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  • #1

    weak-instrument robust inference: Anderson-Rubin stat from -weakiv- is different from that in -ivreghdfe- output

    I am trying to understand the difference between the AR chi-squared statistic that is outputted as a result of -weakiv- command and that is outputted as a result of -ivreghdfe- command.

    Code:
    . sysuse auto, clear
(1978 Automobile Data)

. ivreg2 price (weight=gear) i.turn , partial(i.turn) small

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

Estimates efficient for homoskedasticity only
Statistics consistent for homoskedasticity only

                                                      Number of obs =       74
                                                      F(  1,    55) =     8.24
                                                      Prob > F      =   0.0058
Total (centered) SS     =  436283540.4                Centered R2   =   0.4433
Total (uncentered) SS   =  436283540.4                Uncentered R2 =   0.4433
Residual SS             =  242890767.7                Root MSE      =     2101

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      weight |   4.728751   1.647647     2.87   0.006     1.426792     8.03071
------------------------------------------------------------------------------
Underidentification test (Anderson canon. corr. LM statistic):          14.944
                                                   Chi-sq(1) P-val =    0.0001
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):               13.918
Stock-Yogo weak ID test critical values: 10% maximal IV size             16.38
                                         15% maximal IV size              8.96
                                         20% maximal IV size              6.66
                                         25% maximal IV size              5.53
Source: Stock-Yogo (2005).  Reproduced by permission.
------------------------------------------------------------------------------
Sargan statistic (overidentification test of all instruments):           0.000
                                                 (equation exactly identified)
------------------------------------------------------------------------------
Instrumented:         weight
Excluded instruments: gear_ratio
Partialled-out:       32.turn 33.turn 34.turn 35.turn 36.turn 37.turn 38.turn
                      39.turn 40.turn 41.turn 42.turn 43.turn 44.turn 45.turn
                      46.turn 48.turn 51.turn _cons
                      nb: total SS, model F and R2s are after partialling-out;
                          any small-sample adjustments include partialled-out
                          variables in regressor count K
------------------------------------------------------------------------------

. weakiv 

Weak instrument robust tests and confidence sets for linear IV
H0: beta[price:weight] = 0
------------------------------------------------------------------------------
 Test |       Statistic         p-value |  Conf. level        Conf. Set       
------+---------------------------------+-------------------------------------
   AR | chi2(1)   =     4.59     0.0322 |      95%      [ .8493422, 8.441783] 
------+---------------------------------+-------------------------------------
 Wald | chi2(1)   =     8.24     0.0041 |      95%       [ 1.49942, 7.95808]  
------------------------------------------------------------------------------
Number of obs N = 74.
Method = lagrange multiplier (LM).
Tests assume i.i.d. errors.  Small sample adjustments were used.
Wald statistic in last row is based on ivreg2 estimation and is not robust to weak
instruments.

. ivreghdfe price (weight=gear), a(turn) dof(none) first
(dropped 4 singleton observations)
(MWFE estimator converged in 1 iterations)

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


First-stage regression of weight:

Statistics consistent for homoskedasticity only
Number of obs =                     70
------------------------------------------------------------------------------
      weight |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
  gear_ratio |  -549.1404   147.1975    -3.73   0.000    -844.1309     -254.15
------------------------------------------------------------------------------
F test of excluded instruments:
  F(  1,    55) =    13.92
  Prob > F      =   0.0005
Sanderson-Windmeijer multivariate F test of excluded instruments:
  F(  1,    55) =    13.92
  Prob > F      =   0.0005



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

                                           (Underid)            (Weak id)
Variable     | F(  1,    55)  P-val | SW Chi-sq(  1) P-val | SW F(  1,    55)
weight       |      13.92    0.0005 |       17.71   0.0000 |       13.92

Stock-Yogo weak ID F test critical values for single endogenous regressor:
                                   10% maximal IV size             16.38
                                   15% maximal IV size              8.96
                                   20% maximal IV size              6.66
                                   25% maximal IV size              5.53
Source: Stock-Yogo (2005).  Reproduced by permission.
NB: Critical values are for Sanderson-Windmeijer F statistic.

Underidentification test
Ho: matrix of reduced form coefficients has rank=K1-1 (underidentified)
Ha: matrix has rank=K1 (identified)
Anderson canon. corr. LM statistic       Chi-sq(1)=14.14    P-val=0.0002

Weak identification test
Ho: equation is weakly identified
Cragg-Donald Wald F statistic                                      13.92

Stock-Yogo weak ID test critical values for K1=1 and L1=1:
                                   10% maximal IV size             16.38
                                   15% maximal IV size              8.96
                                   20% maximal IV size              6.66
                                   25% maximal IV size              5.53
Source: Stock-Yogo (2005).  Reproduced by permission.

