The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

(12.4.1)
where 
r
=
n
−
1
t
=
1
x
t
x
t
−
1
/
n
t
=
1
x
2
t
, which can be interpreted as the (sample) correlation
coefficient between successive values of the 
X
’s.
15
If 
ρ
and 
r
are both positive (not an
unlikely assumption for most economic time series), it is apparent from Eq. (12.4.1) that
E
(
ˆ
σ
2
)
< σ
2
; that is, the usual residual variance formula, on average, will underestimate
the true 
σ
2
. In other words, 
ˆ
σ
2
will be biased downward. Needless to say, this bias in 
ˆ
σ
2
will be transmitted to var (
ˆ
β
2
) because in practice we estimate the latter by the formula
ˆ
σ
2
/
x
2
t
.
But even if 
σ
2
is not underestimated, var (
ˆ
β
2
) is a 
biased
estimator of var (
ˆ
β
2
)
AR1
, which
can be readily seen by comparing Eq. (12.2.7) with Eq. (12.2.8),
16
for the two formulas are
not the same. As a matter of fact, if 
ρ
is positive (which is true of most economic time
series) and the 
X
’s are positively correlated (also true of most economic time series), then
it is clear that
var (
ˆ
β
2
)
<
var (
ˆ
β
2
)
AR1
(12.4.2)
that is, the usual OLS variance of 
ˆ
β
2
underestimates its variance under AR(1) (see
Eq. [12.2.9]). Therefore, if we use var (
ˆ
β
2
), we shall inflate the precision or accuracy (i.e.,
underestimate the standard error) of the estimator 
ˆ
β
2
. As a result, in computing the 
t
ratio
as 
t
= ˆ
β
2
/
se (
ˆ
β
2
) (under the hypothesis that 
β
2
=
0), we shall be overestimating the 
t
value
and hence the statistical significance of the estimated 
β
2
. The situation is likely to get worse
if additionally 
σ
2
is underestimated, as noted previously.
To see how OLS is likely to underestimate 
σ
2
and the variance of 
ˆ
β
2
, let us conduct the
following