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Part One
Single-Equation Regression Models
In Section 9.12, interaction terms were created between the education variables
(
DE
2
,
DE
3
, and
DE
4
) and the gender variable (
D
sex
). What happens if we create in-
teraction terms between the education dummies and the permanent worker dummy
variable (
DPT
)?
a.
Estimate the model predicting ln WI containing age, gender, the education
dummy variables, and three new interaction terms:
DE
2
×
DPT
,
DE
3
×
DPT
, and
DE
4
×
DPT
. Does there appear to be a significant interaction effect among the
new terms?
b.
Is there a significant difference between workers with an education level up to pri-
mary and those without a primary education? Assess this with respect to both the
education dummy variable and the interaction term and explain the results. What
about the difference between workers with a secondary level of education and
those without a primary level of education? What about the difference between
those with an education level beyond secondary, compared to those without a pri-
mary level of education?
c.
Now assess the results of deleting the education dummies from the model. Do the
interaction terms change in significance?
Appendix
9A
Semilogarithmic Regression with Dummy Regressor
In Section 9.10 we noted that in models of the type
ln
Y
i
=
β
1
+
β
2
D
i
(1)
the relative change in
Y
(i.e., semielasticity), with respect to the dummy regressor taking values of 1
or 0, can be obtained as (antilog of estimated
β
2
)
−
1 times 100, that is, as
(
e
ˆ
β
2
−
1)
×
100
(2)
The proof is as follows: Since ln and exp (
=
e
) are inverse functions, we can write Eq. (1) as:
ln
Y
i
=
β
1
+
ln(
e
β
2
D
i
)
(3)
Now when
D
=
0,
e
β
2
D
i
=
1 and when
D
=
1,
e
β
2
D
i
=
e
β
2
. Therefore, in going from state 0 to state
1, ln
Y
i
changes by (
e
β
2
−
1). But a change in the log of a variable is a relative change, which after
multiplication by 100 becomes a percentage change. Hence the percentage change is
(
e
β
2
−
1)
×
100, as claimed. (
Note:
ln
e
e
=
1, that is, the log of
e
to base
e
is 1, just as the log of 10
to base 10 is 1. Recall that log to base
e
is called the
natural
log and that log to base 10 is called the
common
log.)
guj75772_ch09.qxd 12/08/2008 04:19 PM Page 314
Part
In Part 1 we considered at length the classical normal linear regression model and showed
how it can be used to handle the twin problems of statistical inference, namely, estimation
and hypothesis testing, as well as the problem of prediction. But recall that this model is
based on several simplifying assumptions, which are as follows.
Assumption 1. The regression model is linear in the parameters.
Assumption 2. The values of the regressors, the
X
’s, are fixed, or
X
values are
independent of the error term. Here, this means we require zero
covariance between
u
i
and each
X
variable.
Assumption 3. For given
X
’s, the mean value of disturbance
u
i
is zero.
Assumption 4. For given
X
’s, the variance of
u
i
is constant or homoscedastic.
Assumption 5. For given
X
’s, there is no autocorrelation, or serial correlation,
between the disturbances.
Assumption 6. The number of observations
n
must be greater than the number of
parameters to be estimated.
Assumption 7. There must be sufficient variation in the values of the
X
variables.
We are also including the following 3 assumptions in this part of the text:
Assumption 8. There is no exact collinearity between the
X
variables.
Assumption 9. The model is correctly specified, so there is no specification bias.
Assumption 10. The stochastic (disturbance) term
u
i
is normally distributed.
Before proceeding further, let us note that most textbooks list fewer than 10 assumptions.
For example, assumptions 6 and 7 are taken for granted rather than spelled out explicitly. We
decided to state them explicitly because distinguishing between the assumptions required
for ordinary least squares (OLS) to have desirable statistical properties (such as BLUE) and
the conditions required for OLS to be useful seems sensible. For example, OLS estimators
are BLUE (best linear unbiased estimators) even if assumption 7 is not satisfied. But in that
case the standard errors of the OLS estimators will be large relative to their coefficients
2
Relaxing
the Assumptions of
the Classical Model
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316
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