Ceris-Cnr, W.P. N° 3 /2008

[36]

ATET

M

⎡

1
NT

=

∑ ⎢⎢ y
i∈T

⎣

T
i

⎤

∑ω

C
ij y j ⎥

−

⎥⎦

j∈C ( i )

where 0 < ωij < 1 is the weight given to the control unit j-th in the comparison with the unit i-th
(with:

∑ω

ij

= 1 ). For each treated unit i, the sum in the square brackets is thus a weighted average

j∈C ( i )

of its (selected) control units. In the case of the “arithmetic mean”, the weights become
ωij = 1/ N iC and the previous formula reduces to:

ATET

ArM

=

1
NT

∑
i∈T

⎡ T
1
⎢ yi − C
Ni
⎢⎣

⎤
y Cj ⎥ .
⎥⎦
j∈C ( i )

∑

C

Therefore, for the nearest neighbour matching, since N i =1 (so that j=i), the formula becomes:

ATET

NN

=

1
NT

⎛
⎜⎜
⎝

∑ y −∑ y
T
i

C
i

i

i

⎞
⎟⎟ ,
⎠

while for the three-nearest neighbours, it takes the following form:

ATET

3NN

=

⎡

1
NT

∑ ⎢⎢ y
i∈T

⎣

T
i

−

1
3

⎤
y Cj ⎥ .
⎥⎦
j∈C ( i )

∑

Furthermore, the kernel matching comes up from equation [36] when:

ωij =

K ( p j − pi )
NiC

∑ K( p

j

,

− pi )

j =1

where K is the kernel function.
Finally, provided that outcomes are considered independent across units, it can be proved that the
analytical variance of the estimator in equation [36] is equal to:
Var ⎛⎜ ATET
⎝
where ω j =

∑ω
i

ij

M

⎞= 1
⎟
⎠ ( N T )2

∑Var ( y

T
i )+

j∈T

∑ (ω ) Var ( y
j

2

C
j )

j∈C

. It is quite clear that there is a penalty for using the same controls more than one

time.

31