Number of instruments in GMM-system estimator

[Marie Blaise]

Dear all,

I am currenlty working on the causal relationship between the informal care decision and the life satisfaction of potential caregivers. As I suspect my first approach, OLS with fixed effect, to biaised due to several sources of endogeneity, I finally use two-step GMM-system estimator. My main specification is the following:


xtabond2 Lifesat lagLS Informal Gender i.Agecat Obj_health i.Marital i.Educ i.Occupation_cat Children Whours ln_st_Hninc i.Urban i.year if RC==2 & gmm_sample==1, h(1) gmm(lagLS, lag(2 3)) gmm(Informal Obj_health i.Marital i.Educ i.Occupation_cat Children Whours ln_st_Hninc i.Urban, lag(1 2)) iv(Gender i.Agecat) iv(i.year, eq(level)) small twostep



Dynamic panel-data estimation, two-step system GMM
------------------------------------------------------------------------------
Group variable: nomem_encr Number of obs = 9180
Time variable : year Number of groups = 1188
Number of instruments = 470 Obs per group: min = 1
F(38, 1187) = 3439.01 avg = 7.73
Prob > F = 0.000 max = 9
-------------------------------------------------------------------------------
Lifesat | Coef. Std. Err. t P>|t| [95% Conf. Interval]
--------------+----------------------------------------------------------------
lagLS | .0661194 .0113218 5.84 0.000 .0439063 .0883324
Informal | -.0857742 .0248276 -3.45 0.001 -.1344851 -.0370632
Gender | -.0503902 .033579 -1.50 0.134 -.1162711 .0154907
_IAgecat_3 | .2282389 .0517332 4.41 0.000 .1267403 .3297376
_IAgecat_4 | .103332 .0522224 1.98 0.048 .0008735 .2057905
_IAgecat_5 | .1005478 .0544976 1.84 0.065 -.0063747 .2074702
_IAgecat_6 | .1969415 .0631799 3.12 0.002 .0729848 .3208982
_IAgecat_7 | .1364177 .0751936 1.81 0.070 -.0111095 .2839449
Obj_health | .1210188 .0424115 2.85 0.004 .037809 .2042286
_IMarital_2 | .2705168 .0652681 4.14 0.000 .1424632 .3985705
_IMarital_3 | .2307198 .0908654 2.54 0.011 .0524452 .4089944
_IMarital_4 | -.472043 .0989332 -4.77 0.000 -.6661464 -.2779397
_IMarital_5 | -.3583853 .0896283 -4.00 0.000 -.5342329 -.1825377
_IEduc_2 | -.2848591 .0605002 -4.71 0.000 -.4035583 -.1661599
_IEduc_3 | -.3615773 .0881303 -4.10 0.000 -.5344858 -.1886688
_IEduc_4 | -.2729954 .0961199 -2.84 0.005 -.4615792 -.0844117
_IEduc_5 | .0225658 .0814172 0.28 0.782 -.1371717 .1823034
_IEduc_6 | .0387999 .0951248 0.41 0.683 -.1478316 .2254314
_IEduc_7 | -.6792458 .0951736 -7.14 0.000 -.8659731 -.4925185
_IEduc_8 | -.2539277 .1004043 -2.53 0.012 -.4509174 -.0569379
_IEduc_9 | -1.15633 .2521884 -4.59 0.000 -1.651115 -.6615454
_IOccupatio_1 | -.2841957 .0604651 -4.70 0.000 -.4028262 -.1655653
_IOccupatio_2 | .1725232 .0570797 3.02 0.003 .0605349 .2845116
Children | .0130852 .0393558 0.33 0.740 -.0641294 .0902998
Whours | -.0043842 .0010849 -4.04 0.000 -.0065128 -.0022556
ln_st_Hninc | -.0886799 .019323 -4.59 0.000 -.1265909 -.0507689
_IUrban_2 | -.1629436 .0972537 -1.68 0.094 -.3537519 .0278647
_IUrban_3 | .425273 .0910808 4.67 0.000 .2465756 .6039704
_IUrban_4 | -.1455187 .0927251 -1.57 0.117 -.327442 .0364046
_IUrban_5 | .5804202 .0825634 7.03 0.000 .4184337 .7424066
_Iyear_2009 | .0304288 .0190255 1.60 0.110 -.0068986 .0677562
_Iyear_2010 | .0105775 .0137473 0.77 0.442 -.0163942 .0375492
_Iyear_2012 | -.0914997 .0176903 -5.17 0.000 -.1262075 -.0567919
_Iyear_2013 | -.0035593 .0195913 -0.18 0.856 -.0419967 .0348781
_Iyear_2014 | -.1069861 .0215754 -4.96 0.000 -.1493162 -.064656
_Iyear_2015 | -.0067729 .0223089 -0.30 0.761 -.0505421 .0369963
_Iyear_2017 | -.0604441 .0254589 -2.37 0.018 -.1103936 -.0104947
_Iyear_2018 | -.1204303 .025664 -4.69 0.000 -.1707822 -.0700784
_cons | 7.48364 .2047738 36.55 0.000 7.081881 7.885399
-------------------------------------------------------------------------------
Warning: Uncorrected two-step standard errors are unreliable.

