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

Part Two
Relaxing the Assumptions of the Classical Model
where 
Y
=
labor’s share and 
t
=
time. Based on annual data for 1949–1964, the
following results were obtained for the primary metal industry:
Model A:
ˆ
Y
t
=
0.4529
−
0.0041
t
R
2
=
0.5284
d
=
0.8252
(
−
3
.
9608)
Model B:
ˆ
Y
t
=
0.4786
−
0.0127
t
+
0.0005
t
2
(
−
3.2724)
(2.7777)
R
2
=
0
.
6629
d
=
1
.
82
where the figures in the parentheses are 
t
ratios.
a.
Is there serial correlation in model A? In model B?
b.
What accounts for the serial correlation?
c.
How would you distinguish between “pure’’ autocorrelation and specification
bias?
12.4.
Detecting autocorrelation: von Neumann ratio test.
*
Assuming that the residual 
ˆ
u
t
are random drawings from normal distribution, von Neumann has shown that for
large n
, the ratio
δ
2
s
2
=
(
ˆ
u
i
− ˆ
u
i
−
1
)
2
/
(
n
−
1)
(
ˆ
u
i
− ¯ˆ
u
)
2
/
n
N ote
:
¯ˆ
u
=
0 in OLS
called the 
von Neumann ratio,
is approximately normally distributed with mean
E
δ
2
s
2
=
2
n
n
−
1
and variance
var
δ
2
s
2
=
4
n
2
n
−
2
(
n
+
1)(
n
−
1)
3
a.
If 
n
is sufficiently large, how would you use the von Neumann ratio to test for
autocorrelation?
b.
What is the relationship between the Durbin–Watson 
d
and the von Neumann
ratio?
c.
The 
d
statistic lies between 0 and 4. What are the corresponding limits for the
von Neumann ratio?
d.
Since the ratio depends on the assumption that the 
ˆ
u
’s are random drawings from
normal distribution, how valid is this assumption for the OLS residuals?
e.
Suppose in an application the ratio was found to be 2.88 with 100 observations.
Test the hypothesis that there is no serial correlation in the data.
Note:
B. I. Hart has tabulated the critical values of the von Neumann ratio for
sample sizes of up to 60 observations.
†
12.5. In a sequence of 17 residuals, 11 positive and 6 negative, the number of runs was 3.
Is there evidence of autocorrelation? Would the answer change if there were 14 runs?
*
J. von Neumann, “Distribution of the Ratio of the Mean Square Successive Difference to the
Variance,’’ 
Annals of Mathematical Statistics,
vol. 12, 1941, pp. 367–395.
†
The table may be found in Johnston, op. cit., 3d ed., p. 559. 
guj75772_ch12.qxd 14/08/2008 10:41 AM Page 454

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Chapter 12
Autocorrelation: What Happens If the Error Terms Are Correlated?
455
12.6.
Theil–Nagar 
ρ
estimate based on d statistic.
Theil and Nagar have suggested that,
in small samples, instead of estimating 
ρ
as (1
−
d
/
2), it should be estimated as
ˆ
ρ
=
n
2
(1
−
d
/
2)
+
k
2
n
2
−
k
2
where 
n
=
total number of observations, 
d
=
Durbin–Watson 
d
, and 
k
=
number
of coefficients (including the intercept) to be estimated.
Show that for large 
n
, this estimate of 
ρ
is equal to the one obtained by the sim-
pler formula (1
−
d
/
2).
12.7.
Estimating 
ρ
: The Hildreth–Lu scanning or search procedure.
*
Since in the first-
order autoregressive scheme
u
t
=
ρ
u
t
−
1
+
ε
t
ρ
is expected to lie between 
−
1 and 
+
1, Hildreth and Lu suggest a systematic
“scanning’’ or search procedure to locate it. They recommend selecting 
ρ
between
−
1 and 
+
1 using, say, 0.1 unit intervals and transforming the data by the generalized
difference equation (12.6.5). Thus, one may choose 
ρ
from 
−
0.9,
−
0.8,
. . .
, 0.8,
0.9. For each chosen 
ρ
we run the generalized difference equation and obtain the as-
sociated RSS: 
ˆ
u
2
t
. Hildreth and Lu suggest choosing that 
ρ
which minimizes the
RSS (hence maximizing the 
R
2
). If further refinement is needed, they suggest using
smaller unit intervals, say, 0.01 units such as
−
0.99,
−
0.98,
. . .
, 0.90, 0.91, and so on.
a.
What are the advantages of the Hildreth–Lu procedure?
b.
How does one know that the 
ρ
value ultimately chosen to transform the data will,
in fact, guarantee minimum 
ˆ
u
2
t
?
12.8.
Estimating 
ρ
: The Cochrane–Orcutt (C–O) iterative procedure.
†
As an illustration
of this procedure, consider the two-variable model:
Y
t
=
β
1
+
β
2
X
t
+
u
t
(1)
and the AR(1) scheme
u
t
=
ρ
u
t
−
1
+
ε
t
,
−
1
< ρ <
1
(2)
Cochrane and Orcutt then recommend the following steps to estimate 
ρ
.
1. Estimate Eq. (1) by the usual OLS routine and obtain the residuals, 
ˆ
u
t
.
Incidentally, note that you can have more than one 
X
variable in the model.
2. Using the residuals obtained in step 1, run the following regression:
ˆ
u
t
= ˆ
ρ
ˆ
u
t
−
1
+
v
t