xtabond2: question on determining appropriate lags

[Siyan Zhao]

Hi everyone,

I have a panel data with n=48, T=40. The data contains people’s subjective experience of their daily social interactions (measured separately by how positive, negative, and meaningful the interactions are) and their daily depression score. Each participant completed 40 days of the study and there were a total of 48 participants.

I want to understand how the positivity, negativity, and meaningfulness of one’s social interactions affect their depression level. Since depression is roughly consistent from day-to-day, I want to include a lagged depression score as a dependent variable. From all the readings, it seems that lagged variable regression is the most appropriate model. So I ran a regression with xtabond2.

The command I ran was:

xtabond2 depression_final L.depression_final pos_si neg_si meaningful, gmm(depression_final, lag(2 3) collapse) gmm(pos_si neg_si meaningful, lag(1 1) collapse) twostep

The results are below:

. xtabond2 depression_final L.depression_final pos_si neg_si meaningful, gmm(depression_final, lag(2 3) colla
> pse) gmm(pos_si neg_si meaningful, lag(1 1) collapse) twostep
Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.

Dynamic panel-data estimation, two-step system GMM
------------------------------------------------------------------------------
Group variable: pid_cat                         Number of obs      =      1161
Time variable : dateval                         Number of groups   =        48
Number of instruments = 10                      Obs per group: min =         3
Wald chi2(4)  =     39.39                                      avg =     24.19
Prob > chi2   =     0.000                                      max =        41
----------------------------------------------------------------------------------
depression_final |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-----------------+----------------------------------------------------------------
depression_final |
             L1. |   .1415712   .0343344     4.12   0.000      .074277    .2088654
                 |
          pos_si |   -.122131   .0306348    -3.99   0.000    -.1821742   -.0620879
          neg_si |  -.0937629    .032896    -2.85   0.004    -.1582378   -.0292879
      meaningful |   .1380227   .0533276     2.59   0.010     .0335025    .2425428
           _cons |    2.86786   .3796965     7.55   0.000     2.123668    3.612051
----------------------------------------------------------------------------------
Warning: Uncorrected two-step standard errors are unreliable.

Instruments for first differences equation
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(2/3).depression_final collapsed
    L.(pos_si neg_si meaningful) collapsed
Instruments for levels equation
  Standard
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    DL.depression_final collapsed
    D.(pos_si neg_si meaningful) collapsed
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -3.51  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =  -1.47  Pr > z =  0.142
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(5)    =  10.11  Prob > chi2 =  0.072
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(5)    =  10.81  Prob > chi2 =  0.055
  (Robust, but can be weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  GMM instruments for levels
    Hansen test excluding group:     chi2(1)    =   0.97  Prob > chi2 =  0.326
    Difference (null H = exogenous): chi2(4)    =   9.84  Prob > chi2 =  0.043
  gmm(depression_final, collapse lag(2 3))
    Hansen test excluding group:     chi2(2)    =   3.69  Prob > chi2 =  0.158
    Difference (null H = exogenous): chi2(3)    =   7.12  Prob > chi2 =  0.068

Given the analysis and the results, my questions are:
1) Do I need the lagged independent term: gmm(pos_si neg_si meaningful, lag(1 1) collapse)? It seems like a good start to include the first lagged independent variables (referring to Sebastian’s response here). But I don’t quite understand why the independent term is needed.

2) Also, for gmm(pos_si neg_si meaningful, lag(1 1) collapse) part, I didn't use iv() for this because the help says iv is for exogenous variables. Since I have priori hypothesis that these 3 variables influence depression score, I don't think it's appropriate to use iv command. Is that correct?

2) Is lag(2 3) the appropriate specification for the lagged dependent variable? I used lag 2 - 3 since my T(40) is not large, compared to n(48). So I can’t afford to use all the lags (i.e., gmm(depression_final, lag(2 .))). However, the coefficient changes if I include all lags for the dependent variable (see result below). Especially for the L1.depression term, the coefficient is larger, more aligned with existing literature. So should I include all lagged dependent variables even though the number of instrument is higher than my N?

