st: How does xtreg calculate clustered standard error?

[Ting JIANG <econjiang@gmail.com>]

Dear all,

I'm using Stata 12, and I'm trying to understand xtreg with simulated
panel data. When I try the following three lines,

reg y x i.id, cluster(id)

areg y x, a(id) cluster(id)

xtreg y x, fe robust

Theoretically they should give identical results. I noticed that they
give the same coefficient estimates as expected, but the clustered
standard errors are different.

The results from OLS (the first and second lines) are in accordance
with the fact that Stata has a multiplier (finite sample adjustment)
for clustered variance G/(G-1) *(n-1)/(n-k), where G is the number of
clusters.

But the clustered standard error reported after xtreg (the third line)
does not match those I got after reg and areg. They are different from
the numbers without finite sample adjustment too.

On the other hand, if conditional homoskedasticity is assumed, reg,
areg, xtreg give exactly the same results.

So my questions are:

(1) How exactly does Stata calculate standard errors in the ``xtreg +
cluster'' case?

(2) Can I expect reg and xtreg to give the same answer when properly specified?

(3) By the way, I also noticed that the t-statistic produced after
xtreg with cluster option does not follow Stock & Watson's (ECMA,
2008) suggestion -- multiply the traditional t-statistic by
sqrt(G/(G-1)). Am I correct?

Thanks.

I'm providing details of my simulation exercise below.

clear all
set seed 20140312
set obs 10
gen id=_n
gen u=rnormal()
expand 4
sort id
gen t=mod(_n-1,4)+1
gen x=rnormal()
gen y=x+u+rnormal()
tab id, gen(d)

xtset id t
reg y x i.id
areg y x, a(id)
xtreg y x, fe

/*
In all three cases, coefficient estimates on x is 1.138147, with
standard error .2372135
*/

reg y x i.id, cluster(id)
areg y x, a(id) cluster(id)
xtreg y x, fe robust

/*
In the first two cases, standard error of the coefficient estimate on
x is .2779968, whereas in the third case, it is .2428549. Coefficient
estimates remain unchanged.
*/

reg y x d*
predict e, residual
mat opaccum A=x d2-d10, group(id) op(e)
mat accum B=x d2-d10
mat V=invsym(B)*A*invsym(B)
di sqrt(V[1,1]*10/(10-1)*(40-1)/(40-11))
di sqrt(V[1,1])

/*
The numbers displayed are .27799676 and .22741949. The first number
coincides with the results after reg and areg when specifying cluster
option. The standard error after xtreg matches neither of the numbers.
×/

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