dependent variable, are non-stationary. However, all variables are stationary in first-differences.

One might in fact expect an error-correction-mechanism.

Below, you can find the output of my regression using xtabond2. Looking at the coefficient of the lagged dependent variable I would not assume a random walk of the dependent variable, or am I overlooking something here ?

The code is :

xtabond2 EnvIm L(1/1).(EnvIm) ln_GDP Sector L(1/1).(Inno) t90 t91 t92 t93 t94 t95 t96 t97 t98 t99 t00 t01 t02 t03 t04 t05 t06 t07 t08 t09 t10, gmm(l.EnvIm, collapse) gmm(ln_GDP, lag(2 3) collapse) iv(t90 t91 t92 t93 t94 t95 t96 t97 t98 t99 t00 t01 t02 t03 t04 t05 t06 t07 t08 t09 t10 Inno Sector) small nodiffsargan robust nolevel orthogonal

I assume GDP to be endogenous, the manufacturing sector as exogenous and Innovation as exogenous (which might be an assumption which I will change).



Dynamic panel-data estimation, one-step difference GMM
------------------------------------------------------------------------------
Group variable: country Number of obs = 388
Time variable : year Number of groups = 21
Number of instruments = 42 Obs per group: min = 16
F(25, 21) = 1957.68 avg = 18.48
Prob > F = 0.000 max = 19
------------------------------------------------------------------------------
| Robust
EnvIm | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
EnvIm |
L1. | .6044269 .0817698 7.39 0.000 .4343772 .7744765
|
ln_GDP | .273699 .1270205 2.15 0.043 .0095455 .5378526
Sector | .5191438 .1754681 2.96 0.007 .154238 .8840496
|
Inno |
L1. | -.0239221 .0185675 -1.29 0.212 -.0625353 .014691
|
t90 | 0 (omitted)
t91 | -.027682 .0379185 -0.73 0.473 -.1065379 .0511739
t92 | 0 (omitted)
t93 | -.0521265 .0261887 -1.99 0.060 -.1065889 .0023359
t94 | -.00291 .0223519 -0.13 0.898 -.0493933 .0435734
t95 | .0028601 .0236798 0.12 0.905 -.0463848 .052105
t96 | -.0231272 .0244325 -0.95 0.355 -.0739375 .027683
t97 | -.0110857 .0278547 -0.40 0.695 -.0690128 .0468413
t98 | .0297274 .0327187 0.91 0.374 -.0383149 .0977696
t99 | .0218161 .0283871 0.77 0.451 -.0372181 .0808502
t00 | .0296256 .0348262 0.85 0.405 -.0427995 .1020506
t01 | .0116483 .0343537 0.34 0.738 -.0597941 .0830907
t02 | .016275 .0349571 0.47 0.646 -.0564224 .0889724
t03 | .0337148 .0348782 0.97 0.345 -.0388184 .106248
t04 | .0647368 .0405541 1.60 0.125 -.0196001 .1490737
t05 | .063651 .0449922 1.41 0.172 -.0299154 .1572174
t06 | .0466327 .0481616 0.97 0.344 -.0535248 .1467903
t07 | .0446187 .0463386 0.96 0.347 -.0517476 .1409851
t08 | .0407832 .0550435 0.74 0.467 -.0736861 .1552525
t09 | -.046943 .0486098 -0.97 0.345 -.1480325 .0541465
t10 | .0062056 .0516641 0.12 0.906 -.1012358 .1136469
------------------------------------------------------------------------------
Instruments for orthogonal deviations equation
Standard
FOD.(t90 t91 t92 t93 t94 t95 t96 t97 t98 t99 t00 t01 t02 t03 t04 t05 t06
t07 t08 t09 t10 Inno Sector)
GMM-type (missing=0, separate instruments for each period unless collapsed)
L(2/3).ln_GDP collapsed
L(1/22).L.EnvIm collapsed
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -2.86 Pr > z = 0.004
Arellano-Bond test for AR(2) in first differences: z = 1.09 Pr > z = 0.275
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(17) = 18.68 Prob > chi2 = 0.347
(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(17) = 0.00 Prob > chi2 = 1.000
(Robust, but weakened by many instruments.)


For further hints on whether my approach is so far appropriate, or whether I should in fact utilize other approaches or estimators I would be very thankful, as this unfortunately is beyond my econometric understanding at this moment.

Thank you so much for the great support
Tobi

[Sebastian Kripfganz]

Your reasoning appears to make sense (although I cannot comment on the particular economic theory).

