In the book 'Mastering 'Metrics', Joshua D. Angrist and Jörn-Steffen Pischke, in an example of the causal effect of having health insurance or not on health levels, they describe "... ...This in turn leads to a simple but important conclusion about the difference in average health by insurance status:
Difference in group means = Avgn[Yi|Di=1]-Avgn[Yi|Di=0]
=Avgn[Y1i|Di=1]-Avgn[Y0i|Di=0], (1.2)"
Next, they prove that Difference in group means differs from causal effects "The constant-effects assumption allows us to write:
Y1i=Y0i+k, (1.3)
or, equivalently, Y1i - Y0i = κ. In other words, κ is both the individual and average causal effect of insurance on health. using the constant- effects model (equation (1.3)) to substitute for
Avgn[Y1i|Di = 1] in equation (1.2), we have:
Avgn[Y1i|Di = 1] - Avgn[Y0i|Di = 0]
={k+ Avgn[Y0i|Di = 1]}- Avgn[Y0i|Di = 0]
=k+ {Avgn[Y0i|Di = 1]-Avgn[Y0i|Di = 0]},”
From this they obtain that: "Difference in group means = Average casual effect + Selection bias."
I think there are some confusing aspects of this process. The difference between Difference in group means and ATE should be described in more detail as,
Difference in group means
= Avgn[Yi|Di=1] - Avgn[Yi|Di=0] = Avgn[Y1i|Di=1] - Avgn[Y0i|Di=0]
=Avgn[Y1i-Y0i]+Avgn[Y0i|Di=1]-Avgn[Y0i|Di=0]+{1-Pr[Di=1]}*{Avgn[Y1i-Y0i|Di=1]-Avgn[Y1i-Y0i|Di=0]}
where Avgn[Y1i - Y0i] is the ATE
or,
= Avgn[Y1i-Y0i|Di=1]+Avgn[Y0i|Di=1]-Avgn[Y0i|Di=0]
where Avgn[Y1i - Y0i|Di=1] is the ATT
or,
= Avgn[Y1i-Y0i|Di=0]+Avgn[Y1i|Di=1]-Avgn[Y1i|Di=0]
where Avgn[Y1i - Y0i|Di=0] is the ATC
Under the assumption of constant-effects (Y1i-Y0i=k), Avgn[Y1i-Y0i] = Avgn[Y1i-Y0i|Di=1] = Avgn[Y1i-Y0i|Di=0].
So,
Difference in group means
= Avgn[Y1i-Y0i]+Avgn[Y0i|Di=1]-Avgn[Y0i|Di=0]+0
= Avgn[Y1i-Y0i]+Avgn[Y0i|Di=1]-Avgn[Y0i|Di=0]
= k+Avgn[Y0i|Di=1]-Avgn[Y0i|Di=0].
or, = Avgn[Y1i-Y0i|Di=1]+Avgn[Y0i|Di=1]-Avgn[Y0i|Di=0]
=Avgn[Y1i-Y0i]+Avgn[Y0i|Di=1]-Avgn[Y0i|Di=0]
=k+Avgn[Y0i|Di=1]-Avgn[Y0i|Di=0].
or, = Avgn[Y1i-Y0i|Di=0]+Avgn[Y1i|Di=1]-Avgn[Y1i|Di=0]
= Avgn[Y1i-Y0i]+Avgn[Y1i|Di=1]-Avgn[Y1i|Di=0]
= k+Avgn[Y1i|Di=1]-Avgn[Y1i|Di=0].
I am not sure if my derivation process above is correct.
As far as I grasp, the derivation seems ok. I would suggest to check in another book to confirm it, such as "Mostly harmless Econometrics" (same authors), in chapter 2.1.
https://www.mostlyharmlesseconometrics.com/
José-Ignacio Antón Thank you for your comments! I checked the derivation of this formula in "Mostly harmless Econometrics" and it is the same as the derivation in "Mastering 'Metrics". I think this process could be described in more detail, just like our derivation process above (if my derivation process is correct).
In econometric research, causality develops explicit regression models, where the causes and effect are investigated. Selection bias is a kind of error that can be represented by ei in sample regression. In my opinion, the sum of squared regression = difference between the predicted value and the mean of depndendent variable
Yadav Sharma Gaudel Thank you for your comment. However, I still have some confusion that the ei in the regression model you mention is more like an error term. The selection bias (in this case omitted variable bias) is due to the fact that the error term contains omitted variables that are correlated with the independent variable, resulting in regression coefficients that have only a correlation interpretation but not a causality interpretation. According to the formula for omitted variable bias (OVB) proposed by Angrist and Pischke, OVB = effect of the omitted variable on the independent variable * effect of the omitted variable on the dependent variable.
I agree the error term, which represents omitted variables or other variables that are not incorporated in regression model.
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