Appendix: Derivation of the simple Moulton factor
Write
and
where Lg is a column vector of ng ones and G is the number of groups. Note that
Let tg = 1 + (ng — 1 )p, so we get
^2X ng t g xg Xg •
With this in hand, we can write
V(b) = (X’X) 1 X’ФХ (X’X) 1
X ngxgx’g) X ngtgxgx’g X ngxgx’g
ggg
We want to compare this with the standard OLS covariance estimator
If the group sizes are equal, ng = n and tg = t = 1 + (n — 1)p, so that
V (b) = ^T ( X nxg x’gI X nxg xg ( X nxg x’g I
g / g V g /
nxgxg
g
which implies (8.2.4).
Table 8.1.1: Monte Carlo results for robust standard errors
