How do we test for significant differences in the coefficients of the same variable between two 2SLS models?

[Xinya Hao]

Hi!

How do we test for significant differences in the coefficients of the same variable between two 2SLS models?

Acemoglu (Acemoglu, Johnson, and Robinson, 2001, AER, page 1395) compared the coefficient differences of the same variable in two instrumental variable (IV) models in his paper "The Colonial Origins of Comparative Development: An Empirical Investigation."

Model 1: ivregress logy (R = S)
Model 2: ivregress logy (R = S M)

How to test [Model 1]R = [Model 2]R?

In his paper, he reported the results of a chi-square test but did not provide specific details on how this step was implemented. (Table 8, Panel C)

In his paper, this test is called the "overidentification test."

The null hypothesis is that the 2SLS coefficients estimated with the instruments indicated in model 1 versus the coefficients estimated using model 2 are significantly different.

How can you replicate this test using Stata? SUR? GMM?

You can find the paper using the DOI (10.1257/aer.91.5.1369).

[Xinya Hao]

Hello everyone. I hope you are having a good day~

Regarding the previous question, I have devised a solution using Seemingly Unrelated Regressions (SUR).

*** toy codes
webuse hsng2, clear
    
    // Model 1 have 4 instruments
    reg hsngval faminc reg1 reg2 reg3
    predict h1, xb

    // Molde 2 have only 1 instruments
    reg hsngval faminc
    predict h2, xb

    // SUR
    sureg (rent = h1) (rent = h2)

    // test for significant differences in the coefficients of the -hsngval- between two models
    test [rent]h1 = [2rent]h2

However, the approach I used for manual computation of the two-stage process seems rather simplistic and may lead to errors in standard errors.

I would greatly appreciate it if you have any alternative methods or suggestions.

Thanks again.

Kind regards,
Hall

[Andrew Musau]

Here is one way to jointly estimate both equations using the gmm command.

webuse hsng2, clear
ivregress 2sls rent pcturban (hsngval = faminc), robust
ivregress 2sls rent pcturban (hsngval = faminc i.region), robust

gmm (eq1: rent - {b1}*hsngval - {b2}*pcturban - {b0}) ///
    (eq2: rent - {c1}*hsngval - {c2}*pcturban - {c0}), ///
    instruments(eq1: pcturban faminc) ///
    instruments(eq2: pcturban faminc i.region) ///
    onestep winitial(unadjusted, indep)

Res.:

. ivregress 2sls rent pcturban (hsngval = faminc), robust

Instrumental variables (2SLS) regression          Number of obs   =         50
                                                  Wald chi2(2)    =      32.55
                                                  Prob > chi2     =     0.0000
                                                  R-squared       =     0.2887
                                                  Root MSE        =     29.517

------------------------------------------------------------------------------
             |               Robust
        rent | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
     hsngval |   .0031938    .000738     4.33   0.000     .0017474    .0046402
    pcturban |  -.5064118   .5428297    -0.93   0.351    -1.570339    .5575149
       _cons |   113.8143   21.62169     5.26   0.000     71.43659    156.1921
------------------------------------------------------------------------------
Instrumented:  hsngval
Instruments:   pcturban faminc

. ivregress 2sls rent pcturban (hsngval = faminc i.region), robust

Instrumental variables (2SLS) regression          Number of obs   =         50
                                                  Wald chi2(2)    =      44.98
                                                  Prob > chi2     =     0.0000
                                                  R-squared       =     0.5989
                                                  Root MSE        =     22.166

------------------------------------------------------------------------------
             |               Robust
        rent | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
     hsngval |   .0022398    .000672     3.33   0.001     .0009227    .0035569
    pcturban |    .081516   .4445938     0.18   0.855     -.789872    .9529039
       _cons |   120.7065   15.25546     7.91   0.000     90.80636    150.6067
------------------------------------------------------------------------------
Instrumented:  hsngval
Instruments:   pcturban faminc 2.region 3.region 4.region

. 
. gmm (eq1: rent - {b1}*hsngval - {b2}*pcturban - {b0}) ///
>     (eq2: rent - {c1}*hsngval - {c2}*pcturban - {c0}), ///
>     instruments(eq1: pcturban faminc) ///
>     instruments(eq2: pcturban faminc i.region) ///
>     onestep winitial(unadjusted, indep)

Step 1
Iteration 0:   GMM criterion Q(b) =  112123.69  
Iteration 1:   GMM criterion Q(b) =  110.91583  
Iteration 2:   GMM criterion Q(b) =  110.91583  

GMM estimation 

Number of parameters =   6
Number of moments    =   9
Initial weight matrix: Unadjusted                 Number of obs   =         50

------------------------------------------------------------------------------
             |               Robust
             | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
         /b1 |   .0031938   .0007379     4.33   0.000     .0017476    .0046401
         /b2 |  -.5064118   .5427822    -0.93   0.351    -1.570245    .5574217
         /b0 |   113.8143   21.62166     5.26   0.000     71.43665     156.192
         /c1 |   .0022398    .000672     3.33   0.001     .0009227    .0035569
         /c2 |    .081516   .4445939     0.18   0.855     -.789872    .9529039
         /c0 |   120.7065   15.25546     7.91   0.000     90.80637    150.6067
------------------------------------------------------------------------------
Instruments for equation eq1: pcturban faminc _cons
Instruments for equation eq2: pcturban faminc 1b.region 2.region 3.region 4.region _cons

[Xinya Hao]

Great. It's exactly what I wanted.
Thanks to Musau for providing the GMM code.
Thanks!