Econometrics

a) Suppose that we are interested in the causal effea of a binary treatment D on an outcome Y.

Assume that s(y°lo = 1,10 = E(Y°ID = 0.1!) where V4 d = 1.0 are the potential outcomes max is a
set ot‘ observed pre-treatment covamtes Let 1? the OLS estimator offi in the following regression
model:
Y;= a+pog+x§6+£h

Show that ph’mw) = [9(1″1 – Y‘olD = 1) if either one crboth of those two conditions are satisfied:

1. 503‘ – yle = 1.2:) = 80:1 – Y,°lo = 1),

2. Y.” – Y.-° = 5‘” l D,- = 1 (Effect homogeneity among the treated).
(Hint use the observational rule Y; = 0;?) + (1 – DOV? and the plim ot‘ the OLS coefficient of a binary
amiable.)

b) Rm a small Monte Carlo simulation study in which compare OLS, a simple regession adjustment and
inverse probability weighting (running two separate regression for treated and non-treated) in those two
scenarios:

(Effect ha’nogmdty) In this scenario the true model is

Y‘= a+fio;+X.6+8.
01′ = 10‘7“” U > 0),
where I (~) is the indicator fimoticn.
(Effect heterogeneity) In this Scenario the true model is
K=d+figog+Z¢8+ee
D; = ((Zflr +0 >0),
Where 2 is a binary indicator and
N(a. b2) in:I – 1 and 0‘ -1
fl; ~N(c,d2) ifz‘ = 0 undo,- =1
0 if D, = 0
Choose c as a and only discus what c = a implies
Examples and hints
Draw two random variablesA and B. For example choose A~N(1, 1) and B~N(2,1), then
3‘: D:”Z“A+D“(1-ZJ‘B.
Note that the true ATT is given by
8(A) Pr(D = II?! = 1) Pr(Z = 1) + 8(8) Pr(D = 1IZ = 0) Pr(Z = 0)
BUM D = 1)
Pr(D – 1)

Hint 1: va~N(0.1), then itis easyto compute Pr(D = 1IZ = 1) and Pr(D = llZ = 0) Earn 0;
I(Z;rr+ v > 0), e. g , Pr(D =1IZ = 1) = Pr(v > -Z;rrlZ =1).
Hint 2: Pr(D = 1) = Pr(D = IIZ = 1) Pr(Z = 1) + Pr(D = 1lZ = 0) Pr(Z = 0).
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