2
$\begingroup$

We have the model:

$HOURS = \beta_1 + \beta_2 ln(WAGE) + \beta_3EDUC+ \beta_4AGE + \beta_5KIDSL6 + \beta_6KIDSL618 + \beta_7NWIFEINC + e$

Does anyone know why using some variables as instruments make the Wu-hausman test indicate the the regressors are endogenous while using other instruments for instrumental variables the model is not considered to be endogenous. Isn't endogeniety something that could exist even in the OLS model and only dependent on the regressors (not the instruments).

As we can see from the log-file below using (exper exper2) and siblings as instruments the Wu-hausman test indicates that there is endogeniety while using mothereduc, fathereduc and heduc as instruments we do not have endogeniety. Why is this?

I'm under the impression that endogeniety exists och does not exists regardless of which instruments you choose.

reg hours lwage $x2list, vce(robust)

Linear regression                                      Number of obs =     428
                                                       F(  6,   421) =    3.93
                                                       Prob > F      =  0.0008
                                                       R-squared     =  0.0670
                                                       Root MSE      =  755.16

------------------------------------------------------------------------------
             |               Robust
       hours |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       lwage |   -17.4078   81.37728    -0.21   0.831    -177.3642    142.5486
        educ |  -14.44486   18.21292    -0.79   0.428    -50.24445    21.35473
         age |  -7.729976   5.849662    -1.32   0.187    -19.22816    3.768206
      kidsl6 |  -342.5048   131.7733    -2.60   0.010    -601.5205   -83.48919
     kids618 |  -115.0205   29.50866    -3.90   0.000    -173.0232   -57.01786
    nwifeinc |  -.0042458   .0032235    -1.32   0.189    -.0105821    .0020904
       _cons |   2114.697   350.3186     6.04   0.000     1426.106    2803.289
------------------------------------------------------------------------------

. estimate store REG

. ivregress 2sls hours (lwage = exper exper2) $x2list, vce(robust) first

First-stage regressions
-----------------------

                                                  Number of obs   =        428
                                                  F(   7,    420) =      12.62
                                                  Prob > F        =     0.0000
                                                  R-squared       =     0.1641
                                                  Adj R-squared   =     0.1502
                                                  Root MSE        =     0.6667

------------------------------------------------------------------------------
             |               Robust
       lwage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        educ |   .0998844   .0141577     7.06   0.000     .0720556    .1277131
         age |  -.0035204   .0061766    -0.57   0.569    -.0156613    .0086205
      kidsl6 |  -.0558725   .1061345    -0.53   0.599    -.2644936    .1527485
     kids618 |  -.0176484   .0295136    -0.60   0.550    -.0756611    .0403642
    nwifeinc |   5.69e-06   2.76e-06     2.07   0.039     2.75e-07    .0000111
       exper |   .0407097   .0153088     2.66   0.008     .0106183    .0708012
      exper2 |  -.0007473   .0004093    -1.83   0.069    -.0015519    .0000572
       _cons |  -.3579972   .3221853    -1.11   0.267    -.9912938    .2752995
------------------------------------------------------------------------------


Instrumental variables (2SLS) regression               Number of obs =     428
                                                       Wald chi2(6)  =   15.41
                                                       Prob > chi2   =  0.0173
                                                       R-squared     =       .
                                                       Root MSE      =  1291.2

------------------------------------------------------------------------------
             |               Robust
       hours |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       lwage |   1544.818   598.8004     2.58   0.010     371.1913    2718.446
        educ |   -177.449   66.84514    -2.65   0.008     -308.463    -46.4349
         age |  -10.78409   10.57756    -1.02   0.308    -31.51573    9.947557
      kidsl6 |  -210.8339   203.9118    -1.03   0.301    -610.4936    188.8258
     kids618 |  -47.55708   56.47944    -0.84   0.400    -158.2547    63.14058
    nwifeinc |  -.0092491   .0052314    -1.77   0.077    -.0195025    .0010042
       _cons |   2432.198    611.223     3.98   0.000     1234.223    3630.173
------------------------------------------------------------------------------
Instrumented:  lwage
Instruments:   educ age kidsl6 kids618 nwifeinc exper exper2

