486
Part Two
Relaxing the Assumptions of the Classical Model
13.6
Incorrect Specification of the Stochastic Error Term
A common problem facing a researcher is the specification of the error term
u
i
that enters
the regression model. Since the error term is not directly observable, there is no easy way
to determine the form in which it enters the model. To see this,
let us return to the models
given in Eqs. (13.2.8) and (13.2.9). For simplicity of exposition, we have assumed that
there is no intercept in the model. We further assume that
u
i
in Eq. (13.2.8) is such that ln
u
i
satisfies the usual OLS assumptions.
If we assume that Eq. (13.2.8) is the “correct” model but estimate Eq. (13.2.9), what are the
consequences? It is shown in Appendix 13.A, Section 13A.4, that if ln
u
i
∼
N
(0,
σ
2
), then
u
i
∼
log normal
e
σ
2
/
2
,
e
σ
2
e
σ
2
−
1
(13.6.1)
As a result,
E
(
ˆ
α
)
=
β
e
σ
2
/
2
(13.6.2)
where
e
is the base of the natural logarithm.
whereas, if we use
Y
i
instead of
Y
i
*,
we obtain
ˆ
Y
i
=
25.00
+
0.6000
X
*
i
(12.218)
(0.0681)
t
=
(2.0461)
(8.8118)
(13.5.12)
R
2
=
0.9066
As these results show, and according to the theory, the estimated coefficients remain
the same. The only effect of errors of measurement in the dependent variable is that
the estimated standard errors of the coefficients tend to be larger (see Eq. [13.5.5]),
which is clearly seen in Eq. (13.5.12). In passing, note that the regression coefficients in
Eqs. (13.5.11) and (13.5.12) are the same because the sample was generated to match
the assumptions of the measurement error model.
Errors of Measurement in
X.
We know that the true regression is Eq. (13.5.11).
Suppose
now that instead of using
X
*
i
, we use
X
i
. (
Note:
In reality
X
*
i
is rarely observable.) The
regression results are as follows:
ˆ
Y
*
i
=
25.992
+
0.5942
X
i
(11.0810)
(0.0617)
(13.5.13)
t
=
(2.3457)
(9.6270)
R
2
=
0.9205
These results are in accord with the theory—when there are measurement errors in the ex-
planatory variable(s), the estimated coefficients are biased. Fortunately, in this example
the bias is rather small—from Eq. (13.5.10) it is evident that the bias depends on
σ
2
w
/σ
2
X
*
,
and in generating the data it was assumed that
σ
2
w
=
36 and
σ
2
X
*
=
3667,
thus making the
bias factor rather small, about 0.98 percent (
=
36
/
3667).
We leave it to the reader to find out what happens when there are errors of measure-
ment in both
Y
and
X
, that is, if we regress
Y
i
on
X
i
rather than
Y
*
i
on
X
*
i
(see Exercise 13.23).
EXAMPLE 13.2
(
Continued
)
guj75772_ch13.qxd 19/08/2008 06:34 PM Page 486
Chapter 13
Econometric Modeling: Model Specification and Diagnostic Testing
487
As you can see,
ˆ
α
is a biased estimator, as its average value is not equal to the true
β
.
We will have more to say about the specification of the stochastic
error term in the chap-
ter on nonlinear-in-the-parameter regression models.
13.7
Nested versus Non-Nested Models
In carrying out specification testing, it is useful to distinguish between
nested
and
non-
nested models.
To distinguish between the two, consider the following models:
Model A:
Y
i
=
β
1
+
β
2
X
2
i
+
β
3
X
3
i
+
β
4
X
4
i
+
β
5
X
5
i
+
u
i
Model B:
Y
i
=
β
1
+
β
2
X
2
i
+
β
3
X
3
i
+
u
i
We say that Model B is nested in Model A because it is a special case of Model A: If we
estimate Model A and test the hypothesis that
β
4
=
β
5
=
0 and do not reject it on the basis
of, say, the
F
test,
32
Model A reduces to Model B. If we add variable
X
4
to Model B, then
Model A will reduce to Model B if
β
5
is zero; here we will use the
t
test
to test the hypoth-
esis that the coefficient of
X
5
is zero.
Without calling them such, the specification error tests that we have discussed previ-
ously and the restricted
F
test that we discussed in Chapter 8 are essentially tests of nested
hypothesis.
Now consider the following models:
Model C:
Y
i
=
α
1
+
α
2
X
2
i
+
α
3
X
3
i
+
u
i
Model D:
Y
i
=
β
1
+
β
2
Z
2
i
+
β
3
Z
3
i
+
v
i
where the
X
’s and
Z
’s are different variables. We say that Models C and D are
non-nested
because one cannot be derived as a special case of the other. In economics, as in other sci-
ences, more than one competing theory may explain a phenomenon. Thus, the monetarists
would emphasize the role of money
in explaining changes in GDP, whereas the Keynesians
may explain them by changes in government expenditure.
It may be noted here that one can allow Models C and D to contain regressors that are
common to both. For example,
X
3
could be included in Model D and
Z
2
could be included
in Model C. Even then these are non-nested models, because Model C does not contain
Z
3
and Model D does not contain
X
2
.
Even if the
same variables enter the model, the functional form may make two models
non-nested. For example, consider the model:
Model E:
Y
i
=
β
1
+
β
2
ln
Z
2
i
+
β
3
ln
Z
3
i
+
w
i
Models D and E are non-nested, as one cannot be derived as a special case of the other.
Since we already have looked at tests of nested models (
t
and
F
tests), in the following
section we discuss some of the tests of non-nested models, which earlier we called model
mis-specification errors.
32
More generally, one can use the likelihood ratio test, or the Wald test
or the Lagrange Multiplier
test, which were discussed briefly in Chapter 8.
guj75772_ch13.qxd 16/08/2008 03:24 PM Page 487
