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

TABLE 12.8
Y
, Personal Consumption
Expenditure, Billions
of 1958 Dollars
X
, Time
ˆ
Y, 
Estimated 
Y
ˆ
u,
Residuals
281.4
1 (
=
1956)
261.4208
19.9791
288.1
2
276.6026
11.4973
290.0
3
291.7844
−
1.7844
307.3
4
306.9661
0.3338
316.1
5
322.1479
−
6.0479
322.5
6
337.3297
−
14.8297
338.4
7
352.5115
−
14.1115
353.3
8
367.6933
−
14.3933
373.7
9
382.8751
−
9.1751
397.7
10
398.0569
−
0.3569
418.1
11
413.2386
4.8613
430.1
12
428.4206
1.6795
452.7
13
443.6022
9.0977
469.1
14
458.7840
10.3159
476.9
15 (
=
1970)
473.9658
2.9341
Note:
Data for
ˆ
Y
obtained from the regression 
Y
t
=
β
0
+
β
1
X
t
+
u
t
. 
*
Henri Theil, 
Introduction to Econometrics,
Prentice Hall, Englewood Cliffs, NJ, 1978, pp. 307–308.
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Chapter 12
Autocorrelation: What Happens If the Error Terms Are Correlated?
463
12.33.
Monte Carlo experiment.
Refer to Tables 12.1 and 12.2. Using 
ε
t
and 
X
t
data given
there, generate a sample of 10 
Y
values from the model
Y
t
=
3
.
0
+
0
.
5
X
t
+
u
t
where 
u
t
=
0
.
9
u
t
−
1
+
ε
t
. Assume 
u
0
=
10.
a.
Estimate the equation and comment on your results.
b.
Now assume 
u
0
=
17. Repeat this exercise 10 times and comment on the results.
c.
Keep the preceding setup intact except now let 
ρ
=
0
.
3 instead of 
ρ
=
0
.
9 and
compare your results with those given in (
b
).
12.34. Using the data given in Table 12.9, estimate the model
Y
t
=
β
1
+
β
2
X
t
+
u
t
where
Y
=
inventories and
X
=
sales, both measured in billions of dollars.
a.
Estimate the preceding regression.
b.
From the estimated residuals find out if there is positive autocorrelation using (
i
) the
Durbin–Watson test and (
ii
) the large-sample normality test given in Eq. (12.6.13).
c.
If 
ρ
is positive, apply the Berenblutt–Webb test to test the hypothesis that 
ρ
=
1.
d.
If you suspect that the autoregressive error structure is of order 
p
, use the
Breusch–Godfrey test to verify this. How would you choose the order of 
p
?
e.
On the basis of the results of this test, how would you transform the data to
remove autocorrelation? Show all your calculations.
TABLE 12.9
Inventories and Sales in U.S. Manufacturing, 1950–1991 (millions of dollars)
Year
Sales*
Inventories
†
Ratio
Year
Sales*
Inventories
†
Ratio
1950
46,486
84,646
1.82
1971
224,619
369,374
1.57
1951
50,229
90,560
1.80
1972
236,698
391,212
1.63
1952
53,501
98,145
1.83
1973
242,686
405,073
1.65
1953
52,805
101,599
1.92
1974
239,847
390,950
1.65
1954
55,906
102,567
1.83
1975
250,394
382,510
1.54
1955
63,027
108,121
1.72
1976
242,002
378,762
1.57
1956
72,931
124,499
1.71
1977
251,708
379,706
1.50
1957
84,790
157,625
1.86
1978
269,843
399,970
1.44
1958
86,589
159,708
1.84
1979
289,973
424,843
1.44
1959
98,797
174,636
1.77
1980
299,766
430,518
1.43
1960
113,201
188,378
1.66
1981
319,558
443,622
1.37
1961
126,905
211,691
1.67
1982
324,984
449,083
1.38
1962
143,936
242,157
1.68
1983
335,991
463,563
1.35
1963
154,391
265,215
1.72
1984
350,715
481,633
1.35
1964
168,129
283,413
1.69
1985
330,875
428,108
1.38
1965
163,351
311,852
1.95
1986
326,227
423,082
1.29
1966
172,547
312,379
1.78
1987
334,616
408,226
1.24
1967
190,682
339,516
1.73
1988
359,081
439,821
1.18
1968
194,538
334,749
1.73
1989
394,615
479,106
1.17
1969
194,657
322,654
1.68
1990
411,663
509,902
1.21
1970
206,326
338,109
1.59
*
Annual data are averages of monthly, not seasonally adjusted, figures.
†
Seasonally adjusted, end of period figures beginning 1982 are not comparable with earlier period. 
Source:
Economic Report of the President,
1993, Table B-53, p. 408. 
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464
Part Two
Relaxing the Assumptions of the Classical Model
f.
Repeat the preceding steps using the following model:
ln
Y
t
=
β
1
+
β
2
ln
X
t
+
u
t
g.
How would you decide between the linear and log–linear specifications? Show
explicitly the test(s) you use.
