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I'm reading Mostly Harmless Econometrics (Available here), and on page 100 they say that 2SLS with dummy instruments is the same as GLS on a set of group means. I don't understand why. From the previous chapters, I got how instrumental variables work in general, but im struggling to differentiate between the Wald Estimator, 2SLS and grouped data. My only explanation so far is that when you use a dummy instrument in the first stage regression, you basically group your second stage regressors according to the dummy first stage instruments. But i still dont get how this relates to group means. Can somebody help?

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The first stage is a regression of endogenous $x$ on binary $d$, so everyone gets with $d=1$ gets the same $\hat x$, and everyone with $d=0$ gets the same $\hat x$. The two values of $\hat x$ will be different as long as the instrument $d$ is relevant.

The second stage is just a regression of $y$ on $\hat x$. This can be done in two ways. The usual way is to regress $y$ on $\hat x$ for the full sample. But you could also just calculate the mean of $y$ for each of the two values of $\hat x$ and do a weighted regression of that mean on $\hat x$, where the weights are number of observations in each of the two $\hat x$ groups. Weighted least squares is a type of GLS. In this case you just have two observations instead of $N$.

Here's an example showing this in Stata with a toy model:

. sysuse auto, clear
(1978 Automobile Data)

. ivreg2 price (mpg = i.foreign)

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

Estimates efficient for homoskedasticity only
Statistics consistent for homoskedasticity only

                                                      Number of obs =       74
                                                      F(  1,    72) =     0.15
                                                      Prob > F      =   0.6987
Total (centered) SS     =  635065396.1                Centered R2   =  -0.1314
Total (uncentered) SS   =   3447834321                Uncentered R2 =   0.7916
Residual SS             =  718514120.5                Root MSE      =     3116

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |   63.13609   160.2388     0.39   0.694    -250.9262    377.1984
       _cons |   4820.629   3431.824     1.40   0.160    -1905.623    11546.88
------------------------------------------------------------------------------
Underidentification test (Anderson canon. corr. LM statistic):          11.452
                                                   Chi-sq(1) P-val =    0.0007
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):               13.183
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:         mpg
Excluded instruments: 1.foreign
------------------------------------------------------------------------------

. /* First Stage */
. qui reg mpg i.foreign

. predict double mpg_hat
(option xb assumed; fitted values)

. tab mpg_hat

     Fitted |
     values |      Freq.     Percent        Cum.
------------+-----------------------------------
   19.82692 |         52       70.27       70.27
   24.77273 |         22       29.73      100.00
------------+-----------------------------------
      Total |         74      100.00

. /* Second Stage */
. reg price mpg_hat

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(1, 72)        =      0.17
       Model |  1507382.66         1  1507382.66   Prob > F        =    0.6802
    Residual |   633558013        72  8799416.85   R-squared       =    0.0024
-------------+----------------------------------   Adj R-squared   =   -0.0115
       Total |   635065396        73  8699525.97   Root MSE        =    2966.4

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     mpg_hat |   63.13609   152.5432     0.41   0.680    -240.9532    367.2254
       _cons |   4820.629   3267.008     1.48   0.144    -1692.032    11333.29
------------------------------------------------------------------------------

. /* Second Stage with WLS/GLS */
. collapse (mean) mean_price = price (count) N = headroom, by(mpg_hat)

. reg mean_price mpg_hat [fweight = N]

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(1, 72)        =         .
       Model |  1507383.12         1  1507383.12   Prob > F        =         .
    Residual |           0        72           0   R-squared       =    1.0000
-------------+----------------------------------   Adj R-squared   =    1.0000
       Total |  1507383.12        73  20649.0838   Root MSE        =         0

------------------------------------------------------------------------------
  mean_price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     mpg_hat |    63.1361          .        .       .            .           .
       _cons |   4820.628          .        .       .            .           .
------------------------------------------------------------------------------

The second regression is just WLS/GLS on this data:

  mpg_hat   mean_price    N  
19.826923      6,072.4   52  
24.772727      6,384.7   22  
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