# IV regression, endogeniety and Wu-hausman question

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
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
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
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.

• 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.
– whuber
Dec 10, 2015 at 18:34
• "I'm under the impression that endogeniety exists och does not exists regardless of which instruments you choose." Should the "och" read "or"? Dec 20, 2015 at 22:50

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.