Regarding the implementation: 19 overidentifying restrictions are too many given your very small cross-sectional sample size. You really should reduce the lag length and use collapsing for the instruments of the lagged depdendent variable (similar to GDP).

Notice that I have said 19 overidentifying restrictions instead of 17 (as wrongly indicated by the Hansen test degrees of freedom). There is a bug in xtabond2 that computes incorrect degrees of freedom when there are omitted coefficients in the regression. As a consequence, the p-value of the Hansen test is incorrect, too. You should hence remove two of your time dummies to avoid getting omitted coefficients.

Even more, you might want to think about replacing the whole set of time dummies just by a linear time trend or time dummies for certain periods rather than years. Your time dimension is relatively large which would introduce a bias because the time dummies cannot be estimated precisely.

[Tobias Wendler]

Thank you so much.
By trying your suggestions I encountered a variety of insights and issues.

Reducing the lag length of the dependent variable drastically helps in terms of number of instruments, while keeping the coefficient and estimation solid.
Yet, I was wondering whether my writing of the lagged DV is correct this way, because I first thought of writing it as

 gmm (EnvIm, lag (1/2) collapse

,
this did not work as since it is already lagged I have to include it similar to an endogenous variable in the instrument set, although the lagged DV is itself predetermined; I hope my approach now is correct for the variable.

Thank you for pointing towards the bug, I just excluded the first and last year and thus came to 19 overidentifying restrictions in the initial estimation (where the lagged DV brought in more instruments).

With the code

xtabond2 EnvIm l.EnvIm ln_GDP Sector l.Inno t91 t92 t93 t94 t95 t96 t97 t98 t99 t00 t01 t02 t03 t04 t05 t06 t07 t08 t09, gmm(EnvIm, lag(2 2) collapse) gmm(ln_GDP, lag(2 2) collapse) iv(t91 t92 t93 t94 t95 t96 t97 t98 t99 t00 t01 t02 t03 t04 t05 t06 t07 t08 t09 Inno Sector) small nodiffsargan robust nolevel orthogonal

I received the following output, where there was no longer need for the Sargan Test as there are as many regressors as instruments:

Dynamic panel-data estimation, one-step difference GMM
------------------------------------------------------------------------------
Group variable: country Number of obs = 388
Time variable : year Number of groups = 21
Number of instruments = 23 Obs per group: min = 16
F(23, 21) = 3322.30 avg = 18.48
Prob > F = 0.000 max = 19
------------------------------------------------------------------------------
| Robust
EnvIm | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
EnvIm |
L1. | .7033147 .0748286 9.40 0.000 .5477001 .8589292
|
ln_GDP | .2189991 .0982136 2.23 0.037 .0147527 .4232455
Sector | .4294698 .1689885 2.54 0.019 .0780391 .7809006
|
Inno |
L1. | -.0359453 .0186113 -1.93 0.067 -.0746496 .0027591
|
t91 | -.0278694 .041782 -0.67 0.512 -.1147598 .0590211
t92 | -.0051707 .0470447 -0.11 0.914 -.1030054 .0926641
t93 | -.0601203 .0455209 -1.32 0.201 -.1547862 .0345457
t94 | -.0060221 .0392478 -0.15 0.880 -.0876423 .0755981
t95 | -.0026365 .0360085 -0.07 0.942 -.0775202 .0722473
t96 | -.0303642 .0324057 -0.94 0.359 -.0977556 .0370273
t97 | -.016564 .0285304 -0.58 0.568 -.0758962 .0427682
t98 | .0240754 .0308307 0.78 0.444 -.0400406 .0881913
t99 | .0124631 .0249674 0.50 0.623 -.0394593 .0643856
t00 | .0210229 .0187623 1.12 0.275 -.0179955 .0600413
t01 | .0014491 .0245177 0.06 0.953 -.0495383 .0524365
t02 | .0059409 .0172094 0.35 0.733 -.0298479 .0417298
t03 | .0232889 .0190832 1.22 0.236 -.0163968 .0629745
t04 | .053801 .0173599 3.10 0.005 .017699 .0899029
t05 | .0473546 .0200804 2.36 0.028 .0055951 .089114
t06 | .0304315 .0144894 2.10 0.048 .000299 .0605639
t07 | .0293202 .0131898 2.22 0.037 .0018905 .0567499
t08 | .0261669 .0171282 1.53 0.142 -.0094532 .061787
t09 | -.0657626 .0149022 -4.41 0.000 -.0967535 -.0347717
------------------------------------------------------------------------------
Instruments for orthogonal deviations equation
Standard
FOD.(t91 t92 t93 t94 t95 t96 t97 t98 t99 t00 t01 t02 t03 t04 t05 t06 t07
t08 t09 Inno Sector)
GMM-type (missing=0, separate instruments for each period unless collapsed)
L2.ln_GDP collapsed
L2.EnvIm collapsed
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -2.87 Pr > z = 0.004
Arellano-Bond test for AR(2) in first differences: z = 1.10 Pr > z = 0.273
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(0) = 0.00 Prob > chi2 = .
(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(0) = 0.00 Prob > chi2 = .
(Robust, but weakened by many instruments.)