3) Why is the L1.depression coefficient so low even though depression and L1.depression are highly correlated (~0.80 using pwcorr)? While I understand that Pearson's correlation and system GMM are completely different tests, I can’t understand on a high level why the high correlation of the lagged dependent variable is not present in xtabond2 result.


I’ve read the following resources on xtabond2:

• Stata help on xtabond2
• Roodman's Stata journal chapter on how to do xtabond,
• Estimating dynamic panel data

I sincerely apologize for the long post. Thank you very much for your time and for your help!

Siyan

. xtabond2 depression_final L.depression_final pos_si neg_si meaningful, gmm(L.depression_final,  collapse) g
> mm(pos_si neg_si meaningful, lag(1 1) collapse) twostep
Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.
Warning: Number of instruments may be large relative to number of observations.
Warning: Two-step estimated covariance matrix of moments is singular.
  Using a generalized inverse to calculate optimal weighting matrix for two-step estimation.
  Difference-in-Sargan statistics may be negative.

Dynamic panel-data estimation, two-step system GMM
------------------------------------------------------------------------------
Group variable: pid_cat                         Number of obs      =      1161
Time variable : dateval                         Number of groups   =        48
Number of instruments = 49                      Obs per group: min =         3
Wald chi2(4)  =  37100.72                                      avg =     24.19
Prob > chi2   =     0.000                                      max =        41
----------------------------------------------------------------------------------
depression_final |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-----------------+----------------------------------------------------------------
depression_final |
             L1. |   .1674352    .002132    78.53   0.000     .1632565    .1716139
                 |
          pos_si |  -.0777748   .0031215   -24.92   0.000    -.0838929   -.0716567
          neg_si |  -.0721509   .0006684  -107.94   0.000     -.073461   -.0708407
      meaningful |    .072521   .0045241    16.03   0.000     .0636539    .0813882
           _cons |   2.530163   .0422862    59.83   0.000     2.447284    2.613043
----------------------------------------------------------------------------------
Warning: Uncorrected two-step standard errors are unreliable.

Instruments for first differences equation
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(1/.).L.depression_final collapsed
    L.(pos_si neg_si meaningful) collapsed
Instruments for levels equation
  Standard
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    D.L.depression_final collapsed
    D.(pos_si neg_si meaningful) collapsed
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -3.45  Pr > z =  0.001
Arellano-Bond test for AR(2) in first differences: z =  -1.06  Pr > z =  0.290
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(44)   =  74.52  Prob > chi2 =  0.003
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(44)   =  43.79  Prob > chi2 =  0.480
  (Robust, but can be weakened by many instruments.)

[Sebastian Kripfganz]

1. If you just use the instruments from the lagged dependent variable, these instruments may not be strong enough for the independent variables. So, it is almost always recommended to also include the lagged independent variables as instruments.
2. Technically, gmm() with collapse is having the same effect as iv() [depending on exact specification of these options]. The key decisions are about the lag range and whether those instruments shall be differenced or not. If you do not want to assume that the independent variables are strictly exogenous, then the default settings of gmm() should be fine, while the default for iv() is typically not.
3. Actually, T=40 is quite large for such an estimation. These GMM estimators are designed for situations where N is typically much larger than T. Given that T is large, it is indeed a good idea not to use all lags. I cannot see much of a difference in the coefficients from the two specifications. The standard errors appear to be too small (in particular in the second specification) which is likely a consequence of not using the robust option to get Windmeijer-corrected standard errors.
4. One reason for a high unconditional autocorrelation of the dependent variable could be that time-invariant group-specific effects play an important role. This is taken into account by the GMM estimator but not by a simple correlation coefficient.

