Stationarity in System GMM

[Jose Valente]

Hey everyone,

I am trying to estimate a model with System GMM, but it happens that my time series variables (these are four out of 13 explanatory variables in total) are non-stationary. Is it ok to use the first differences of these four variables and keeping the remaining variables in levels or is there a problem?

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

Thanks and best

José Valente.

[Sebastian Kripfganz]

System GMM estimators are usually used when T is small. In such a situation, nonstationarity is not a concern, in particular when you estimate a dynamic model. Replacing the levels of these variables by their first differences changes the interpretation of the respective coefficients which may not be desired.

Edit: Nonstationarity may, however, turn some of your instruments for the level model invalid.

[Jose Valente]

Thank you very much for the timely and helpful response, Sebastian.

Two (perhaps naive) doubts:

1) 28 quarters can be considered a small T (insofar as nonstationarity is not a concern)?

2) Removing the nonstationary variables from the group of instruments used solves the problem of invalid instruments?

Thanks in advance.

[Sebastian Kripfganz]

1) There is no definite answer, whether T=28 is still "small" or not. With regard to nonstationarity, the question is why you are worried about it. When your model is dynamic (with a lagged dependent variable), there is no risk of a spurious regression problem even if the regressors are nonstationary. If your regressors are trending over time, the preferred way is to include either a linear time trend or a set of time dummies as regressors. In any case, with such a T that is not too small anymore, you need to be careful that the total number of instruments is not becoming too large.

2) Not necessarily. While you might remove invalid instruments, the remaining instruments might be weak. Again, there is no general answer here. Overidentification tests can guide you to some extent.

[Jose Valente]

Thank you again, Sebastian. Your comments are quite enlightening. Some more information about what I am running through:

1) I am also estimating a static model in addition to the dynamic version. In that case, what do you think about removing the nonstationary variables from the static model and keeping them in the dynamic version? Does that make sense? I know that I can have some omitted variable bias in the static model, but since I also estimate the dynamic (and complete) version for comparison purposes, I may have some insights. What do you think?

2) Another problem with the dynamic version is that the coefficient of the lagged dependent variable is roughly one. I am afraid that, in this case, the GMM estimator becomes weak. Is the nonlinear Ahn-Schmidt GMM estimator an alternative, in this case?

[Sebastian Kripfganz]

1) As you said, there is likely to be an omitted variable bias in the static model. Since the AR coefficient in the dynamic model is close to one, I would suggest to run the static model completely in first differences (including the differences of the nonstationary variables). Note that this requires that all you variables are strictly exogenous, if estimated by OLS. Otherwise, you would need to use instrumental variables again.

2) The Ahn-Schmidt GMM estimator might indeed be helpful in this case. Yet another alternative might be to use a maximum likelihood estimator, if you are willing to assume that all of your variables (besides the lagged dependent variable) are strictly exogenous:
XTDPDQML: new Stata command for quasi-maximum likelihood estimation of linear dynamic panel models

[Jose Valente]

I tried your command xtdpdgmm. As an example, the code was

 xtdpdgmm Spread Spreadt Spreadt2 RkAv1 CrRka, noserial gmmiv( Spreadt , collapse model(difference)) iv(RkAv RkAv1 RkAv3 RkAv4 CrRka CrRka1 CrRka2 CrRka3 CrRka4, difference model(difference)) twostep vce(robust)

Generalized method of moments estimation

Step 1       
initial:       f(b) =   13258742
alternative:   f(b) =   13207126
rescale:       f(b) =  1603811.2
Iteration 0:   f(b) =  1603811.2  
Iteration 1:   f(b) =  34588.919  
Iteration 2:   f(b) =  2960.9712  
Iteration 3:   f(b) =  1217.9874  
Iteration 4:   f(b) =  1205.4847  
Iteration 5:   f(b) =  1205.4262  
Iteration 6:   f(b) =   1205.426  
Iteration 7:   f(b) =   1205.426  