Instruments for first differences equation
Standard
D.(Gender _IAgecat_3 _IAgecat_4 _IAgecat_5 _IAgecat_6 _IAgecat_7)
GMM-type (missing=0, separate instruments for each period unless collapsed)
L(1/2).(Informal Obj_health _IMarital_2 _IMarital_3 _IMarital_4
_IMarital_5 _IEduc_2 _IEduc_3 _IEduc_4 _IEduc_5 _IEduc_6 _IEduc_7 _IEduc_8
_IEduc_9 _IOccupatio_1 _IOccupatio_2 Children Whours ln_st_Hninc _IUrban_2
_IUrban_3 _IUrban_4 _IUrban_5)
L(2/3).lagLS
Instruments for levels equation
Standard
_Iyear_2009 _Iyear_2010 _Iyear_2011 _Iyear_2012 _Iyear_2013 _Iyear_2014
_Iyear_2015 _Iyear_2017 _Iyear_2018
Gender _IAgecat_3 _IAgecat_4 _IAgecat_5 _IAgecat_6 _IAgecat_7
_cons
GMM-type (missing=0, separate instruments for each period unless collapsed)
D.(Informal Obj_health _IMarital_2 _IMarital_3 _IMarital_4 _IMarital_5
_IEduc_2 _IEduc_3 _IEduc_4 _IEduc_5 _IEduc_6 _IEduc_7 _IEduc_8 _IEduc_9
_IOccupatio_1 _IOccupatio_2 Children Whours ln_st_Hninc _IUrban_2
_IUrban_3 _IUrban_4 _IUrban_5)
DL.lagLS
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -11.20 Pr > z = 0.000
Arellano-Bond test for AR(2) in first differences: z = 1.51 Pr > z = 0.131
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(431) = 451.36 Prob > chi2 = 0.240
(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(431) = 429.72 Prob > chi2 = 0.508
(Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
GMM instruments for levels
Hansen test excluding group: chi2(261) = 259.75 Prob > chi2 = 0.510
Difference (null H = exogenous): chi2(170) = 169.97 Prob > chi2 = 0.486
gmm(lagLS, lag(2 3))
Hansen test excluding group: chi2(416) = 393.68 Prob > chi2 = 0.778
Difference (null H = exogenous): chi2(15) = 36.04 Prob > chi2 = 0.002
iv(Gender _IAgecat_3 _IAgecat_4 _IAgecat_5 _IAgecat_6 _IAgecat_7)
Hansen test excluding group: chi2(425) = 418.60 Prob > chi2 = 0.578
Difference (null H = exogenous): chi2(6) = 11.12 Prob > chi2 = 0.085
iv(_Iyear_2009 _Iyear_2010 _Iyear_2011 _Iyear_2012 _Iyear_2013 _Iyear_2014 _Iyear_2015 _Iyear_2017 _Iyear_2018, eq(level))
Hansen test excluding group: chi2(422) = 420.59 Prob > chi2 = 0.510
Difference (null H = exogenous): chi2(9) = 9.13 Prob > chi2 = 0.426




My question is about the number of instruments and whether should I be worried. I have 470 instruments and 1188 units of individuals meaning that the arbitrary rule-of-thumb mentioned by Roodman (2009) is respected: the number of instruments should not outnumber the number of units of individuals. However, Roodman (2009) also says that a too high number of instruments might weaken the Hansen test, which my case, is higher than 0.25 (and it might be a sign of trouble). I have also used the command collapse. The number of instruments has slighlty decreased (roughly 450) but the Hansen test is still above 0.25.

Overall, should I be worried about the results I post here, especially the number of instruments (470) or could I consider that 470 is too low compared with the size sample and the number of units ?

I hope my explanation is clear and I thank you in advance for your time and consideration,

Marie

[Sebastian Kripfganz]

My personal opinion is that it is not very reasonable to consider an upper bound of 0.25 for the acceptable Hansen test p-values. Nevertheless, 431 overidentifying restrictions are still a lot and reducing the instrument count could help.

I would worry more about the still relatively low p-value of the AR(2) test and the rejection of the Difference-in-Hansen test for the gmm(lagLS, lag(2 3)) instruments. The latter can be a consequence of remaining serial correlation. It might help to add further lags of the dependent and independent variables as regressors to the model to allow for richer dynamics.

Further information that might be useful:

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

[Marie Blaise]

Dear Mr Kripfganz,

Thank you so much for your answer. Following your recommandations, I add more lags of both the dependent and independent variables using the command collapse and the AR(2) is much higher (around 0.5).

I have another concern about my regressions. In another section, I analyse whether the results are sensitive to different specifications. As the GMM estimator allows the use of external instruments, I add two external variables from macroeconomic database (the reason for doing so is that my study focuses on the Netherlands where a reform on the long-term care entered into force in 2015). These variables do not vary neither between nor within individuals; they only change through years (for instance: number of formal carers in institution in 2008, in 2009 and so on). I think I should exclude year dummies as it seems extremely collinear but actually I am not quite sure. Please, may I ask you a piece of advice ?


Thanks again for your time and your precious help,

Marie

[Sebastian Kripfganz]

These additional external variables are indeed collinear with the time dummies. Unless you are explicitly interested in the effects of those two variables, I would not include them and stick to the time dummies. Otherwise, if you exclude the time dummies, the question remains to what extent the measured effect is really a "causal" effect of these two variables or just picks up effects of other variables that are constant across individuals (and thus captured by the time dummies).

[Marie Blaise]

Dear Mr Kripfganz,

Thank you again for this comment.
Marie