Weak-instrument-robust inference
Tests of joint significance of endogenous regressors B1 in main equation
Ho: B1=0 and orthogonality conditions are valid
Anderson-Rubin Wald test           F(1,55)=        5.00     P-val=0.0294
Anderson-Rubin Wald test           Chi-sq(1)=      6.37     P-val=0.0116
Stock-Wright LM S statistic        Chi-sq(1)=      5.84     P-val=0.0157

Number of observations               N  =         70
Number of regressors                 K  =          1
Number of endogenous regressors      K1 =          1
Number of instruments                L  =          1
Number of excluded instruments       L1 =          1

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

Estimates efficient for homoskedasticity only
Statistics consistent for homoskedasticity only

                                                      Number of obs =       70
                                                      F(  1,    55) =     8.24
                                                      Prob > F      =   0.0058
Total (centered) SS     =  436283540.4                Centered R2   =   0.4433
Total (uncentered) SS   =  436283540.4                Uncentered R2 =   0.4433
Residual SS             =  242890767.7                Root MSE      =     2101

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      weight |   4.728751   1.647647     2.87   0.006     1.426792     8.03071
------------------------------------------------------------------------------
Underidentification test (Anderson canon. corr. LM statistic):          14.136
                                                   Chi-sq(1) P-val =    0.0002
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):               13.918
Stock-Yogo weak ID test critical values: 10% maximal IV size             16.38
                                         15% maximal IV size              8.96
                                         20% maximal IV size              6.66
                                         25% maximal IV size              5.53
Source: Stock-Yogo (2005).  Reproduced by permission.
------------------------------------------------------------------------------
Sargan statistic (overidentification test of all instruments):           0.000
                                                 (equation exactly identified)
------------------------------------------------------------------------------
Instrumented:         weight
Excluded instruments: gear_ratio
Partialled-out:       _cons
                      nb: total SS, model F and R2s are after partialling-out;
                          any small-sample adjustments include partialled-out
                          variables in regressor count K
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
        turn |        14           0          14     |
-----------------------------------------------------+

.
    Tags: None
  • #2
    sample sizes are different

    Comment

    • #3
      Thanks for noticing it George Ford ! Do you confirm that the AR chi2 stats outputted in -weakiv- & -ivreg2- refer to the exact same test?

      I have now re-estimated them with the same sample - they are still not identical (although they are closer than before). I don't want to be picky, I was just trying to understand if they refer to the exact same test and if I am safe using the output displayed by -ivreghdfe- instead of that from -weakiv-.

      Code:
      . sysuse auto, clear
(1978 Automobile Data)

. ivreghdfe price (weight=gear), a(turn) dof(none) first
(dropped 4 singleton observations)
(MWFE estimator converged in 1 iterations)

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


First-stage regression of weight:

Statistics consistent for homoskedasticity only
Number of obs =                     70
------------------------------------------------------------------------------
      weight |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
  gear_ratio |  -549.1404   147.1975    -3.73   0.000    -844.1309     -254.15
------------------------------------------------------------------------------
F test of excluded instruments:
  F(  1,    55) =    13.92
  Prob > F      =   0.0005
Sanderson-Windmeijer multivariate F test of excluded instruments:
  F(  1,    55) =    13.92
  Prob > F      =   0.0005



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

                                           (Underid)            (Weak id)
Variable     | F(  1,    55)  P-val | SW Chi-sq(  1) P-val | SW F(  1,    55)
weight       |      13.92    0.0005 |       17.71   0.0000 |       13.92

Stock-Yogo weak ID F test critical values for single endogenous regressor:
                                   10% maximal IV size             16.38
                                   15% maximal IV size              8.96
                                   20% maximal IV size              6.66
                                   25% maximal IV size              5.53
Source: Stock-Yogo (2005).  Reproduced by permission.
NB: Critical values are for Sanderson-Windmeijer F statistic.

Underidentification test
Ho: matrix of reduced form coefficients has rank=K1-1 (underidentified)
Ha: matrix has rank=K1 (identified)
Anderson canon. corr. LM statistic       Chi-sq(1)=14.14    P-val=0.0002

Weak identification test
Ho: equation is weakly identified
Cragg-Donald Wald F statistic                                      13.92

Stock-Yogo weak ID test critical values for K1=1 and L1=1:
                                   10% maximal IV size             16.38
                                   15% maximal IV size              8.96
                                   20% maximal IV size              6.66
                                   25% maximal IV size              5.53
Source: Stock-Yogo (2005).  Reproduced by permission.

