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Estimation of Error Variance

The coverage probability of this conÖdence interval is
P

 2 Cb

= P (jT()j  c) ! P (jZj  c) = 1
where the limit is taken as n ! 1, and holds since T() is asymptotically jZj by Theorem 7.11. We
call the limit the asymptotic coverage probability, and call Cb an asymptotic 1 % conÖdence
interval for . Since the t-ratio is asymptotically pivotal, the asymptotic coverage probability is
independent of the parameter :
It is useful to contrast the conÖdence interval (7.35) with (5.11) for the normal regression
model. They are similar, but there are di§erences. The normal regression interval (5.11) only
applies to regression coe¢ cients , not to functions  of the coe¢ cients. The normal interval
(5.11) also is constructed with the homoskedastic standard error, while (7.35) can be constructed
with a heteroskedastic-robust standard error. Furthermore, the constants c in (5.11) are calculated
using the student t distribution, while c in (7.35) are calculated using the normal distribution. The
di§erence between the student t and normal values are typically small in practice (since sample sizes
are large in typical economic applications). However, since the student t values are larger, it results
in slightly larger conÖdence intervals, which is probably reasonable. (A practical rule of thumb is
that if the sample sizes are su¢ ciently small that it makes a di§erence, then probably neither (5.11)
nor (7.35) should be trusted.) Despite these di§erences, the coincidence of the intervals means that
inference on regression coe¢ cients is generally robust to using either the exact normal sampling
assumption or the asymptotic large sample approximation, at least in large samples.
In Stata, by default the program reports 95% conÖdence intervals for each coe¢ cient where
the critical values c are calculated using the tnk distribution. This is done for all standard error
methods even though it is only justiÖed for homoskedastic standard errors and under normality.
The standard coverage probability for conÖdence intervals is 95%, leading to the choice c = 1:96
for the constant in (7.35). Rounding 1.96 to 2, we obtain what might be the most commonly used
conÖdence interval in applied econometric practice
Cb =
h
b 2s(b); b+ 2s(b)
i
:
This is a useful rule-of thumb. This asymptotic 95% conÖdence interval Cb is simple to compute and
can be roughly calculated from tables of coe¢ cient estimates and standard errors. (Technically, it
is an asymptotic 95.4% interval, due to the substitution of 2.0 for 1.96, but this distinction is overly
prec