Econ 103
UCLA, Fall 2010
Problem Set 3 Solutions
by Anthony Keats and Sarolta Laczó
Part 1: True or False and explain briefly why.
1. The assumption that
E
(
u
i

X
i
=
x
i
) = 0
says that the expected value of
u
i
changes
depending on the value of
X
i
.
FALSE
. This assumption says that the expected value of
u
i
does not change depending
on the value of
X
i
, that is,
X
i
and
u
i
are uncorrelated. This is the first assumption
of OLS. If
E
(
u
i

X
i
=
x
i
)
6
= 0
, then
X
i
and
u
i
are correlated and the coefficient
b
β
associated with this regressor will be inconsistent.
2. If
Cov
(
X
i
, u
i
)
>
0
, then the OLS estimator
ˆ
β
1
will tend to be higher than
β
1
.
TRUE
. To see this, refer to the formula for omitted variable bias:
b
β
1
p
→
β
1
+
ρ
X,u
σ
u
σ
X
and to the forumula for the correlation coefficient
ρ
:
ρ
X,u
=
Cov
(
X
i
, u
i
)
σ
X
σ
u
Substituting into the first equation we have:
b
β
1
p
→
β
1
+
Cov
(
X
i
, u
i
)
σ
2
X
Since the term in the denominator is just the variance of
X
i
, it is always positive. Thus,
if
Cov
(
X
i
, u
i
)
>
0
the direction of the bias will be positive as well.
3. Consider an omitted variable
V
i
that is negatively correlated with
X
i
.
Also suppose
that
V
i
positively affects
Y
i
. Then the OLS estimator
ˆ
β
1
is negatively biased.
TRUE
. Since
V
i
is omitted from the regression, it enters the error term
u
i
. The formula
given above (question 2) for the omitted variable bias tells us that when
ρ
X,u
<
0
then
the OLS estimator
b
β
1
will be negatively biased.
4. Suppose you run a regression and obtain the estimate
ˆ
β
1
= 3
.
4
. STATA tells you that
the
t
statistic for the null hypothesis that
β
1
= 0
is equal to 1.7.
This implies that
SE
(
ˆ
β
1
)
is equal to 2.
TRUE
. The equation for the
t
statistic is:
t
=
b
β
1

β
1
,
0
SE
(
b
β
1
)
1
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Econ 103
UCLA, Fall 2010
where
β
1
,
0
is the value under the null. In this case the null hypothesis is
β
1
,
0
= 0
so the
equation becomes:
t
=
b
β
1
SE
(
b
β
1
)
and rearranging this gives us:
SE
(
b
β
1
) =
b
β
1
t
=
3
.
4
1
.
7
= 2
5. In the regression model
Y
i
=
β
0
+
β
1
Female
i
+
β
2
Education
i
+
u
i
,
β
1
represents the
intercept for females.
FALSE
.
β
0
+
β
1
represents the intercept for females. The coefficient
β
1
on the dummy
Female
i
represents the
difference
between the intercept for males and the intercept for
females.
6. In the regression model
Y
i
=
β
0
+
β
1
Female
i
+
β
2
Education
i
+
β
3
Education
i
·
Female
i
+
u
i
,
β
2
+
β
3
represents the return to education for females.
TRUE
.
β
2
+
β
3
represents the change in
Y
i
associated with a marginal change in
Education for females.
β
1
represents the associated change in
Y
i
given a marginal change
in Education for men. As with the intercept coefficients in the previous question,
β
3
represents the difference between men and women in the return to education
7. Under perfect multicollinearity, the OLS estimator cannot be computed.
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 Econometrics, Linear Regression, Regression Analysis, Null hypothesis, Errors and residuals in statistics

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