Overidentified Model

[Arnola Putri]

Hi there, i would like to ask for opinions regarding overidentified model. I want to do a research on child labour on district level in Indonesia. I have a small panel data consist of 34 N and 8 T, with 1 dependent variable (child labour) and 3 independent variables (unemployement, average household expenditure, average length of school). I use Arellano-Bond estimation and Sargan Test and found that my model is overidentified.

my questions are:
- am i using the right method to do this research? and if i am, am i doing the steps right?
- what is overidentified and what should i do if the Sargan Test result shows that my model is overidentified?

below is my command:

encode Provinsi, gen(Prov)

xtset Prov Tahun
       panel variable:  Prov (strongly balanced)
        time variable:  Tahun, 2012 to 2019
                delta:  1 unit

. xtabond C P R T, lags(1) artests(2)

Arellano-Bond dynamic panel-data estimation     Number of obs     =        204
Group variable: Prov                            Number of groups  =         34
Time variable: Tahun
                                                Obs per group:
                                                              min =          6
                                                              avg =          6
                                                              max =          6

Number of instruments =     21                  Wald chi2(4)      =      42.56
                                                Prob > chi2       =     0.0000
One-step results
------------------------------------------------------------------------------
           C |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           C |
         L1. |   .3403482   .0642567     5.30   0.000     .2144075     .466289
             |
           P |   .0090138   .0048503     1.86   0.063    -.0004926    .0185202
           R |  -2367.249   3238.778    -0.73   0.465    -8715.138     3980.64
           T |  -70.68445   397.1617    -0.18   0.859     -849.107    707.7381
       _cons |    15609.8   22564.17     0.69   0.489    -28615.15    59834.76
------------------------------------------------------------------------------
Instruments for differenced equation
        GMM-type: L(2/.).C
        Standard: D.P D.R D.T
Instruments for level equation
        Standard: _cons

. estat sargan
Sargan test of overidentifying restrictions
        H0: overidentifying restrictions are valid

        chi2(16)     =  78.47937
        Prob > chi2  =    0.0000

and below is my dataset (i use ipolate to manipulate the missing values)

