Differences in estimation of SEs between xtivreg & xtivreg2

[Ariel Karlinsky]

This post discusses the following well known external ado programs: xtoverid , xtivreg2, ivreg2 .
I have that latest versions of each of these. I'm running Stata 12 but on my work station with Stata 14.1 it's the same.

Iv'e noticed through xtoverid undocumented noi option, that the estimation of SEs between Stata's built in xtivreg command and xtoverid differs.
xtoverid utilizees xtivreg2 for estimation so these can be traced back to differences in SEs between xtivreg & xtivreg. The question is which one is preferable and why is there a difference?

Here's a silly do that will replicate this:

sysuse nlswide1, clear
gen id = _n
reshape long age count collgrad c_city union ttl_exp tenure hours wage, i(id) j(year)
xtset i year
xtivreg wage (tenure = hours) collgrad, fe
estimates store xtivreg
xtivreg2 wage (tenure = hours) collgrad, fe
estimates store xtivreg2

estimates table xtivreg xtivreg2, se

and the output clearly shows the differnce between the SEs (in red):

----------------------------------------
    Variable |   xtivreg        xtivreg2      
-------------+--------------------------
      tenure | -.23157191   -.23157191  
             |  .86449019    .76240868 
    collgrad |  30.426044    30.426044  
             |  27.655421    24.389789  
       _cons |  1.4035523              
             |  2.6004731              
----------------------------------------

[Emad Shehata]

use small option in xtivreg2 to get the same SEs as in xtivreg

sysuse nlswide1, clear
gen id = _n
reshape long age count collgrad c_city union ttl_exp tenure hours wage, i(id) j(year)
xtset i year
xtivreg wage (tenure = hours) collgrad, fe
estimates store xtivreg
xtivreg2 wage (tenure = hours) collgrad, fe small
estimates store xtivreg2

estimates table xtivreg xtivreg2, se

With large-sample statistics, the covariance matrix has the adjustment (N-N_g);
with small-sample statistics, the covariance matrix has the adjustment (N-N_g-K),
where K is the number of regressors.

[Ariel Karlinsky]

Thanks. this issue arose in my research which has about 1.3 Million observations and 150K panel groups - so I probably don't need to adjust for small sample, am I right?

Also, as I said in the original post - the issue first came about while looking at the output of xtoverid after running xtivreg Random Effects model. and I noticed that xtoverid's estimation (which relies on xtivreg2) returns different SEs:

. xtivreg wage (tenure = hours age) collgrad, first

First-stage G2SLS regression
                                                 Number of obs    =         18
                                                 Wald chi(3)      =        124
                                                 Prob > chi2      =     0.0000
------------------------------------------------------------------------------
      tenure |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    collgrad |    .765833   1.656756     0.46   0.644     -2.48135    4.013016
       hours |     .20045   .0874043     2.29   0.022     .0291407    .3717593
         age |   .2684181    .026844    10.00   0.000      .215805    .3210313
       _cons |  -12.53995   3.324454    -3.77   0.000    -19.05577   -6.024144
------------------------------------------------------------------------------

G2SLS random-effects IV regression              Number of obs      =        18
Group variable: id                              Number of groups   =         9

R-sq:  within  = 0.7942                         Obs per group: min =         2
       between = 0.5361                                        avg =       2.0
       overall = 0.6614                                        max =         2

                                                Wald chi2(2)       =     40.14
corr(u_i, X)       = 0 (assumed)                Prob > chi2        =    0.0000

------------------------------------------------------------------------------
        wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      tenure |   .6361275   .1322349     4.81   0.000      .376952    .8953031
    collgrad |   3.301938    2.08222     1.59   0.113    -.7791381    7.383014
       _cons |   3.634081   .6243019     5.82   0.000     2.410472    4.857691
-------------+----------------------------------------------------------------
     sigma_u |  1.0911996
     sigma_e |  1.2966394
         rho |  .41459632   (fraction of variance due to u_i)
------------------------------------------------------------------------------
Instrumented:   tenure
Instruments:    collgrad hours age
------------------------------------------------------------------------------

. xtoverid, noi

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


First-stage regression of __00000H:

Statistics consistent for homoskedasticity only
Number of obs =                     18
------------------------------------------------------------------------------
    __00000H |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    __00000K |     .20045   .0874043     2.29   0.038     .0129864    .3879136
    __00000N |   .2684181    .026844    10.00   0.000     .2108436    .3259927
    __00000Q |    .765833   1.656756     0.46   0.651    -2.787556    4.319222
    __00000D |  -12.53995   3.324454    -3.77   0.002     -19.6702   -5.409709
------------------------------------------------------------------------------
F test of excluded instruments:
  F(  2,    14) =    50.09
  Prob > F      =   0.0000
Sanderson-Windmeijer multivariate F test of excluded instruments:
  F(  2,    14) =    50.09
  Prob > F      =   0.0000



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

                                           (Underid)            (Weak id)
Variable     | F(  2,    14)  P-val | SW Chi-sq(  2) P-val | SW F(  2,    14)
__00000H     |      50.09    0.0000 |      128.81   0.0000 |       50.09

Stock-Yogo weak ID F test critical values for single endogenous regressor:
                                   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 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(2)=15.79    P-val=0.0004

Weak identification test
Ho: equation is weakly identified
Cragg-Donald Wald F statistic                                      50.09

Stock-Yogo weak ID test critical values for K1=1 and L1=2:
                                   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.

