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

(3)
which is the empirical counterpart of Eq. (2).
‡
3. Using 
ˆ
ρ
obtained in Eq. (3), estimate the generalized difference equation (12.9.6).
*
G. Hildreth and J. Y. Lu, “Demand Relations with Autocorrelated Disturbances,’’ Michigan State
University, 
Agricultural Experiment Station,
Tech. Bull. 276, November 1960.
†
D. Cochrane and G. H. Orcutt, “Applications of Least-Squares Regressions to Relationships
Containing Autocorrelated Error Terms,” 
Journal of the American Statistical Association,
vol. 44, 1949,
pp 32–61.
‡
Note that 
ˆ
ρ
=
ˆ
u
t
ˆ
u
t
−
1
/
ˆ
u
2
t
(why?). Although biased, 
ˆ
ρ
is a consistent estimator of the true 
ρ
.
guj75772_ch12.qxd 14/08/2008 10:41 AM Page 455

---

456
Part Two
Relaxing the Assumptions of the Classical Model
4. Since a priori it is not known if the 
ˆ
ρ
obtained from Eq. (3) is the best estimate
of 
ρ
, substitute the values of 
ˆ
β
∗
1
and 
ˆ
β
∗
2
obtained in step (3) in the original re-
gression Eq. (1) and obtain the new residuals, say, 
ˆ
u
∗
t
as
ˆ
u
∗
t
=
Y
t
− ˆ
β
∗
1
− ˆ
β
∗
2
X
t
(4)
which can be easily computed since 
Y
t
,
X
t
,
ˆ
β
∗
1
, and 
ˆ
β
∗
2
are all known.
5. Now estimate the following regression:
ˆ
u
∗
t
= ˆ
ρ
∗
ˆ
u
∗
t
−
1
+
w
t
(5)
which is similar to Eq. (3) and thus provides the second-round estimate of 
ρ
.
Since we do not know whether this second-round estimate of
ρ
is the best estimate
of the true
ρ
, we go into the third-round estimate, and so on. That is why the C–O
procedure is called an iterative procedure. But how long should we go on this
(merry-) go-round? The general recommendation is to stop carrying out iterations
when the successive estimates of
ρ
differ by a small amount, say, by less than 0.01 or
0.005. In our wages–productivity example, it took about three iterations before we
stopped.
a.
Use the Cochrane–Orcutt iterative procedure to estimate 
ρ
for the wages–
productivity regression, Eq. (12.5.2). How many iterations were involved before
you obtained the “final” estimate of 
ρ
?
b.
Using the final estimate of 
ρ
obtained in (
a
), estimate the wages–productivity re-
gression, dropping the first observation as well as retaining the first observation.
What difference you see in the results?
c.
Do you think that it is important to keep the first observation in transforming the
data to solve the autocorrelation problem?
12.9.
Estimating
ρ
: The Cochrane–Orcutt two-step procedure.
This is a shortened ver-
sion of the C–O iterative procedure. In step 1, we estimate 
ρ
from the first iteration,
that is from Eq. (3) in the preceding exercise, and in step 2 we use that estimate of
ρ
to run the generalized difference equation, as in Eq. (4) in the preceding exercise.
Sometimes in practice, this two-step method gives results quite similar to those
obtained from the more elaborate C–O iterative procedure.
Apply the C–O two-step method to the illustrative wages–productivity
regression (12.5.1) given in this chapter and compare your results with those ob-
tained from the iterative method. Pay special attention to the first observation in the
transformation.
12.10.
Estimating
ρ
: 
Durbin’s two-step method.
*
To explain this method, we can write the
generalized difference equation (12.9.5) equivalently as follows:
Y
t
=
β
1
(1
−
ρ
)
+
β
2
X
t
−
β
2
ρ
X
t
−
1
+
ρ
Y
t
−
1
+
ε
t
(1)
Durbin suggests the following two-step procedure to estimate 
ρ
. 
First,
treat Eq. (1)
as a multiple regression model, regressing 
Y
t
on 
X
t
,
X
t
−
1
, and 
Y
t
−
1
and treat the
estimated value of the regression coefficient of 
Y
t
−
1
(
= ˆ
ρ
) as an estimate of 
ρ
.
Second,
having obtained 
ˆ
ρ
, use it to estimate the parameters of generalized differ-
ence equation (12.9.5) or its equivalent, Eq. (12.9.6).
*
J. Durbin, “Estimation of Parameters in Time-Series Regression Models,” 
Journal of the Royal Statistical
Society,
series B, vol. 22, 1960, p. 139–153.
guj75772_ch12.qxd 14/08/2008 10:41 AM Page 456

---

Chapter 12
Autocorrelation: What Happens If the Error Terms Are Correlated?
457
a.
Apply the Durbin two-step method to the wages–productivity example discussed
in this chapter and compare your results with those obtained from the
Cochrane–Orcutt iterative procedure and the C–O two-step method. Comment
on the “quality” of your results.
b.
If you examine Eq. (1) above, you will observe that the coefficient of
X
t
−
1
(
= −
ρβ
2
) is equal to minus 1 times the product of the coefficient of
X
t
(
=
β
2
) and the coefficient of 
Y
t
−
1
(
=
ρ
). How would you test that coeffi-
cients obey the preceding restriction?
12.11. In measuring returns to scale in electricity supply, Nerlove used cross-sectional
data of 145 privately owned utilities in the United States for the period 1955 and re-
gressed the log of total cost on the logs of output, wage rate, price of capital, and
price of fuel. He found that the residuals estimated from this regression exhibited
“serial’’ correlation, as judged by the Durbin–Watson 
d
. To seek a remedy, he plot-
ted the estimated residuals on the log of output and obtained Figure 12.11.
a.
What does Figure 12.11 show?
b.
How can you get rid of “serial’’ correlation in the preceding situation?
12.12. The residuals from a regression when plotted against time gave the scattergram in
Figure 12.12. The encircled “extreme’’ residual is called an
outlier
. An outlier is an
observation whose value exceeds the values of other observations in the sample by a
log (output)
0
u
i
Regression residuals
×
×
× × ×
×
×
× × ×× × ×× × × ×
×
× × × ×
× × ×
× ×
×
×
×
×
××× ×
×
×
× × ×
× × ×
×
×