* Example generated by -dataex-. To install: ssc install dataex
clear
input str20 Provinsi int Tahun double(P R C T) long Prov
"Aceh"               2012    584100 8.36    2096  9.06  1
"Aceh"               2013    627381 8.44  2326.5 10.12  1
"Aceh"               2014    679850 8.71    2557  9.02  1
"Aceh"               2015    752118 8.77    1884  9.93  1
"Aceh"               2016    808094 8.86     453  7.57  1
"Aceh"               2017 902994.59 8.98    5556  6.57  1
"Aceh"               2018 973817.35 9.09    5454  6.34  1
"Aceh"               2019 993433.33 9.18    2602  6.17  1
"Bali"               2012    818697 7.25   13642  3.43  2
"Bali"               2013    939726 7.32 11156.5  3.65  2
"Bali"               2014   1047711 7.35    8671  5.14  2
"Bali"               2015   1118101 7.46    8026  6.29  2
"Bali"               2016   1211879 7.62   13605   2.6  2
"Bali"               2017 1360994.3 7.78    8221  3.78  2
"Bali"               2018 1419495.2 7.84   12357  3.61  2
"Bali"               2019 1509974.2 7.98    7904  3.58  2
"Banten"             2012    705098 7.85    4030  7.03  3
"Banten"             2013    818267  8.1  2704.5  6.58  3
"Banten"             2014    931436 8.35    1379  6.13  3
"Banten"             2015   1044605 8.36    2465  5.68  3
"Banten"             2016   1157774 8.49    9971  5.23  3
"Banten"             2017 1303765.8 8.62    3885  5.54  3
"Banten"             2018 1414572.8 8.87    5135  5.11  3
"Banten"             2019 1455946.9 8.94    4273  4.49  3
"Bengkulu"           2012    542220 6.92    1695  4.47  4
"Bengkulu"           2013    580271 6.96    1339  4.15  4
"Bengkulu"           2014    644011 6.97     983  4.18  4
"Bengkulu"           2015    667401 7.05     359  4.65  4
"Bengkulu"           2016    774525 7.12    3530  2.76  4
"Bengkulu"           2017 898382.51 7.28    1162  4.28  4
"Bengkulu"           2018 899727.88 7.46    1601   3.7  4
"Bengkulu"           2019 1002864.4 7.69    2676  3.76  4
"DI Yogyakarta"      2012    699727 7.73    3602  3.14  5
"DI Yogyakarta"      2013    784864 7.79    2297     3  5
"DI Yogyakarta"      2014    900699 7.82     992  3.24  5
"DI Yogyakarta"      2015    920786 8.03    1034  4.54  5
"DI Yogyakarta"      2016   1044770 8.13    1709  4.82  5
"DI Yogyakarta"      2017 1134979.4 8.29    1423  4.23  5
"DI Yogyakarta"      2018 1224306.5 8.37    3830  3.91  5
"DI Yogyakarta"      2019 1287201.4 8.51    2212  4.04  5
"DKI Jakarta"        2012    498094 6.85    3000  4.11  6
"DKI Jakarta"        2013    571752  6.9    2333   4.3  6
"DKI Jakarta"        2014    659839 7.05    1666  4.19  6
"DKI Jakarta"        2015    830472 7.14    3154  4.47  6
"DKI Jakarta"        2016    870412 7.23    2901  4.21  6
"DKI Jakarta"        2017 938801.32 7.34    2834     4  6
"DKI Jakarta"        2018 1006077.5 7.39     881  3.91  6
"DKI Jakarta"        2019 1036176.7 7.59    1183  3.82  6
"Gorontalo"          2012    584341 7.73    1237  3.95  7
"Gorontalo"          2013    648554 7.82   958.5  4.19  7
"Gorontalo"          2014    700073 7.89     680  3.68  7
"Gorontalo"          2015    760612 7.97    1000   4.1  7
"Gorontalo"          2016    842912 8.12    2204  3.29  7
"Gorontalo"          2017 918349.42 8.29    1562  3.81  7
"Gorontalo"          2018    940635 8.52    2189  3.37  7
"Gorontalo"          2019 983640.61 8.75    1542  3.11  7
"Jambi"              2012    700296 8.63    2034   3.9  8
"Jambi"              2013    777409 8.72  1463.5  3.24  8
"Jambi"              2014    780346 8.84     893  3.33  8
"Jambi"              2015    928602    9    2280  4.07  8
"Jambi"              2016   1070962 9.12    3540  2.72  8
"Jambi"              2017 1140166.4 9.19    3187  3.02  8
"Jambi"              2018 1302661.1 9.32    1905  3.37  8
"Jambi"              2019 1339725.6 9.38    1457  3.18  8
"Jawa Barat"         2012    613273 6.62    8402  3.54  9
"Jawa Barat"         2013    672211 6.69  9172.5  3.99  9
"Jawa Barat"         2014    786711 6.83    9943  4.04  9
"Jawa Barat"         2015    783050 6.93    3533  5.15  9
"Jawa Barat"         2016    860227 6.98   15784  4.23  9
"Jawa Barat"         2017 929135.41 7.05   28737  4.36  9
"Jawa Barat"         2018 1028671.8 7.12   27270  4.18  9
"Jawa Barat"         2019 1080369.7 7.31   17438  4.35  9
"Jawa Tengah"        2012    751833 7.48   19508  5.19 10
"Jawa Tengah"        2013    813926 7.59 17249.5  3.66 10
"Jawa Tengah"        2014    880425  7.6   14991   3.8 10
"Jawa Tengah"        2015    956156 7.76    8664  4.92 10
"Jawa Tengah"        2016   1047247 7.89    7941  5.45 10
"Jawa Tengah"        2017   1157806 7.99   14108  4.77 10
"Jawa Tengah"        2018   1226469    8   20514  4.35 10
"Jawa Tengah"        2019 1250362.5  8.2   13868  4.18 10
"Jawa Timur"         2012    949152 8.83   12613  9.02 11
"Jawa Timur"         2013   1065917 8.87 11645.5  7.95 11
"Jawa Timur"         2014   1127400 9.04   10678  7.38 11
"Jawa Timur"         2015   1193642 9.15   11328   7.5 11
"Jawa Timur"         2016   1296926 9.24   16338  7.95 11
"Jawa Timur"         2017 1443927.9 9.36   20800  6.91 11
"Jawa Timur"         2018 1560354.5 9.48   15523  6.41 11
"Jawa Timur"         2019 1617640.1  9.7   20899  5.94 11
"Kalimantan Barat"   2012    597163  8.8    4227  7.71 12
"Kalimantan Barat"   2013    649515 8.81  2732.5  9.91 12
"Kalimantan Barat"   2014    748665 9.15    1238 10.51 12
"Kalimantan Barat"   2015    794355 9.16    1746  9.93 12
"Kalimantan Barat"   2016    846106 9.27    1505  7.05 12
"Kalimantan Barat"   2017  903859.3 9.38    4537  9.29 12
"Kalimantan Barat"   2018    965837 9.58    2524  6.95 12
"Kalimantan Barat"   2019 1002239.1 9.81    3910  6.69 12
"Kalimantan Selatan" 2012    484661 6.33    6162  5.23 13
"Kalimantan Selatan" 2013    547748 6.54  5186.5   5.3 13
"Kalimantan Selatan" 2014    636019 6.67    4211  5.75 13
"Kalimantan Selatan" 2015    668499 6.71    2106  5.69 13
end
label values Prov Prov
label def Prov 1 "Aceh", modify
label def Prov 2 "Bali", modify
label def Prov 3 "Banten", modify
label def Prov 4 "Bengkulu", modify
label def Prov 5 "DI Yogyakarta", modify
label def Prov 6 "DKI Jakarta", modify
label def Prov 7 "Gorontalo", modify
label def Prov 8 "Jambi", modify
label def Prov 9 "Jawa Barat", modify
label def Prov 10 "Jawa Tengah", modify
label def Prov 11 "Jawa Timur", modify
label def Prov 12 "Kalimantan Barat", modify
label def Prov 13 "Kalimantan Selatan", modify

Thank you in advanced

[Sebastian Kripfganz]

Let's first clarify the terminology a bit: The Sargan test does not test whether your model is overidentified. It test whether the so-called overidentifying restrictions in an overidentified model are valid. The model is overidentified whenever you have more instruments than regressors. Here, you have 21 instruments for 5 regressors. That yields 21-5=16 overidentifying restrictions (which equals the degrees of freedom of the Sargan test).

What are these overidentifying restrictions? You only need 5 instruments to identify (or, simplified: to estimate) your model with 5 regressors. Every additional instrument adds another restriction that the estimator needs to satisfy, a so-called overidentifying restriction (because its more than what is needed for identification). The Sargan test checks whether these overidentifying restrictions are valid (or, simplified: whether the additional instruments are valid, assuming that there are at least 5 valid instruments included). Effectively, it compares the overidentified model with 21 instruments to a just-identified model with 5 instruments. (It does not make an explicit assumption about which 5 out of your 21 instruments are assumed to be valid. Just keep in mind that the Sargan test can never test the validity of all 21 instruments jointly.) If the estimates from the overidentified model are very different from the just-identified model, this suggests that the additional instruments are invalid. All of this gets summarized in the p-value of the Sargan test statistic.

The following presentation might also be helpful:

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