More on dynamic panel data GMM estimation:

• Kripfganz, S. (2019). Generalized method of moments estimation of linear dynamic panel data models. Proceedings of the 2019 London Stata Conference.
• XTDPDGMM: new Stata command for GMM estimation of linear (dynamic) panel data models

The following topic highlights some general problems with xtabond2:

• A collection of bugs in xtabond2, xtabond, xtdpd, xtdpdsys, gmm

[Siyan Zhao]

Thank you so so SO much for the detailed response Sebastian! Your answer clarified the questions that have plagued me for weeks. I will definitely check out the resources you linked.

I have a few follow-up questions:

2. Technically, gmm() with collapse is having the same effect as iv() [depending on exact specification of these options]. The key decisions are about the lag range and whether those instruments shall be differenced or not. If you do not want to assume that the independent variables are strictly exogenous, then the default settings of gmm() should be fine, while the default for iv() is typically not.

I used collapse to reduce the the number of instruments (from 722 to 10). However, given that I don't believe the independent are strictly exogenous, should I still use default gmm(), i.e., gmm(depression_final, lag(2 3)) gmm(pos_interaction neg_interaction meaningful, lag(1 1)), even if it explodes the number of instruments? Even with eq(level) to only use level instruments, I still get 400+ instruments. It also seems like using level only instruments change the coefficient of the lagged dependent variable drastically (result attached below; I also added robust to the command).

4. One reason for a high unconditional autocorrelation of the dependent variable could be that time-invariant group-specific effects play an important role. This is taken into account by the GMM estimator but not by a simple correlation coefficient.

This makes a lot of sense. A stupid follow-up.: if I want to examine, say, the effect of gender on depression, since it's a time-invariant term, I should do a random effect model with xtreg with i.gender as the dependent variable. Is that right?

Thank you again for your attention!
Siyan

. xtabond2 depression_final L.depression_final pos_interaction neg_interaction meaningful, gmm(depression_fin
> al, lag(2 3) eq(level)) gmm(pos_interaction neg_interaction meaningful, lag(1 1) eq(level)) twostep robust
Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.
Warning: Number of instruments may be large relative to number of observations.
Warning: Two-step estimated covariance matrix of moments is singular.
  Using a generalized inverse to calculate optimal weighting matrix for two-step estimation.
  Difference-in-Sargan statistics may be negative.

Dynamic panel-data estimation, two-step system GMM
------------------------------------------------------------------------------
Group variable: pid_cat                         Number of obs      =      1161
Time variable : dateval                         Number of groups   =        48
Number of instruments = 436                     Obs per group: min =         3
Wald chi2(4)  =     59.13                                      avg =     24.19
Prob > chi2   =     0.000                                      max =        41
----------------------------------------------------------------------------------
                 |              Corrected
depression_final |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-----------------+----------------------------------------------------------------
depression_final |
             L1. |   .6496806   .0969365     6.70   0.000     .4596886    .8396726
                 |
 pos_interaction |  -.1891746   .1491851    -1.27   0.205    -.4815719    .1032228
 neg_interaction |  -.1043089   .1243378    -0.84   0.402    -.3480066    .1393887
      meaningful |  -.0440572   .1095981    -0.40   0.688    -.2588655    .1707511
           _cons |   1.605658   .8417413     1.91   0.056    -.0441245    3.255441
----------------------------------------------------------------------------------
Instruments for levels equation
  Standard
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    DL(2/3).depression_final
    DL.(pos_interaction neg_interaction meaningful)
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -3.37  Pr > z =  0.001
Arellano-Bond test for AR(2) in first differences: z =   1.19  Pr > z =  0.235
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(431)  = 609.19  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(431)  =  44.61  Prob > chi2 =  1.000
  (Robust, but can be weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  gmm(depression_final, eq(level) lag(2 3))
    Hansen test excluding group:     chi2(254)  =  35.41  Prob > chi2 =  1.000
    Difference (null H = exogenous): chi2(177)  =   9.20  Prob > chi2 =  1.000
  gmm(pos_interaction neg_interaction meaningful, eq(level) lag(1 1))
    Hansen test excluding group:     chi2(173)  =  43.05  Prob > chi2 =  1.000
    Difference (null H = exogenous): chi2(258)  =   1.56  Prob > chi2 =  1.000

[Sebastian Kripfganz]

You can use the gmm() option with appropriate lags to account for endogenous regressors. Just use the collapse option to reduce the number of instruments. 400+ instruments is definitely way too many in your case.

