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
