mixed - Stata

16 mixed — Multilevel mixed-effects linear regression
Examining the REML output, we find that the estimates of the variance components are slightly
larger than the ML estimates. This is typical, because ML estimates, which do not incorporate the
degrees of freedom used to estimate the fixed effects, tend to be biased downward.
Three-level models
The clustered-data representation of the mixed model given in (2) can be extended to two nested
levels of clustering, creating a three-level model once the observations are considered. Formally,
y jk = X jk β + Z (3)
jk u(3) k
+ Z (2)
jk u(2) jk + ɛ jk (7)
for i = 1, . . . , n jk first-level observations nested within j = 1, . . . , M k second-level groups, which
are nested within k = 1, . . . , M third-level groups. Group j, k consists of n jk observations, so y jk ,
X jk , and ɛ jk each have row dimension n jk . Z (3)
jk
is the n jk × q 3 design matrix for the third-level
random effects u (3)
k
, and Z(2)
jk is the n jk × q 2 design matrix for the second-level random effects u (2)
jk .
Furthermore, assume that
u (3)
k
∼ N(0, Σ 3 ); u (2)
jk ∼ N(0, Σ 2); ɛ jk ∼ N(0, σ 2 ɛ I)
and that u (3)
k
, u(2) jk , and ɛ jk are independent.
Fitting a three-level model requires you to specify two random-effects equations: one for level
three and then one for level two. The variable list for the first equation represents Z (3)
jk
second equation represents Z (2)
jk
; that is, you specify the levels top to bottom in mixed.
Example 4
and for the
Baltagi, Song, and Jung (2001) estimate a Cobb–Douglas production function examining the
productivity of public capital in each state’s private output. Originally provided by Munnell (1990),
the data were recorded over 1970–1986 for 48 states grouped into nine regions.