mixed - Stata

mixed — Multilevel mixed-effects linear regression 37
where y jk represents the logarithms of gross state products for the n jk = 17 observations from state
j in region k, X jk is a set of regressors, u (3)
k
is a random intercept at the region level, and u (2)
jk
is
a random intercept at the state (nested within region) level. We assume that u (3)
k
∼ N(0, σ3) 2 and
u (2)
jk
∼ N(0, σ2 2) independently. Define
⎡
v k = ⎢
⎣
u (3)
k
+ u (2)
1k
u (3)
k
+ u (2)
2k
.
.
u (3)
k
+ u (2)
M k ,k
where M k is the number of states in region k. Making this substitution, we can stack the observations
for all the states within region k to get
⎤
⎥
⎦
y k = X k β + Z k v k + ɛ k
where Z k is a set of indicators identifying the states within each region; that is,
for a k-column vector of 1s J k , and
Z k = I Mk ⊗ J 17
⎡
σ3 2 + σ2 2 σ3 2 · · · σ 2 ⎤
3
σ3 2 σ3 2 + σ2 2 · · · σ3
2 Σ = Var(v k ) = ⎢
⎣
.
.
. ..
. ⎥ .. ⎦
σ3 2 σ3 2 σ3 2 σ3 2 + σ2
2
M k ×M k
Because Σ is an exchangeable matrix, we can fit this alternative form of the model by specifying the
exchangeable covariance structure.