Step 2       
Iteration 0:   f(b) =          1  
Iteration 1:   f(b) =  .95622147  
Iteration 2:   f(b) =  .95621172  
Iteration 3:   f(b) =   .9562117  

Group variable: Bk_dummy                     Number of obs         =       945
Time variable: Trimestre                     Number of groups      =        35

Moment conditions:     linear =      36      Obs per group:    min =        27
                    nonlinear =      25                        avg =        27
                        total =      61                        max =        27

                              (Std. Err. adjusted for 35 clusters in Bk_dummy)
------------------------------------------------------------------------------
             |              WC-Robust
      Spread |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     Spreadt |   .9557091   .0227023    42.10   0.000     .9112134    1.000205
    Spreadt2 |  -1.387442   .4282301    -3.24   0.001    -2.226758   -.5481265
       RkAv1 |  -10.24243   3.252095    -3.15   0.002    -16.61642    -3.86844
       CrRka |  -1.353812   .7186651    -1.88   0.060     -2.76237    .0547454
       _cons |   22.26624   6.508548     3.42   0.001     9.509719    35.02276
------------------------------------------------------------------------------

. estat serial

Arellano-Bond test for autocorrelation of the first-differenced residuals
H0: no autocorrelation of order 1:     z =   -2.7568   Prob > |z|  =    0.0058
H0: no autocorrelation of order 2:     z =   -0.8701   Prob > |z|  =    0.3842

. estat overid

Sargan-Hansen test of the overidentifying restrictions
H0: overidentifying restrictions are valid

2-step moment functions, 2-step weighting matrix       chi2(56)    =   33.4674
                                                       Prob > chi2 =    0.9927

2-step moment functions, 3-step weighting matrix       chi2(56)    =   35.0000
                                                       Prob > chi2 =    0.9875

Still the autorregressive coefficient approaches unity. Shouldn't I be worried?

[Sebastian Kripfganz]

What is Spreadt2? The second lag of the dependent variable? In that case, the AR(1)-Koeffizient on its own is not very informative. You would need to consider the sum of the two coefficients, which is negative. This in turn is not a reasonable result.

Given the large number of time periods, I would recommend to restrict the lag depth for the GMM-type instruments. Usually, up to 4 or 5 lags should be sufficient.

[Jose Valente]

Spreadt2 is an interaction between Spreadt (which, in turn, is the lagged dependent variable) and a dummy. This interaction has the purpose of distinguishing among the spreads of five loan categories.

Explaining: In order to work with only one panel (instead of five panels, one for each loan category), I stacked up the five panels and used four dummies to distinguish among the five loan categories. So I interacted the four dummies with the explanatory variables. This means that Spreadt2 is the interaction between Spreadt and the dummy related to the second loan category (that dummy takes the value 1 if the spread refers to the second loan category and zero otherwise).

But I indeed intend to sum the two coefficients in order to interpret the impact of the lagged spread for the second loan category. Does that sound reasonable? In this case, a negative result for the sum is not really plausible?

[Sebastian Kripfganz]

I guess one can do that, although you might ask a bit too much from your data given that you only have 35 groups. This makes it difficult to obtain precise estimates.

[Sarah Magd]

Dear Prof. Sebastian Kripfganz,

Do we need to care about the high multicollinearity in our sys-GMM model? If yes, how do we test for multicollinearity in our sys-GMM model after estimating it with xtdpdgmm

[Sebastian Kripfganz]

Multicollinearity is a potential problem irrespective of the estimator you are using. There is nothing special about it when using the system GMM estimator. If there is perfect collinearity, most commands will omit one or more of the regressors. In general, you might simply want to have a look at the correlation among your regressors.

[Sarah Magd]

Thanks, Prof. Sebastian Kripfganz for your reply.

- The mutual correlation between the variables is less than 0.5.
- Do I need to run the VIF command? If yes, Should I use the usual "reg Y X1 X2..Xn" and then type VIF? Should I also control for fixed effects and robust standard errors?

[Sebastian Kripfganz]

I would suggest to follow the usual practice in your field of research. I am afraid I cannot give you any specific advice.