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Demonstrating the Frisch–Waugh–Lovell theorem in Stata
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clear | |
set seed 10009 | |
set obs 100 | |
gen x1 = rnormal() | |
* Induce positive correlation between x1 and x2 | |
gen x2 = rnormal() + .2*x1 | |
* TRUE data-generating process | |
gen y = 1 + x1 + 5*x2 + rnormal() | |
* Step 1: Residualize x2 | |
reg x2 x1 | |
predict resid_x2, res | |
* Step 2: Residualize y | |
reg y x1 | |
predict resid_y, res | |
reg resid_y resid_x2, noci cformat(%9.2f) pformat(%5.2f) sformat(%8.2f) | |
* Correcty specified regression | |
reg y x1 x2, noci cformat(%9.2f) pformat(%5.2f) sformat(%8.2f) | |
* The coef. on x2 is overstated, due to OVB: | |
reg y x2, noci cformat(%9.2f) pformat(%5.2f) sformat(%8.2f) | |
* Just using residualized x2: | |
reg y resid_x2, noci cformat(%9.2f) pformat(%5.2f) sformat(%8.2f) | |
* Plugging in both: | |
reg y x1 resid_x2, noci cformat(%9.2f) pformat(%5.2f) sformat(%8.2f) | |
scatter y x2, /// | |
yscale(r(-10 15)) /// | |
name(n1, replace) title("Correlation between Y and X2" "(uncontrolled)") | |
scatter resid_y resid_x2, /// | |
yscale(r(-10 15)) /// | |
name(n2, replace) title("FWL" "(controls for X1)") /// | |
xtitle("x2_residuals") ytitle("y_residuals") | |
gr combine n1 n2 |
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