Aggregate versus Disaggregate Data in Measuring School Quality

(9) ˆ − 1 ˆ −1
−1
[ X V X ( m V m)]
( X Vˆ
−1
β = ′ − E ′
′ Y )
)
.
When measurement error is not taken into account, the matrix ( m V m)
1 −
E ′ is omitted.
Using Goldstein’s derivation of ( m V m)
1 −
E ′ and realizing that the inverse of V is also a block
2 2
( n j −1)
σ u + σ e
diagonal with elements
in the diagonal, each element ( 2 2 2
1, 2 )
σ ( n σ + σ )
h h of the H × H
matrix ( m V m)
1 −
E ′ can be expressed as
e
j
2
J ⎧ σ ⎫ σ
u
( h1
, h2
) m
(10) ∑ n
= ⎨ − +
j j ( n
1 j 1)
1⎬
.
2
2 2
⎩ σ e ⎭ n jσ
u + σ e
u
e
Now let us see how this omitted matrix, ( m V m)
1 −
E ′ , compares with the one obtained when
aggregating the model. Aggregating the true disaggregate model, we obtain:
(11) = x β ) + u + e , j = 1,
K,
J
where notation is as in model (8).
y. j ( . j j . j
Y . j = y.
j + q.
j
X h.
j = xh.
j + mh.
j
cov( q . j , q.
j'
) = 0
E ( q.
j ) = E(
mh.
j ) = 0
m m = σ / n
cov( h ij , h ij ) h , h ) m
1
2
11
( 1 2
Notice how the covariance of measurement error between any two fixed explanatory
variables is reduced in the aggregate model. Now the covariance matrix of the true model is a
diagonal matrix with elements defined in the first part of this section; and which will be denoted
j