If you specify iv(i.gender, eq(level)), the underlying assumption is similar to a random-effects model. The gender dummy is assumed to be uncorrelated with the unobserved effects. Please see the following article for more information about time-invariant regressors in dynamic panel models:

• 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.

[Siyan Zhao]

Thanks a bunch!!! Your responses are super helpful.

[John Sgr]

Dear Sebastian,

I have further questions to you.

I have a panel data with n=769, T=12. The data contains information about individuals’ depression level(dep_score), average income(avg_inc), health status(health_st), and injuries(injury) in a monthly basis and time dummies(w_*) for each month. Individualid variable is identifier for individuals.

I want to understand how income, health status and injuries affect the depression level. Since depression status is roughly stable, I used first lag of depression score as an independent variable. Moreover I use dynamic model as depression could affect average income and health status in the subsequent period. Injuries and time dummies are exogenous variables in my model.

I ran a regression with xtabond2. I used robust option for Windmeijer’s correction. I used cluster to allow serial correlation within observations. Small options stands for getting t values instead of z. noleveleq used to estimate difference GMM model. Collapse is for decreasing the number of instruments.

The command I ran was:
Code:

 xtabond2 L(0/1).dep_score avg_inc health_st injury w_*,  gmm(L.dep_score avg_inc health_st, lag(1 3) collapse) iv(injury w_*) robust cluster(individualid) small two noleveleq

Given the background and code, my questions are

1. Is there any advantage of using xtdpdgmm instead of xtabond2 in my case and in general?

2. I did not get what exactly the intrument for dependent variable is. Does it the first difference of first lag of Y (delta Y_{i,t-1}) or first lag itself(Y_{i,t-1})?

3. Can I just specify exogenous variables as iv(injury w_*) which are not instruments of Y but explanatory variables of Y? What's the correct way to include them in the command?

4. Does it make sense to specify robust and cluster() at the same time?

5. Do I need to specify e(diff) in the gmm parentheses while I’m using noleveleq option? It seems e(diff) is option for how to use instruments while nolevelq is about model itself.

6. I decided to use diff option to eliminate unobserved heterogeneity like eliminating fixed-effects. Does it makes sense?

Sincere thanks for your time.

Best regards,
John

[Sebastian Kripfganz]

1. For such a difference GMM estimation, it should not make much of a difference whether you use xtdpdgmm or xtabond2. In general, the following could be some benefits of using xtdpdgmm, depending on what you would like to do:

• You could use an iterated GMM estimator to limit the influence of the first-step weighting matrix.
• You could add nonlinear moment conditions valid under the absence of serial correlation to obtain more efficient estimates and to avoid possible identification issues when the dependent variable is quite persistent.
• There are some additional less often used options and postestimation features.
• If you prefer to continue working with xtabond2, make sure to update the command to the latest version to avoid a couple of issues that were discussed in the following topic (with a follow-up discussion in another topic linked therein): https://www.statalist.org/forums/for...d-xtdpdsys-gmm

2. The way you specified the model, lags 1 to 3 of the first lag (i.e. effectively lags 2 to 4) of the dependent variable are used as instruments for the first-differenced model.

3. If the variables injury w* are strictly exogenous, you can specify them as instruments in the iv() option. When you use dummy variables in the w* notation, make sure to update xtabond2 to the latest version to avoid a bug that was present in an earlier version (see link above).

