Aggregate versus Disaggregate Data in Measuring School Quality

⎛u
⎞
⎜
⎛
′ ′
⎜ ⎟ ~ N 0,
A
⎟
⎞
2 A1
Vu,e
A1
A2
.
⎝ w ⎠ ⎝
⎠
Having the joint distribution of u and w = u + e a , our estimator is easily derived (Moser, theorem
2.2.1) as:
E(
u | w)
= Cov(
u,
w)
Cov(
w)
1
−1
⎡
⎢
( w)
= ⎢
⎢
⎢⎣
2
u
σ + σ / n
2
u
σ
σ
2
u
2
e
2
u
2
e
M
σ + σ / n
Derivation of the conditional covariance matrix Cov ( u / uˆ
)
Disaggregate data: Recall equation (1.a):
Y = Xβ
+ Ζ u + e
⎡1n1
⎢
Ζ = ⎢
⎢
⎣
0
O
Ζ u + e ~ N
1
0
n
J
( 0,V )
⎤
⎥
⎥
⎥
⎦
J
1
w ⎤
1
⎥
⎥
w ⎥
J ⎥⎦
The shrinkage estimator of school effects (equation 2) in matrix notation is:
2 −1
uˆ
σ Z'V
(Y X ˆ
−1
−1
−1
−1
= u − β ) or u = Z'V
(I − X ( X'V
X ) X'V
)Y
2 ˆ σ u
(*)
This shows clearly that the shrinkage estimator is a linear combination of the independent
variable vector.
Thus, we can derive the joint distribution of ( u, uˆ
)' by knowing the distribution of ( u, Y )' .
The distribution of ( u, Y )' is:
2
⎡u
⎤ ⎛ ⎡
⎜
⎡0
⎤ σ u I
⎢ ⎥ ~ N
⎜ ⎢ ⎥,
⎢ 2
⎣Y
⎦ ⎝ ⎣Xβ
⎦ ⎣σ
u Z
2
σ ⎤⎞
u Z'
⎥⎟
.
V ⎟
⎦⎠
In general, u and any linear combination of Y of the form u ˆ = AY , will be jointly distributed as
follows: