Weak-instrument-robust inference test with xtivreg2

[Ellen Sterk]

Dear all,

I am using the command xtivreg2, fe with Stata17. I think I found two good instruments for my endogenous regressor. I am only unsure about the test results of the weak-instrument-robust inference tests. I have read the help file, but don't quite understand it. It states "The null hypothesis tested in both cases
is that the coefficients of the endogenous regressors in the structural equation are jointly equal to zero, and, in addition, that the overidentifying restrictions are valid." I feel like I want the null hypothesis to be rejected for the first part of this but not the second... Could someone explain what rejecting the null hypothesis means here?

Thank you in advance!

Here's my output:

xtivreg2 log_Res_Prod_own_E_E_PPP log_Population_Density log_GDP_Euro_CLV_PPP Year_rank (RTR = Tax_Level_E Reg_Quality), fe endog(RTR) first bw(3) robust small 

FIXED EFFECTS ESTIMATION
------------------------
Number of groups =        26                    Obs per group: min =        13
                                                               avg =      20.0
                                                               max =        23

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

First-stage regression of RTR:

FIXED EFFECTS ESTIMATION
------------------------
Number of groups =        26                    Obs per group: min =        13
                                                               avg =      20.0
                                                               max =        23

OLS estimation
--------------

Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity and autocorrelation
  kernel=Bartlett; bandwidth=     3
  time variable (t):  Year
  group variable (i): Country_Code

                                                      Number of obs =      520
                                                      F(  5,   489) =     5.39
                                                      Prob > F      =   0.0001
Total (centered) SS     =  243.8229584                Centered R2   =   0.2345
Total (uncentered) SS   =  243.8229584                Uncentered R2 =   0.2345
Residual SS             =  186.6363613                Root MSE      =    .6178

----------------------------------------------------------------------------------------
                       |               Robust
                   RTR | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-----------------------+----------------------------------------------------------------
log_Population_Density |   6.056206   1.907159     3.18   0.002     2.308969    9.803443
  log_GDP_Euro_CLV_PPP |   .8751533   .2703399     3.24   0.001     .3439821    1.406324
             Year_rank |  -.0257033    .011946    -2.15   0.032    -.0491752   -.0022313
           Tax_Level_E |    .046613   .0269594     1.73   0.084    -.0063576    .0995836
           Reg_Quality |  -.6487274   .2963532    -2.19   0.029     -1.23101   -.0664446
----------------------------------------------------------------------------------------
Included instruments: log_Population_Density log_GDP_Euro_CLV_PPP Year_rank
                      Tax_Level_E Reg_Quality
------------------------------------------------------------------------------
Partial R-squared of excluded instruments:   0.0480
Test of excluded instruments:
  F(  2,   489) =     3.63
  Prob > F      =   0.0271



Summary results for first-stage regressions
-------------------------------------------

Variable    | Shea Partial R2 |   Partial R2    |  F(  2,   489)    P-value
RTR         |     0.0480      |     0.0480      |        3.63       0.0271

NB: first-stage F-stat heteroskedasticity and autocorrelation-robust

Underidentification tests
Ho: matrix of reduced form coefficients has rank=K1-1 (underidentified)
Ha: matrix has rank=K1 (identified)
Kleibergen-Paap rk LM statistic             Chi-sq(2)=6.31     P-val=0.0427
Kleibergen-Paap rk Wald statistic           Chi-sq(2)=7.34     P-val=0.0255

Weak identification test
Ho: equation is weakly identified
Kleibergen-Paap Wald rk F statistic                 3.63
See main output for Cragg-Donald weak id test critical values

Weak-instrument-robust inference
Tests of joint significance of endogenous regressors B1 in main equation
Ho: B1=0 and overidentifying restrictions are valid
Anderson-Rubin Wald test     F(2,489)= 26.14     P-val=0.0000
Anderson-Rubin Wald test     Chi-sq(2)=52.82     P-val=0.0000
Stock-Wright LM S statistic  Chi-sq(2)=27.24     P-val=0.0000

NB: Underidentification, weak identification and weak-identification-robust
    test statistics heteroskedasticity and autocorrelation-robust

Number of observations               N  =        520
Number of regressors                 K  =          4
Number of instruments                L  =          5
Number of excluded instruments       L1 =          2

IV (2SLS) estimation
--------------------

Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity and autocorrelation
  kernel=Bartlett; bandwidth=     3
  time variable (t):  Year
  group variable (i): Country_Code

                                                      Number of obs =      520
                                                      F(  4,   490) =    33.96
                                                      Prob > F      =   0.0000
Total (centered) SS     =  34.62532019                Centered R2   =  -0.3334
Total (uncentered) SS   =  34.62532019                Uncentered R2 =  -0.3334
Residual SS             =  46.16854371                Root MSE      =     .307

----------------------------------------------------------------------------------------
                       |               Robust
log_Res_Prod_own_E_E~P | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-----------------------+----------------------------------------------------------------
                   RTR |   .4537421   .1597346     2.84   0.005     .1398928    .7675914
log_Population_Density |  -1.967359   1.187216    -1.66   0.098    -4.300022    .3653037
  log_GDP_Euro_CLV_PPP |   .5442732   .1780607     3.06   0.002     .1944164      .89413
             Year_rank |   .0132872   .0045773     2.90   0.004     .0042937    .0222807
----------------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic):              6.308
                                                   Chi-sq(2) P-val =    0.0427
------------------------------------------------------------------------------
Weak identification test (Kleibergen-Paap rk Wald F statistic):          3.634
Stock-Yogo weak ID test critical values: 10% maximal IV size             19.93
                                         15% maximal IV size             11.59
                                         20% maximal IV size              8.75
                                         25% maximal IV size              7.25
Source: Stock-Yogo (2005).  Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments):         0.700
                                                   Chi-sq(1) P-val =    0.4029
-endog- option:
Endogeneity test of endogenous regressors:                              27.299
                                                   Chi-sq(1) P-val =    0.0000
Regressors tested:    RTR
------------------------------------------------------------------------------
Instrumented:         RTR
Included instruments: log_Population_Density log_GDP_Euro_CLV_PPP Year_rank
Excluded instruments: Tax_Level_E Reg_Quality
------------------------------------------------------------------------------

[George Ford]

You want to not reject both.

The OR test is essentially a test of whether the IVs are correlated with the the error term.

[Ellen Sterk]

George Ford Thank you for your reply! Is it possible that you are referring to the test of over overidentifying restrictions (the Hansen J statistic in my case)? I was unsure about the Anderson-Rubin Wald test and the Stock-Wright LM S statistic (weak-instrument robust interference). My understanding in the meantime is that a rejection of the null there means my endogenous regressor is relevant. Is that correct?

[George Ford]

For everything but the "Endogeneity test of endogenous regressors", the joint null is that the instruments are valid (or at least weakly so) based on various criteria. So you don't want to reject any of these.

The "Endogeneity test of endogenous regressors" has a null of exogeneity. If you don't reject this, then you can proceed with OLS (or least, would have an argument for doing so). If you do reject, then you need to deal with endogeneity somehow.