4. The robust option is redundant when you specify the cluster() option.

5. The eq(diff) suboption is implied by the noleveleq option, so you do not have to use it. However, I personally prefer to always specify the eq() suboption to avoid any unintended consequences in case you want to extend the model later on.

6. Yes, using the eq(diff) suboption or the noleveleq option is the most straightforward way to eliminate fixed effects.

[John Sgr]

Thanks for this clear explanation!

1. From you explanation I understand that option eq(diff) is not option for defining instruments as a first differenced form (like using delta Y as an instrument). So instruments are always in level rather than lagged form, right?

2. If I use Y_t-1 as an explanatory variable can I use Y_t-1 as instrument for Y_t or should I start with Y_t-2? Application of this question in my case is: Is it true to define my command as gmm(dep_score, lag(1 3)) instead of gmm(L.dep_score, lag(1 3)) while using L.dep_score as explanatory variable?

[Sebastian Kripfganz]

1. xtabond2 by default transforms instruments specified with the iv() option for eq(diff) into first differences, i.e. D.Y is used as an instrument for the first-differenced model. If you just want to use Y itself, you need to specify the passthru suboption. This is different for the gmm() option, which by default uses Y for the first-differenced model and not D.Y. If this is confusing, my xtdpdgmm command follows a what-you-type-is-what-you-get approach, i.e. the variables are only differenced when you explicitly specify it.

2. Y_t-1 never works as an instrument for the first-differenced model because it is correlated by construction with the first-differenced error term. For eq(diff), you always need to start with Y_t-2. gmm(L.dep_score, lag(1 3)) is valid, while gmm(dep_score, lag(1 3)) is not.

[John Sgr]

Dear Sebastian,

I owed you too much, thanks for your kind help!

[Sabrin Rahman]

I am trying to see the impact of Foreign Direct Investment (FDI) on Gender Equality. Please I need help!

I am struggling to choose Exogenous Variables for GMM ivstyle? My Dependent variable is Gender Inequality Index(GII) and Independent Variable is FDI.

I also wanted to check whether GMM is fine or not I have 80 countries and my T is 24.
I am having difficulties to find exogenous variable for my model. My model is given below.

I am going to use Dynamic Panel Model. My lagged Dependent Variable is lag of GII which I am going to use in GMMSTYLE() but I am not sure what to include in the IVSTYLE(). I am not sure whether I should use lags of control variables in the ivstyle(external Instruments) command under XTABOND2. If I use lags of control variables I have around 9 control variables which ones shall I use?

My Control Variables are: Fertility rate, GDP growth,tradeopenness, natural resource rent, Polity IV, secondary education, govt expenditure etc.

My Main Explanatory variable is FDI Inflows

Model:
Gender Inequality Index(GII) = a+GIIt-1+bFDI+ (ControlV)+U

[Adam Abdulrahman]

Hello! I'm currently running GMM tests in Stata but I keep getting both a very high number of instruments (100-200+) and an even higher Wald Chi2 score (1.71e+08).

Below is the code that I ran and its following output:

xtdpdsys PFAGDP LNGDPPcap SDGDum LNGDPPcapxSDGDum DependencyRatio Inflation CMReturns PopGrowth LFParticRate, lags(1) twostep artests(2).
System dynamic panel-data estimation Number of obs = 603
Group variable: Country Number of groups = 35
Time variable: Year
Obs per group:
min = 1
avg = 17.22857
max = 20

