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

Monte Carlo experiment.
Suppose in the two-variable model we “know’’ that
the true 
β
1
=
1 and 
β
2
=
0
.
8. Therefore, the stochastic PRF is
Y
t
=
1
.
0
+
0
.
8
X
t
+
u
t
(12.4.3)
424
Part Two
Relaxing the Assumptions of the Classical Model
15
See S. M. Goldfeld and R. E. Quandt, 
Nonlinear Methods in Econometrics
, North Holland Publishing
Company, Amsterdam, 1972, p. 183. In passing, note that if the errors are positively autocorrelated,
the 
R
2
value tends to have an upward bias, that is, it tends to be larger than the 
R
2
in the absence of
such correlation.
16
For a formal proof, see Kmenta, op. cit., p. 281.
guj75772_ch12.qxd 14/08/2008 10:40 AM Page 424

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Chapter 12
Autocorrelation: What Happens If the Error Terms Are Correlated?
425
Hence,
E
(
Y
t
|
X
t
)
=
1
.
0
+
0
.
8
X
t
(12.4.4)
which gives the true population regression line. Let us assume that 
u
t
are generated by the
first-order autoregressive scheme as follows:
u
t
=
0
.
7
u
t
−
1
+
ε
t
(12.4.5)
where 
ε
t
satisfy all the OLS assumptions. We assume further for convenience that the 
ε
t
are
normally distributed with zero mean and unit (
=
1) variance. Equation (12.4.5) postulates
that the successive disturbances are positively correlated, with a coefficient of autocorrela-
tion of 
+
0.7, a rather high degree of dependence.
Now, using a table of random normal numbers with zero mean and unit variance, we
generated 10 random numbers shown in Table 12.1 and then by the scheme (12.4.5) we
generated 
u
t
. To start off the scheme, we need to specify the initial value of 
u
, say, 
u
0
=
5.
Plotting the 
u
t
generated in Table 12.1, we obtain Figure 12.5, which shows that initially
each successive 
u
t
is higher than its previous value and subsequently it is generally smaller
than its previous value showing, in general, a positive autocorrelation.
Now suppose the values of 
X
are fixed at 1, 2, 3, . . . , 10. Then, given these
X
’s, we can
generate a sample of 10 
Y
values from Eq. (12.4.3) and the values of 
u
t
given in Table 12.1.
The details are given in Table 12.2. Using the data of Table 12.2, if we regress 
Y
on 
X
, we
obtain the following (sample) regression:
ˆ
Y
t
=
6.5452
+
0.3051
X
t
(0.6153)
(0.0992)
t
=
(10.6366)
(3.0763)