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

(12.9.13)
Thus, in reasonably large samples one can obtain 
ρ
from Eq. (12.9.13) and use it to trans-
form the data as shown in the generalized difference equation (12.9.5). Keep in mind that
the relationship between 
ρ
and 
d
given in Eq. (12.9.13) may not hold true in small samples,
for which Theil and Nagar have proposed a modification, which is given in Exercise 12.6.
In our wages–productivity regression (12.5.2), we obtain a 
d
value of 0.2176. Using this
value in Eq. (12.9.13), we obtain 
ˆ
ρ
≈
0
.
8912. Using this estimated 
ρ
value, we can esti-
mate regression (12.9.5). All we have to do is subtract 0.8912 times the previous value of 
Y
from its current value and similarly subtract 0.8912 times the previous value of 
X
from
its current value and run the OLS regression on the variables thus transformed as in
Eq. (12.9.6), where 
Y
∗
t
=
(
Y
t
−
0
.
8912
Y
t
−
1
) and
X
∗
t
=
(
X
t
−
0
.
8912
X
t
−
1
) .
ρ
Estimated from the Residuals
If the AR(1) scheme 
u
t
=
ρ
u
t
−
1
+
ε
t
is valid, a simple way to estimate 
ρ
is to regress the
residuals 
ˆ
u
t
on 
ˆ
u
t
−
1
, for the 
ˆ
u
t
are consistent estimators of the true 
u
t
, as noted previously.
That is, we run the following regression:
ˆ
u
t
=
ρ .
ˆ
u
t
−
1
+
v
t