. estimate store REGIV

. esttab REG REGIV , se(%12.4f) b(%12.5f) star(* 0.10 ** 0.05 *** 0.01) mtitles("OLS" "IV") title("Model test")

Model test
--------------------------------------------
                      (1)             (2)   
                      OLS              IV   
--------------------------------------------
lwage           -17.40780      1544.81848***
                (81.3773)      (598.8004)   

educ            -14.44486      -177.44896***
                (18.2129)       (66.8451)   

age              -7.72998       -10.78409   
                 (5.8497)       (10.5776)   

kidsl6         -342.50482***   -210.83387   
               (131.7733)      (203.9118)   

kids618        -115.02051***    -47.55708   
                (29.5087)       (56.4794)   

nwifeinc         -0.00425        -0.00925*  
                 (0.0032)        (0.0052)   

_cons          2114.69725***   2432.19773***
               (350.3186)      (611.2230)   
--------------------------------------------
N                     428             428   
--------------------------------------------
Standard errors in parentheses
* p<0.10, ** p<0.05, *** p<0.01

. 
end of do-file

. do "C:\Users\Patrick\AppData\Local\Temp\STD03000000.tmp"

. 
. estat endogenous

  Tests of endogeneity
  Ho: variables are exogenous

  Robust score chi2(1)            =  22.2071  (p = 0.0000)
  Robust regression F(1,420)      =   26.355  (p = 0.0000)

. estat overid

  Test of overidentifying restrictions:

  Score chi2(1)          =  1.23424  (p = 0.2666)

. 
end of do-file

Above I do IV regress with exper and exper2 as instruments for lwage. We find the variables are exogenous.

. ivregress 2sls hours (lwage = mothereduc) $x2list, vce(robust) first

First-stage regressions
-----------------------

                                                  Number of obs   =        428
                                                  F(   6,    421) =      12.33
                                                  Prob > F        =     0.0000
                                                  R-squared       =     0.1380
                                                  Adj R-squared   =     0.1257
                                                  Root MSE        =     0.6762

------------------------------------------------------------------------------
             |               Robust
       lwage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        educ |   .1154201   .0155732     7.41   0.000     .0848091     .146031
         age |  -4.87e-06   .0053543    -0.00   0.999    -.0105294    .0105196
      kidsl6 |   -.095265   .1084798    -0.88   0.380    -.3084945    .1179645
     kids618 |  -.0433942    .028462    -1.52   0.128    -.0993396    .0125511
    nwifeinc |   3.21e-06   2.64e-06     1.22   0.225    -1.98e-06    8.41e-06
  mothereduc |  -.0199982   .0115711    -1.73   0.085    -.0427425    .0027461
       _cons |  -.0692528   .3233721    -0.21   0.831    -.7048779    .5663723
------------------------------------------------------------------------------


Instrumental variables (2SLS) regression               Number of obs =     428
                                                       Wald chi2(6)  =   23.90
                                                       Prob > chi2   =  0.0005
                                                       R-squared     =  0.0610
                                                       Root MSE      =  751.35

------------------------------------------------------------------------------
             |               Robust
       hours |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       lwage |  -106.4097    590.134    -0.18   0.857    -1263.051    1050.232
        educ |  -5.158324   64.91348    -0.08   0.937    -132.3864    122.0698
         age |   -7.55598   5.979866    -1.26   0.206     -19.2763    4.164342
      kidsl6 |  -350.0063   135.5988    -2.58   0.010     -615.775   -84.23754
     kids618 |   -118.864   37.42259    -3.18   0.001    -192.2109   -45.51706
    nwifeinc |  -.0039608   .0038792    -1.02   0.307    -.0115639    .0036424
       _cons |   2096.609   382.8154     5.48   0.000     1346.304    2846.913
------------------------------------------------------------------------------
Instrumented:  lwage
Instruments:   educ age kidsl6 kids618 nwifeinc mothereduc

. estimate store IVREGmother

. estat endogenous

  Tests of endogeneity
  Ho: variables are exogenous

  Robust score chi2(1)            =  .023462  (p = 0.8783)
  Robust regression F(1,420)      =  .023051  (p = 0.8794)

. esttab REG IVREGmother , se(%12.4f) b(%12.5f) star(* 0.10 ** 0.05 *** 0.01) mtitles("ols" "IVmother") title("Model test")