12.35. Table 12.10 gives data on real rate of return on common stocks at time (RR
t
), out-
put growth in period (
t
+
1), (OG
t
+
1
), and inflation in period 
t
(Inf
t
), all in percent
form, for the U.S. economy for the period 1954–1981.
a.
Regress RR
t
on inflation.
b.
Regress RR
t
on OG
t
+
1
and Inf
t
c.
Comment on the two regression results in view of Eugene Fama’s observation
that “the negative simple correlation between real stock returns and inflation is
spurious because it is the result of two structural relationships: a positive relation
between current real stock returns and expected output growth [measured by
OG
t
+
1
], and a negative relationship between expected output growth and current
inflation.”
d.
Would you expect autocorrelation in either of the regressions in (
a
) and (
b
)?
Why or why not? If you do, take the appropriate corrective action and present the
revised results.
TABLE 12.10
Rate of Return,
Output Growth and
Inflation, United
States, 1954–1981
Observation
RR
Growth
Inflation
1954
53.0
6.7
−
0.4
1955
31.2
2.1
0.4
1956
3.7
1.8
2.9
1957
−
13.8
−
0.4
3.0
1958
41.7
6.0
1.7
1959
10.5
2.1
1.5
1960
−
1.3
2.6
1.8
1961
26.1
5.8
0.8
1962
−
10.5
4.0
1.8
1963
21.2
5.3
1.6
1964
15.5
6.0
1.0
1965
10.2
6.0
2.3
1966
−
13.3
2.7
3.2
1967
21.3
4.6
2.7
1968
6.8
2.8
4.3
1969
−
13.5
−
0.2
5.0
1970
−
0.4
3.4
4.4
1971
10.5
5.7
3.8
1972
15.4
5.8
3.6
1973
−
22.6
−
0.6
7.9
1974
−
37.3
−
1.2
10.8
1975
31.2
5.4
6.0
1976
19.1
5.5
4.7
1977
−
13.1
5.0
5.9
1978
−
1.3
2.8
7.9
1979
8.6
−
0.3
9.8
1980
−
22.2
2.6
10.2
1981
−
12.2
−
1.9
7.3 
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Chapter 12
Autocorrelation: What Happens If the Error Terms Are Correlated?
465
12.36.
The Durbin h statistic.
Consider the following model of wage determination:
Y
t
=
β
1
+
β
2
X
t
+
β
3
Y
t
−
1
+
u
t
where
Y
=
wages 
=
index of real compensation per hour
X
=
productivity 
=
index of output per hour.
a.
Using the data in Table 12.4, estimate the above model and interpret your results.
b.
Since the model contains lagged regressand as a regressor, the Durbin–Watson 
d
is not appropriate to find out if there is serial correlation in the data. For
such models, called autoregressive models, Durbin has developed the so-called
h 
statistic
to test for first-order autocorrelation, which is defined as:
*
h
= ˆ
ρ
n
1
−
n
[var (
ˆ
β
3
)]
where 
n
=
sample size, var (
ˆ
β
3
)
=
variance of the coefficient of the lagged 
Y
t
−
1
,
and 
ˆ
ρ
=
estimate of the first-order serial correlation.
For large sample size (technically, asymptotic), Durbin has shown that, under
the null hypothesis that 
ρ
=
0,
h
∼
N
(0, 1)
that is, the 
h
statistic follows the standard normal distribution. From the proper-
ties of the normal distribution we know that the probability of 
|
h
|
>
1
.
96 is
about 5 percent. Therefore, if in an application 
|
h
|
>
1
.
96, we can reject the null
hypothesis that 
ρ
=
0, that is, there is evidence of first-order autocorrelation in
the autoregressive model given above.
To apply the test, we proceed as follows: 
First,
estimate the above model by
OLS (don’t worry about any estimation problems at this stage). 
Second,
note
var (
ˆ
β
3
) in this model as well as the routinely computed 
d
statistic. 
Third,
using
the 
d
value, obtain 
ˆ
ρ
≈
(1
−
d
/
2). It is interesting to note that although we can-
not use the 
d
value to test for serial correlation in this model, we can use it to ob-
tain an estimate of 
ρ
. 
Fourth,
now compute the 
h
statistic. 
Fifth,
if the sample
size is reasonably large and if the computed 
|
h
|
exceeds 1.96, we can conclude
that there is evidence of first-order autocorrelation. Of course, you can use any
level of significance you want.
Apply the 
h 
test to the autoregressive wage determination model given earlier
and draw appropriate conclusions and compare your results with those given in
regression (12.5.1).
12.37.
Dummy variables and autocorrelation.
Refer to the savings–income regression dis-
cussed in Chapter 9. Using the data given in Table 9.2, and assuming an AR(1)
scheme, reestimate the savings–income regression, taking into account autocorre-
lation. Pay close attention to the transformation of the dummy variable. Compare
your results with those presented in Chapter 9.
12.38. Using the wages–productivity data given in Table 12.4, estimate model (12.9.8) and
compare your results with those given in regression (12.9.9). What conclusion(s)
do you draw?
*
J. Durbin, “Testing for Serial Correlation in Least-squares Regression When Some of the Regressors
Are Lagged Dependent Variables,” 
Econometrica,
vol. 38, pp. 410–421.
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