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(2,14)=       20.19     P-val=0.0001
Anderson-Rubin Wald test           Chi-sq(2)=     51.91     P-val=0.0000
Stock-Wright LM S statistic        Chi-sq(2)=     13.37     P-val=0.0013

Number of observations               N  =         18
Number of regressors                 K  =          3
Number of endogenous regressors      K1 =          1
Number of instruments                L  =          4
Number of excluded instruments       L1 =          2

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

Estimates efficient for homoskedasticity only
Statistics consistent for homoskedasticity only

                                                      Number of obs =       18
                                                      F(  3,    15) =    71.22
                                                      Prob > F      =   0.0000
Total (centered) SS     =  85.16833116                Centered R2   =   0.7121
Total (uncentered) SS   =    368.81094                Uncentered R2 =   0.9335
Residual SS             =  24.52038882                Root MSE      =    1.167

------------------------------------------------------------------------------
    __00000F |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    __00000H |   .6361275   .1207134     5.27   0.000     .3995337    .8727214
    __00000Q |   3.301938   1.900798     1.74   0.082    -.4235578    7.027434
    __00000D |   3.634081   .5699071     6.38   0.000     2.517084    4.751079
------------------------------------------------------------------------------
Underidentification test (Anderson canon. corr. LM statistic):          15.793
                                                   Chi-sq(2) P-val =    0.0004
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):               50.093
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.
------------------------------------------------------------------------------
Sargan statistic (overidentification test of all instruments):           3.509
                                                   Chi-sq(1) P-val =    0.0610
------------------------------------------------------------------------------
Instrumented:         __00000H
Included instruments: __00000Q __00000D
Excluded instruments: __00000K __00000N
------------------------------------------------------------------------------

Test of overidentifying restrictions: 
Cross-section time-series model: xtivreg g2sls   
Sargan-Hansen statistic   3.509  Chi-sq(1)    P-value = 0.0610

What's intreseting here also is that the First Stage estimation is identical, but the second stage is not. Since Random Effects is not documented on xtivreg2, I don't know what causes this.

[Ariel Karlinsky]

Mark Schaffer - Can you shed some light on this? Iv'e also encountered this old thread where you question what the "small" adjustment in xtreg\xtivreg does - and I don't see you quite got an answer there...

[Emad Shehata]

small option is just to correct degrees of freedom
So, se. will differ between xtivreg and xtivreg2

[Ariel Karlinsky]

but xtivreg (built in) also has "small" option, and the only thing it does is report t stats instead of Z stats. SE's remain the same:

----------------------------------------------------------------------------------------------------------------------------------------
      name:  <unnamed>
       log:  C:\Users\karli_000\Desktop\statalist.log
  log type:  text
 opened on:   9 May 2016, 12:15:55

. sysuse nlswide1, clear
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

. gen id = _n

. reshape long age count collgrad c_city union ttl_exp tenure hours wage, i(id) j(year)
(note: j = 68 88)

Data                               wide   ->   long
-----------------------------------------------------------------------------
Number of obs.                        9   ->      18
Number of variables                  20   ->      12
j variable (2 values)                     ->   year
xij variables:
                            age68 age88   ->   age
                        count68 count88   ->   count
                  collgrad68 collgrad88   ->   collgrad
                      c_city68 c_city88   ->   c_city
                        union68 union88   ->   union
                    ttl_exp68 ttl_exp88   ->   ttl_exp
                      tenure68 tenure88   ->   tenure
                        hours68 hours88   ->   hours
                          wage68 wage88   ->   wage
-----------------------------------------------------------------------------

. xtset i year
       panel variable:  id (strongly balanced)
        time variable:  year, 68 to 88, but with gaps
                delta:  1 unit

. xtivreg wage (tenure = hours) collgrad, fe

Fixed-effects (within) IV regression         Number of obs      =           18
Group variable: id                           Number of groups   =            9

R-sq:  within  = 0.4748                      Obs per group: min =            2
       between = 0.3393                                     avg =          2.0
       overall = 0.2588                                     max =            2