Model test
--------------------------------------------
                      (1)             (2)   
                      ols        IVmother   
--------------------------------------------
lwage           -17.40780      -106.40969   
                (81.3773)      (590.1340)   

educ            -14.44486        -5.15832   
                (18.2129)       (64.9135)   

age              -7.72998        -7.55598   
                 (5.8497)        (5.9799)   

kidsl6         -342.50482***   -350.00627***
               (131.7733)      (135.5988)   

kids618        -115.02051***   -118.86399***
                (29.5087)       (37.4226)   

nwifeinc         -0.00425        -0.00396   
                 (0.0032)        (0.0039)   

_cons          2114.69725***   2096.60887***
               (350.3186)      (382.8154)   
--------------------------------------------
N                     428             428   
--------------------------------------------
Standard errors in parentheses
* p<0.10, ** p<0.05, *** p<0.01

. 
end of do-file

. do "C:\Users\Patrick\AppData\Local\Temp\STD03000000.tmp"

. ivregress 2sls hours (lwage = fathereduc) $x2list, vce(robust) first

First-stage regressions
-----------------------

                                                  Number of obs   =        428
                                                  F(   6,    421) =      12.90
                                                  Prob > F        =     0.0000
                                                  R-squared       =     0.1364
                                                  Adj R-squared   =     0.1241
                                                  Root MSE        =     0.6768

------------------------------------------------------------------------------
             |               Robust
       lwage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        educ |   .1141899   .0151482     7.54   0.000     .0844144    .1439654
         age |   .0010194   .0052505     0.19   0.846    -.0093011    .0113399
      kidsl6 |  -.0875674   .1083449    -0.81   0.419    -.3005317    .1253969
     kids618 |  -.0458046   .0285587    -1.60   0.109    -.1019399    .0103308
    nwifeinc |   3.48e-06   2.68e-06     1.30   0.195    -1.79e-06    8.75e-06
  fathereduc |  -.0163147    .010337    -1.58   0.115    -.0366333    .0040039
       _cons |  -.1432883   .3155694    -0.45   0.650    -.7635762    .4769996
------------------------------------------------------------------------------


Instrumental variables (2SLS) regression               Number of obs =     428
                                                       Wald chi2(6)  =   19.99
                                                       Prob > chi2   =  0.0028
                                                       R-squared     =       .
                                                       Root MSE      =  856.52

------------------------------------------------------------------------------
             |               Robust
       hours |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       lwage |    599.808   775.6417     0.77   0.439    -920.4217    2120.038
        educ |  -78.84572   84.45428    -0.93   0.351    -244.3731    86.68163
         age |  -8.936616    6.77463    -1.32   0.187    -22.21465    4.341414
      kidsl6 |  -290.4833   159.9655    -1.82   0.069      -604.01    23.04338
     kids618 |  -88.36656   46.33123    -1.91   0.056    -179.1741    2.440976
    nwifeinc |  -.0062226   .0042903    -1.45   0.147    -.0146314    .0021863
       _cons |   2240.138    416.825     5.37   0.000     1423.176      3057.1
------------------------------------------------------------------------------
Instrumented:  lwage
Instruments:   educ age kidsl6 kids618 nwifeinc fathereduc

. estimate store IVREGfather

. estat endogenous

  Tests of endogeneity
  Ho: variables are exogenous

  Robust score chi2(1)            =  .763741  (p = 0.3822)
  Robust regression F(1,420)      =  .756369  (p = 0.3850)

. esttab REG IVREGfather , se(%12.4f) b(%12.5f) star(* 0.10 ** 0.05 *** 0.01) mtitles("ols" "IVfather") title("Model test")

Model test
--------------------------------------------
                      (1)             (2)   
                      ols        IVfather   
--------------------------------------------
lwage           -17.40780       599.80803   
                (81.3773)      (775.6417)   

educ            -14.44486       -78.84572   
                (18.2129)       (84.4543)   

age              -7.72998        -8.93662   
                 (5.8497)        (6.7746)   

kidsl6         -342.50482***   -290.48330*  
               (131.7733)      (159.9655)   

kids618        -115.02051***    -88.36656*  
                (29.5087)       (46.3312)   

nwifeinc         -0.00425        -0.00622   
                 (0.0032)        (0.0043)   