In this context, I was now asking myself whether there is a rule of thumb on how many overidentifying restrictions there need to be before it makes any sense to consider the Sargan Test ?
(In one estimation the p-value was 0.00, yet there was only one overidentifying restriction, should this bother then ?)

Thanks for the tip with including a time-trend or specific periods instead of years.
Trying a time-trend

xtabond2 EnvIm l.EnvIm ln_GDP Sector l.Inno c.year, gmm(EnvIm, lag(2 2) collapse) gmm(ln_GDP, lag(2 2) collapse) iv(c.year Inno Sector) small nodiffsargan robust nolevel orthogonal

the results became implausible as GDP became insignificant (and negative), other coefficients as well and also the time-trend itself was insignificant.
Also worse became the results when I included dummies for the starting point of the financial crisis, or the introduction of the Euro (I am looking at EU countries), or generating dummies for each time period of 5 years.
I can not yet explain why a time-trend distorts the results so severely.

Thus, I would consider to stay with the specification above, and refine it.

Is there any other potentially severe problem that I need to consider ?

If you can recommend any practical source to look on (beyond the xtabond2 Paper) to prevent me asking stupid question, please let me know.

Thank you very much for the support, I highly appreciate and am thankful for your readiness to help.
Please let me know if the wealth of questions I am coming up with is bothering or accessible elsewhere.

Thanks again and best
Tobi

[Sebastian Kripfganz]

Some comments:

1. As you have realized yourself, the first valid instrument is indeed the second lag of the dependent variable. I would usually at least consider two lags (i.e. the second and third lag). Otherwise, you are essentially using an Anderson-Hsiao estimator that may break down in some situations, which led to the development of the Arellano-Bond estimator who added further lags. (This will automatically give you overidentifying restrictions.)

2. Unless you want to assume the standard error-components structure with homoskedasticity, forget about the Sargan test and just focus on the Hansen test (given that you are taking all necessary precautions to avoid a too-many-instruments problem). If the Hansen test rejects the null hypothesis already for a single overidentifying restriction, this should worry you, yes.

3. The distortion may not come from the time trend itself but from the lack of the time dummies. Time effects are a tricky matter, in particular given your large time horizon. I do not have a one-size-fits-all answer for you on that matter.

Just ask. I may, however, not respond to every question or keep some responses rather short.

[Tobias Wendler]

Thank you so much, this is such a great help

[Tobias Wendler]

Dear Sebastian,

I stumbled across a further question during my latest research steps.

In which way does multicollinearity become a problem in GMM, is it relevant to check for multicollinearity in levels or in differences ?

Thanks and best
Tobi

[Sebastian Kripfganz]

Mulitcollinearity among the regressors creates the same problems for GMM as for any other regression method.

[Tobias Wendler]

Thanks, is it sufficient if there is no multicollinearity in differences or do we need to check for the lagged levels (or the regressor with its level value and lag as used in the model ?) with which we instrument?

[Sebastian Kripfganz]

There should be no multicollinearity among the regressors (in first differences for that matter) and among the instruments.

[Tobias Wendler]

Then, among the regressors it is not a problem, but how do I account for the instruments ?, because in levels I would have multicollinearity, so now I do not really know how to proceed with this issue of potential multicollinearity among the instruments (both identifying and getting rid). Especially because I have seen that at least when using System GMM other authors have appearingly not been worried by multicollinearity in levels.

Another problem I stumbled across is that when using first-difference transform instead of orthogonal deviations my results change dramatically, and I am now insecure, whether staying in orthogonal deviations is sound, although Hayakawa (2009) argues to use FOD. In terms, of the OLS-LSDV range of the lagged DV the value of the orthogonal deviation transform is almost exactly to the LSDV estimate, while the First difference transform estimate is significantly lower, although the LSDV should be the lower bound. do these thoughts justify staying with the FOD transform, or am I on the wrong track here ?

Thank you again
Tobi

[Sebastian Kripfganz]

It is difficult to give general advise on this matter, so I am afraid that I will not be able to give more help here.

Similarly, I do not have an answer for your second question. Sorry.