Weak-instrument-robust inference
Tests of joint significance of endogenous regressors B1 in main equation
Ho: B1=0 and orthogonality conditions are valid
Anderson-Rubin Wald test           F(1,55)=        5.00     P-val=0.0294
Anderson-Rubin Wald test           Chi-sq(1)=      6.37     P-val=0.0116
Stock-Wright LM S statistic        Chi-sq(1)=      5.84     P-val=0.0157

Number of observations               N  =         70
Number of regressors                 K  =          1
Number of endogenous regressors      K1 =          1
Number of instruments                L  =          1
Number of excluded instruments       L1 =          1

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

Estimates efficient for homoskedasticity only
Statistics consistent for homoskedasticity only

                                                      Number of obs =       70
                                                      F(  1,    55) =     8.24
                                                      Prob > F      =   0.0058
Total (centered) SS     =  436283540.4                Centered R2   =   0.4433
Total (uncentered) SS   =  436283540.4                Uncentered R2 =   0.4433
Residual SS             =  242890767.7                Root MSE      =     2101

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      weight |   4.728751   1.647647     2.87   0.006     1.426792     8.03071
------------------------------------------------------------------------------
Underidentification test (Anderson canon. corr. LM statistic):          14.136
                                                   Chi-sq(1) P-val =    0.0002
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):               13.918
Stock-Yogo weak ID test critical values: 10% maximal IV size             16.38
                                         15% maximal IV size              8.96
                                         20% maximal IV size              6.66
                                         25% maximal IV size              5.53
Source: Stock-Yogo (2005).  Reproduced by permission.
------------------------------------------------------------------------------
Sargan statistic (overidentification test of all instruments):           0.000
                                                 (equation exactly identified)
------------------------------------------------------------------------------
Instrumented:         weight
Excluded instruments: gear_ratio
Partialled-out:       _cons
                      nb: total SS, model F and R2s are after partialling-out;
                          any small-sample adjustments include partialled-out
                          variables in regressor count K
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
        turn |        14           0          14     |
-----------------------------------------------------+

. ivreg2 price (weight=gear) i.turn if e(sample)==1, partial(i.turn) small

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

Estimates efficient for homoskedasticity only
Statistics consistent for homoskedasticity only

                                                      Number of obs =       70
                                                      F(  1,    55) =     8.24
                                                      Prob > F      =   0.0058
Total (centered) SS     =  436283540.4                Centered R2   =   0.4433
Total (uncentered) SS   =  436283540.4                Uncentered R2 =   0.4433
Residual SS             =  242890767.7                Root MSE      =     2101

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      weight |   4.728751   1.647647     2.87   0.006     1.426792     8.03071
------------------------------------------------------------------------------
Underidentification test (Anderson canon. corr. LM statistic):          14.136
                                                   Chi-sq(1) P-val =    0.0002
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):               13.918
Stock-Yogo weak ID test critical values: 10% maximal IV size             16.38
                                         15% maximal IV size              8.96
                                         20% maximal IV size              6.66
                                         25% maximal IV size              5.53
Source: Stock-Yogo (2005).  Reproduced by permission.
------------------------------------------------------------------------------
Sargan statistic (overidentification test of all instruments):           0.000
                                                 (equation exactly identified)
------------------------------------------------------------------------------
Instrumented:         weight
Excluded instruments: gear_ratio
Partialled-out:       34.turn 35.turn 36.turn 37.turn 38.turn 40.turn 41.turn
                      42.turn 43.turn 44.turn 45.turn 46.turn 48.turn _cons
                      nb: total SS, model F and R2s are after partialling-out;
                          any small-sample adjustments include partialled-out
                          variables in regressor count K
------------------------------------------------------------------------------

. weakiv 

Weak instrument robust tests and confidence sets for linear IV
H0: beta[price:weight] = 0
------------------------------------------------------------------------------
 Test |       Statistic         p-value |  Conf. level        Conf. Set       
------+---------------------------------+-------------------------------------
   AR | chi2(1)   =     4.59     0.0322 |      95%      [ .8493422, 8.441783] 
------+---------------------------------+-------------------------------------
 Wald | chi2(1)   =     8.24     0.0041 |      95%       [ 1.49942, 7.95808]  
------------------------------------------------------------------------------
Number of obs N = 70.
Method = lagrange multiplier (LM).
Tests assume i.i.d. errors.  Small sample adjustments were used.
Wald statistic in last row is based on ivreg2 estimation and is not robust to weak
instruments.

      Comment

      • #4
        And_Rubin is used in weakiv. There's a small sample adjustment in weakiv I'm not sure the other is making. see what happens if you add "small" as an option for ivheghdfe.

        Comment

        • #5
          Also look at -ereturn list- after ivreg2 and ivreghdfe to see if multiple A-R results are provided. I know ivreg2 does, and ivreghdfe is essentially a wrapper using ivreg2.

          Comment

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