                                             Wald chi2(2)       =       158.46
corr(u_i, Xb)  = -0.9454                     Prob > chi2        =       0.0000

------------------------------------------------------------------------------
        wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      tenure |  -.2315719   .8644902    -0.27   0.789    -1.925942    1.462798
    collgrad |   30.42604   27.65542     1.10   0.271    -23.77759    84.62967
       _cons |   1.403552   2.600473     0.54   0.589    -3.693281    6.500386
-------------+----------------------------------------------------------------
     sigma_u |   6.350528
     sigma_e |  2.1424307
         rho |  .89781635   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F  test that all u_i=0:     F(8,7) =     0.87             Prob > F    = 0.5811
------------------------------------------------------------------------------
Instrumented:   tenure
Instruments:    collgrad hours
------------------------------------------------------------------------------

. xtivreg wage (tenure = hours) collgrad, fe small

Fixed-effects (within) IV regression         Number of obs      =           18
Group variable: id                           Number of groups   =            9

R-sq:  within  = 0.4748                      Obs per group: min =            2
       between = 0.3393                                     avg =          2.0
       overall = 0.2588                                     max =            2

                                                F(11,7)         =        4.57
corr(u_i, Xb)  = -0.9454                        Prob > F        =       0.0538

------------------------------------------------------------------------------
        wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      tenure |  -.2315719   .8644902    -0.27   0.797    -2.275766    1.812623
    collgrad |   30.42604   27.65542     1.10   0.308    -34.96864    95.82072
       _cons |   1.403552   2.600473     0.54   0.606    -4.745589    7.552694
-------------+----------------------------------------------------------------
     sigma_u |   6.350528
     sigma_e |  2.1424307
         rho |  .89781635   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F  test that all u_i=0:     F(8,7) =     0.87             Prob > F    = 0.5811
------------------------------------------------------------------------------
Instrumented:   tenure
Instruments:    collgrad hours
------------------------------------------------------------------------------

. xtivreg2 wage (tenure = hours) collgrad, fe

FIXED EFFECTS ESTIMATION
------------------------
Number of groups =         9                    Obs per group: min =         2
                                                               avg =       2.0
                                                               max =         2

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

Estimates efficient for homoskedasticity only
Statistics consistent for homoskedasticity only

                                                      Number of obs =       18
                                                      F(  2,     7) =     4.57
                                                      Prob > F      =   0.0538
Total (centered) SS     =  61.18027552                Centered R2   =   0.4748
Total (uncentered) SS   =  61.18027552                Uncentered R2 =   0.4748
Residual SS             =  32.13006463                Root MSE      =    1.889

------------------------------------------------------------------------------
        wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      tenure |  -.2315719   .7624087    -0.30   0.761    -1.725865    1.262722
    collgrad |   30.42604   24.38979     1.25   0.212    -17.37706    78.22915
------------------------------------------------------------------------------
Underidentification test (Anderson canon. corr. LM statistic):           1.084
                                                   Chi-sq(1) P-val =    0.2979
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):                0.958
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:         tenure
Included instruments: collgrad
Excluded instruments: hours
------------------------------------------------------------------------------

. xtivreg2 wage (tenure = hours) collgrad, fe small

FIXED EFFECTS ESTIMATION
------------------------
Number of groups =         9                    Obs per group: min =         2
                                                               avg =       2.0
                                                               max =         2

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

Estimates efficient for homoskedasticity only
Statistics consistent for homoskedasticity only

                                                      Number of obs =       18
                                                      F(  2,     7) =     4.57
                                                      Prob > F      =   0.0538
Total (centered) SS     =  61.18027552                Centered R2   =   0.4748
Total (uncentered) SS   =  61.18027552                Uncentered R2 =   0.4748
Residual SS             =  32.13006463                Root MSE      =    2.142

------------------------------------------------------------------------------
        wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      tenure |  -.2315719   .8644902    -0.27   0.797    -2.275766    1.812623
    collgrad |   30.42604   27.65542     1.10   0.308    -34.96864    95.82072
------------------------------------------------------------------------------
Underidentification test (Anderson canon. corr. LM statistic):           1.084
                                                   Chi-sq(1) P-val =    0.2979
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):                0.958
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:         tenure
Included instruments: collgrad
Excluded instruments: hours
------------------------------------------------------------------------------

. log close
      name:  <unnamed>
       log:  C:\Users\karli_000\Desktop\statalist.log
  log type:  text
 closed on:   9 May 2016, 12:16:22
----------------------------------------------------------------------------------------------------------------------------------------

[Ariel Karlinsky]

Sorry for the bump, but is there anyway to go the other way around? replicate the SEs reported by xtivreg2/xtoverid with xtivreg? from what I gather it seems that the corrections used by xtivreg2/xtoverid are more generally acceptable