_cons          2114.69725***   2240.13767***
               (350.3186)      (416.8250)   
--------------------------------------------
N                     428             428   
--------------------------------------------
Standard errors in parentheses
* p<0.10, ** p<0.05, *** p<0.01

. 
end of do-file

. do "C:\Users\Patrick\AppData\Local\Temp\STD03000000.tmp"

. ivregress 2sls hours (lwage = heduc) $x2list, vce(robust) first

First-stage regressions
-----------------------

                                                  Number of obs   =        428
                                                  F(   6,    421) =      12.07
                                                  Prob > F        =     0.0000
                                                  R-squared       =     0.1354
                                                  Adj R-squared   =     0.1230
                                                  Root MSE        =     0.6772

------------------------------------------------------------------------------
             |               Robust
       lwage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        educ |   .1180822   .0159596     7.40   0.000     .0867118    .1494527
         age |   .0014976   .0051916     0.29   0.773    -.0087071    .0117022
      kidsl6 |  -.0807521   .1109522    -0.73   0.467    -.2988414    .1373372
     kids618 |  -.0438408    .028807    -1.52   0.129    -.1004642    .0127827
    nwifeinc |   4.35e-06   2.80e-06     1.55   0.121    -1.15e-06    9.85e-06
       heduc |  -.0195687   .0125636    -1.56   0.120    -.0442638    .0051265
       _cons |  -.1325044   .3141308    -0.42   0.673    -.7499645    .4849558
------------------------------------------------------------------------------


Instrumental variables (2SLS) regression               Number of obs =     428
                                                       Wald chi2(6)  =   20.02
                                                       Prob > chi2   =  0.0027
                                                       R-squared     =       .
                                                       Root MSE      =  874.46

------------------------------------------------------------------------------
             |               Robust
       hours |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       lwage |   652.9985   904.9896     0.72   0.471    -1120.748    2426.745
        educ |  -84.39566   97.88184    -0.86   0.389    -276.2405    107.4492
         age |  -9.040602   7.200103    -1.26   0.209    -23.15254    5.071341
      kidsl6 |  -286.0002   173.2299    -1.65   0.099    -625.5245    53.52414
     kids618 |  -86.06958   51.85856    -1.66   0.097    -187.7105    15.57133
    nwifeinc |  -.0063929   .0045117    -1.42   0.156    -.0152356    .0024498
       _cons |   2250.948   447.9813     5.02   0.000     1372.921    3128.975
------------------------------------------------------------------------------
Instrumented:  lwage
Instruments:   educ age kidsl6 kids618 nwifeinc heduc

. estimate store IVREGhusband

. estat endogenous

  Tests of endogeneity
  Ho: variables are exogenous

  Robust score chi2(1)            =  .646754  (p = 0.4213)
  Robust regression F(1,420)      =  .641961  (p = 0.4235)

Now using both mothereduc, fathereduc and heduc we have endogeniety.

. esttab REG IVREGhusband , se(%12.4f) b(%12.5f) star(* 0.10 ** 0.05 *** 0.01) mtitles("ols" "IVhusband") title("Model test")

Model test
--------------------------------------------
                      (1)             (2)   
                      ols       IVhusband   
--------------------------------------------
lwage           -17.40780       652.99854   
                (81.3773)      (904.9896)   

educ            -14.44486       -84.39566   
                (18.2129)       (97.8818)   

age              -7.72998        -9.04060   
                 (5.8497)        (7.2001)   

kidsl6         -342.50482***   -286.00018*  
               (131.7733)      (173.2299)   

kids618        -115.02051***    -86.06958*  
                (29.5087)       (51.8586)   

nwifeinc         -0.00425        -0.00639   
                 (0.0032)        (0.0045)   

_cons          2114.69725***   2250.94790***
               (350.3186)      (447.9813)   
--------------------------------------------
N                     428             428   
--------------------------------------------
Standard errors in parentheses
* p<0.10, ** p<0.05, *** p<0.01

. 
end of do-file

. do "C:\Users\Patrick\AppData\Local\Temp\STD03000000.tmp"

. ivregress 2sls hours (lwage = siblings) $x2list, vce(robust) first

First-stage regressions
-----------------------

                                                  Number of obs   =        428
                                                  F(   6,    421) =      11.25
                                                  Prob > F        =     0.0000
                                                  R-squared       =     0.1326
                                                  Adj R-squared   =     0.1202
                                                  Root MSE        =     0.6783

