Cerulli G., Working Paper Cnr-Ceris, N° 04/2014

and

where:
N1

y   ij  1  x jβ1  e1 j  



ij 1 j

j 1

N1

N1

j 1

j 1

ei    ij e1 j  e0i  wi (e1i  e0i )  wi  ij e1 j

N1

j 1

N1

N1

N1

j 1

j 1

j 1

1  ij   ij x jβ1   ij e1 j 


N1



N1



j 1



j 1

(12)

1    ij x j  β1   ij e1 j

and by developing ATE further using Eq.
(11), we finally get the result in (10).

The proof is in Appendix. See A3.
Proposition 4. Ordinary Least Squares
(OLS) consistency. Under assumption 1
(CMI), 2 and 3, the error tem of regression
(14) has zero mean conditional on (wi, xi), i.e.:

N1
 N1

E  ei wi ,xi   E    ij e1 j  e0i  wi (e1i  e0i )  wi  ij e1 j wi ,xi   0
Proposition 2. Formula of ATE(xi) with
j 1
 j 1

(15)
N1
N1


neighbourhood
interactions.
Given
E  ei wi ,xi   E    ij e1 j  e0i  wi (e1i  e0i )  wi  ij e1 j wi ,xi   0
assumptions 2 and 3 and the result  inj 1
j 1

proposition 1, we have that:

thus implying that Eq. (14) is a regression
whose parameters can be consistently
ATE(xi ) = ATE  (xi  x )δ  ij ( x  x model
j ) β1
j

1
(13) estimated by Ordinary Least Squares (OLS).
N1
ATE(xi ) = ATE  (xi  x )δ  ij ( x  x j ) β1
The proof is in Appendix. See A4.
j 1
Once a consistent estimation of the
parameters of (14) is obtained, we can
where it is now easy to see that estimate ATE directly from the regression,
ATE =Ex{ATE(x)}. The proof is in Appendix. and ATE(xi) by plugging the estimated
See A2.
parameters into formula (11). This is because
ATE(xi) becomes a function of consistent
Proposition 3. Baseline random-coefficient estimates, and thus consistent itself:
regression. By substitution of equations (7)
into the POM of Eq. (6), we obtain the
plim ATE(xi )  ATE(xi )
following random-coefficient
regression
model (Wooldridge, 1997):
where ATE(xi ) is the plug-in estimator of
N1





N

ATE(xi).

Observe,

however,

that

the

yi    wi  ATE+xiβ0  wi (xi  x )δ  wi ij w j ((exogenous)
x  x j ) β1  eiweighting matrix(14)
Ω=[ωij] needs
N

   wi  ATE+xiβ0  wi (xi  x )δ  wi ij w j ( x  x j ) β1  ei
j 1

where,

  0  1

δ  β1  β0

j 1

(14)

to be provided in advance.
(14)
Once the formulas for ATE and ATE(xi) are
available, it is also possible to recover the
Average Treatment Effect on Treated (ATET)
and on non-Treated (ATENT) as:

11