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

ρ
may be appropriate:
a.
First-difference regression.
b.
Moving average regression.
c.
Theil–Nagar transform.
d.
Cochrane and Orcutt iterative procedure.
e.
Hildreth–Lu scanning procedure.
f.
Durbin two-step procedure.
12.17. Consider the model:
Y
t
=
β
1
+
β
2
X
t
+
u
t
where
u
t
=
ρ
1
u
t
−
1
+
ρ
2
u
t
−
2
+
ε
t
*
Source: Prices and Earnings in 1951–1969: An Econometric Assessment,
Department of Employment,
Her Majesty’s Stationery Office, 1971, Table C, p. 37, Eq. 63.
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Chapter 12
Autocorrelation: What Happens If the Error Terms Are Correlated?
459
that is, the error term follows an AR(2) scheme and 
ε
t
is a white noise error term.
Outline the steps you would take to estimate the model taking into account the
second-order autoregression.
12.18. Including the correction factor 
C
, the formula for 
ˆ
β
GLS
2
given in Eq. (12.3.1) is
ˆ
β
GLS
2
=
(1
−
ρ
2
)
x
1
y
1
+
n
t
=
2
(
x
t
−
ρ
x
t
−
1
)(
y
t
−
ρ
y
t
−
1
)
(1
−
ρ
2
)
x
2
1
+
n
t
=
2
(
x
t
−
ρ
x
t
−
1
)
2
Given this formula and Eq. (12.3.1), find the expression for the correction factor 
C
.
12.19. Show that estimating Eq. (12.9.5) is equivalent to estimating the GLS discussed in
Section 12.3, excluding the first observation on 
Y
and 
X
.
12.20. For regression (12.9.9), the estimated residuals have the following signs, which for
ease of exposition are bracketed.
(
++++
)(
−
)(
+++++++
)(
−
)(
++++
)(
−−
)(
+
)(
−−
)(
+
)(
−−
)(
++
)(
−
)
(
+
)(
−−−−−−−−−
)(
+
)
On the basis of the runs test, do you reject the null hypothesis that there is no auto-
correlation in the residuals?
*
12.21.
Testing for higher-order serial correlation.
Suppose we have time series data on a
quarterly basis. In regression models involving quarterly data, instead of using the
AR(1) scheme given in Eq. (12.2.1), it may be more appropriate to assume an
AR(4) scheme as follows:
u
t
=
ρ
4
u
t
−
4
+
ε
t
that is, to assume that the current disturbance term is correlated with that of the
same quarter in the previous year rather than that of the preceding quarter.
To test the hypothesis that 
ρ
4
=
0, Wallis
†
suggests the following modified
Durbin–Watson 
d
test:
d
4
=
n
t
=
5
(
ˆ
u
t
− ˆ
u
t
−
4
)
2
n
t
=
1
ˆ
u
2
t
The testing procedure follows the usual 
d
test routine discussed in the text. Wallis
has prepared 
d
4
tables, which may be found in his original article.
Suppose now we have monthly data. Could the Durbin–Watson test be
generalized to take into account such data? If so, write down the appropriate 
d
12
formula.
12.22. Suppose you estimate the following regression:

ln output
t
=
β
1
+
β
2

ln
L
t
+
β
3

ln
K
t
+
u
t
where 
Y
is output, 
L
is labor input, 
K
is capital input, and 

is the first-difference
operator. How would you interpret 
β
1
in this model? Could it be regarded as an es-
timate of technological change? Justify your answer.
*Optional.
†
Kenneth Wallis, “Testing for Fourth Order Autocorrelation in Quarterly Regression Equations,’’ 
Economet-
rica,
vol. 40, 1972, pp. 617–636. Tables of 
d
4
can also be found in J. Johnston, op. cit., 3d ed., p. 558.
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460