------------------------------------------------------------------------------
             |               Robust
       lwage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        educ |   .1048841   .0146161     7.18   0.000     .0761544    .1336138
         age |   .0018242   .0052106     0.35   0.726    -.0084178    .0120661
      kidsl6 |  -.0842532   .1111352    -0.76   0.449    -.3027022    .1341957
     kids618 |  -.0439312    .028611    -1.54   0.125    -.1001695    .0123071
    nwifeinc |   3.17e-06   2.70e-06     1.17   0.242    -2.14e-06    8.48e-06
    siblings |  -.0110307   .0136049    -0.81   0.418    -.0377727    .0157113
       _cons |  -.1668875   .3101863    -0.54   0.591    -.7765943    .4428193
------------------------------------------------------------------------------


Instrumental variables (2SLS) regression               Number of obs =     428
                                                       Wald chi2(6)  =    4.81
                                                       Prob > chi2   =  0.5689
                                                       R-squared     =       .
                                                       Root MSE      =    2100

------------------------------------------------------------------------------
             |               Robust
       hours |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       lwage |   2896.553   3671.989     0.79   0.430    -4300.413    10093.52
        educ |  -318.4901   381.4712    -0.83   0.404     -1066.16    429.1796
         age |  -13.42669   17.76516    -0.76   0.450    -48.24577    21.39239
      kidsl6 |  -96.90406   471.7734    -0.21   0.837    -1021.563    827.7549
     kids618 |   10.81645   187.3789     0.06   0.954    -356.4395    378.0724
    nwifeinc |  -.0135783   .0138106    -0.98   0.326    -.0406465    .0134899
       _cons |   2706.919   1153.148     2.35   0.019     446.7899    4967.048
------------------------------------------------------------------------------
Instrumented:  lwage
Instruments:   educ age kidsl6 kids618 nwifeinc siblings

. estimate store IVREGsib

. estat endogenous

  Tests of endogeneity
  Ho: variables are exogenous

  Robust score chi2(1)            =  4.30171  (p = 0.0381)
  Robust regression F(1,420)      =   4.2976  (p = 0.0388)

. esttab REG IVREGsib , se(%12.4f) b(%12.5f) star(* 0.10 ** 0.05 *** 0.01) mtitles("ols" "IVsibling") title("Model test")

Model test
--------------------------------------------
                      (1)             (2)   
                      ols       IVsibling   
--------------------------------------------
lwage           -17.40780      2896.55290   
                (81.3773)     (3671.9886)   

educ            -14.44486      -318.49015   
                (18.2129)      (381.4712)   

age              -7.72998       -13.42669   
                 (5.8497)       (17.7652)   

kidsl6         -342.50482***    -96.90406   
               (131.7733)      (471.7734)   

kids618        -115.02051***     10.81645   
                (29.5087)      (187.3789)   

nwifeinc         -0.00425        -0.01358   
                 (0.0032)        (0.0138)   

_cons          2114.69725***   2706.91871** 
               (350.3186)     (1153.1481)   
--------------------------------------------
N                     428             428   
--------------------------------------------
Standard errors in parentheses
* p<0.10, ** p<0.05, *** p<0.01

. 
end of do-file

. do "C:\Users\Patrick\AppData\Local\Temp\STD03000000.tmp"

. esttab REG REGIV IVREGmother IVREGfather IVREGhusband IVREGsib , se(%12.4f) b(%12.5f) star(* 0.10 ** 0.05 *** 0.01) mtitles("ols" "IV" "IVmother" "IVfather" "IVhusband" "IVsibl
> ing") title("Model test")