Number of instruments = 218 Wald chi2(9) = 1.71e+08
Prob > chi2 = 0.0000
Two-step results
----------------------------------------------------------------------------------
PFAGDP | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-----------------+----------------------------------------------------------------
PFAGDP |
L1. | . 9695742 .003495 277.42 0.000 .9627241 .9764244
|
LNGDPPcap | 35.98271 1.9354 18.59 0.000 32.18939 39.77602
SDGDum | 16.02427 5.916328 2.71 0.007 4.428483 27.62006
LNGDPPcapxSDGDum | -3.615439 1.266003 -2.86 0.004 -6.096759 -1.134119
DependencyRatio | 3.062755 .2688706 11.39 0.000 2.535778 3.589732
Inflation | -.1901946 .0337445 -5.64 0.000 -.2563326 -.1240565
CMReturns | 11.93403 .2071649 57.61 0.000 11.528 12.34007
PopGrowth | -2.534041 .3077528 -8.23 0.000 -3.137226 -1.930857
LFParticRate | -.3305291 .0430299 -7.68 0.000 -.4148661 -.246192
_cons | -152.4423 8.000879 -19.05 0.000 -168.1238 -136.7609
----------------------------------------------------------------------------------
Warning: gmm two-step standard errors are biased; robust standard
errors are recommended.
Instruments for differenced equation
GMM-type: L(2/.).PFAGDP
Standard: D.LNGDPPcap D.SDGDum D.LNGDPPcapxSDGDum D.DependencyRatio
D.Inflation D.CMReturns D.PopGrowth D.LFParticRate
Instruments for level equation
GMM-type: LD.PFAGDP
Standard: _cons

When i tried running the following code, i was able to lower the number of instruments and Wald Chi2 score, but both are still relatively high:
xtdpdsys PFAGDP LNGDPPcap SDGDum LNGDPPcapxSDGDum DependencyRatio Inflation CMReturns PopGrowth LFParticRate, lags(1) maxldep(1) maxlags(1) pre(LNGDPPcap, lagstruct(1,1)) artests(2)

xtdpdsys PFAGDP LNGDPPcap SDGDum LNGDPPcapxSDGDum DependencyRatio Inflation CMReturns PopGrowth LFParticRate, lags(1) maxldep(1) maxlags(1) pre(LNGDPPcap, lags
> truct(1,1)) artests(2)
note: LNGDPPcap dropped because of collinearity

System dynamic panel-data estimation Number of obs = 602
Group variable: Country Number of groups = 35
Time variable: Year
Obs per group:
min = 1
avg = 17.2
max = 20

Number of instruments = 85 Wald chi2(10) = 5043.31
Prob > chi2 = 0.0000
One-step results
----------------------------------------------------------------------------------
PFAGDP | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-----------------+----------------------------------------------------------------
PFAGDP |
L1. | 1.000401 .0301854 33.14 0.000 .9412387 1.059563
|
LNGDPPcap |
--. | 19.03887 27.25736 0.70 0.485 -34.38458 72.46231
L1. | -13.77005 26.78573 -0.51 0.607 -66.26912 38.72902
|
SDGDum | -8.781053 47.83211 -0.18 0.854 -102.5303 84.96817
LNGDPPcapxSDGDum | 1.667439 10.34791 0.16 0.872 -18.61409 21.94897
DependencyRatio | -1.056532 1.346547 -0.78 0.433 -3.695715 1.582651
Inflation | -.1494649 .2983697 -0.50 0.616 -.7342587 .4353289
CMReturns | 9.951293 2.3603 4.22 0.000 5.32519 14.5774
PopGrowth | -1.691797 1.676526 -1.01 0.313 -4.977727 1.594133
LFParticRate | .0269194 .3792645 0.07 0.943 -.7164253 .7702641
_cons | -18.29174 33.14363 -0.55 0.581 -83.25207 46.66859
----------------------------------------------------------------------------------
Instruments for differenced equation
GMM-type: L(2/2).PFAGDP L(1/1).L.LNGDPPcap
Standard: D.LNGDPPcap D.SDGDum D.LNGDPPcapxSDGDum D.DependencyRatio
D.Inflation D.CMReturns D.PopGrowth D.LFParticRate
Instruments for level equation
GMM-type: LD.PFAGDP LD.LNGDPPcap
Standard: _cons

Any ideas on how to fix this, and also specify the use of only 2 variables (lag of the dependent variable PFAGDP, and LNGDPPcap) as the only instruments?

Thank you so much!