Model test
------------------------------------------------------------------------------------------------------------
                      (1)             (2)             (3)             (4)             (5)             (6)   
                      ols              IV        IVmother        IVfather       IVhusband       IVsibling   
------------------------------------------------------------------------------------------------------------
lwage           -17.40780      1544.81848***   -106.40969       599.80803       652.99854      2896.55290   
                (81.3773)      (598.8004)      (590.1340)      (775.6417)      (904.9896)     (3671.9886)   

educ            -14.44486      -177.44896***     -5.15832       -78.84572       -84.39566      -318.49015   
                (18.2129)       (66.8451)       (64.9135)       (84.4543)       (97.8818)      (381.4712)   

age              -7.72998       -10.78409        -7.55598        -8.93662        -9.04060       -13.42669   
                 (5.8497)       (10.5776)        (5.9799)        (6.7746)        (7.2001)       (17.7652)   

kidsl6         -342.50482***   -210.83387      -350.00627***   -290.48330*     -286.00018*      -96.90406   
               (131.7733)      (203.9118)      (135.5988)      (159.9655)      (173.2299)      (471.7734)   

kids618        -115.02051***    -47.55708      -118.86399***    -88.36656*      -86.06958*       10.81645   
                (29.5087)       (56.4794)       (37.4226)       (46.3312)       (51.8586)      (187.3789)   

nwifeinc         -0.00425        -0.00925*       -0.00396        -0.00622        -0.00639        -0.01358   
                 (0.0032)        (0.0052)        (0.0039)        (0.0043)        (0.0045)        (0.0138)   

_cons          2114.69725***   2432.19773***   2096.60887***   2240.13767***   2250.94790***   2706.91871** 
               (350.3186)      (611.2230)      (382.8154)      (416.8250)      (447.9813)     (1153.1481)   
------------------------------------------------------------------------------------------------------------
N                     428             428             428             428             428             428   
------------------------------------------------------------------------------------------------------------
Standard errors in parentheses
* p<0.10, ** p<0.05, *** p<0.01

. 
end of do-file

. 

And lastly using siblings as instrument gives no endogenous. Why is this? I am doing the same regression for all the models only using different instruments.

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  • 6
    $\begingroup$ The extensive, unedited dump of software output is likely to repel all but the most intensely interested readers. I would suggest that greatly abbreviating your post would increase the chance of getting a useful answer. $\endgroup$
    – whuber
    Dec 10 '15 at 18:34
  • $\begingroup$ "I'm under the impression that endogeniety exists och does not exists regardless of which instruments you choose." Should the "och" read "or"? $\endgroup$
    – Silverfish
    Dec 20 '15 at 22:50
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To understand your problem you first need to understand how the endogeneity test works. Suppose you have an outcome $y$ and an explanatory variable $x$ which you think is endogenous because it has some correlation with the error term, i.e. $$\begin{matrix}y_i &=& \alpha &+& \beta x_i &+& \epsilon_i & \\ & && & & \hspace{-1cm}\nwarrow & \hspace{-0.8cm} \nearrow \\ & & & & & corr & \end{matrix}$$ then you can use an instrument ($z$) to test whether this is actually true.

When you regress your endogenous variable on the instrument, this splits up the variation of $x$ into an explained part (which we know is exogenous because the instrument $z$ is supposed to be exogenous), and an unexplained part

$$x_i \quad = \underbrace{a \quad + \quad \pi z_i}_{\text{good variation} } \quad + \underbrace{\eta_i}_{\text{bad variation}}$$

Now it is important to understand the required assumptions for a valid instrument:

  1. the instrument must affect the endogenous variable, $\text{corr}(x_i,z_i)\neq 0$
  2. the instrument must not be correlated with the structural error $\text{corr}(z_i,\epsilon_i)=0$

If either of these conditions fail, we are not successful in separating out the exogenous variation in $x$ using our instrument either because it is weak or it is not exogenous itself.

Your endogeneity test then takes the residuals from this regression, $\widehat{\eta}$, and regresses $$y_i = \alpha + \beta x_i + \delta \widehat{\eta}_i + \epsilon_i$$ and tests $H_0:\delta=0$. If we reject this null then it must be the case that the "bad" part in the variation of $x$ (which we separated out before into $\widehat{\eta}$) significantly affects the outcome and therefore we suspect endogeneity

Now you see why it matters what instrument we use for this test. In your example a couple of instruments fail to meet condition 1. as they are not sufficiently highly correlated with the endogenous variable. For example, mother's education, father's education, and husband's education have first stage F-statistics (i.e. the square of the t-stat in case of one instrument) of 2.99, 2.5, and 2.43, respectively. Typical we worry about instruments with F-statistics of less than 10, so these three are unlikely to be good instruments and therefore any endogeneity test built on them will not be reliable either.

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