Econ 7331
Econometrics 1
Spring 2010
Professor Chris Murray
McElhinney 211 D
Email: cmurray@mail.uh.edu
Problem Set 1
Due date: Tuesday, February 2, in class.
1. Consider the linear regression model Yi = α + βX i + u i . The OLS estimators of α and
β are as follows:
βˆ =
αˆ =
N ∑ X i Yi − ∑ X i ∑ Yi
(1)
N ∑ X i2 − (∑ X i )
2
∑ X ∑Y − ∑ X ∑ X Y
N ∑ X − (∑ X )
2
i
i
i
i i
2
2
i
.
(2)
i
Demonstrate that αˆ and βˆ can be more compactly written as:
βˆ =
∑x y
∑x
i
i
(1)’
αˆ = Y − βˆX
(2)’
2
i
where xi ≡ X i − X and yi ≡ Yi − Y .
Now demonstrate that if you regressed yi on xi, the estimator for β would be the same as
in equation (1)’.
This result is a rudimentary form of the Frisch-Waugh theorem, which we will prove
more generally later. In this context, the Frisch-Waugh theorem states that there are two
equivalent methods of obtaining β̂ :
•
Regress Yi on a constant and Xi. β̂ is then the regression coefficient on Xi.
•
Remove the mean from both Yi and Xi then regress yi on xi. β̂ is then the regression
coefficient on xi .
2. Prove the following:
E (αˆ ) = α
var(αˆ ) =
σ 2 ∑ X i2
N ∑ xi2
.
− Xσ 2
ˆ
ˆ
cov(α , β ) =
∑ xi2
3. Prove that if Yi is regressed only on a constant that the OLS estimator is equal to the
sample mean of the dependent variable. What is the R 2 from this regression?
4. Consider a regression without a constant term, i.e. Yi = βX i + u i . Yi and X i are not
in deviations from their respective means. Argue that the R2 from this regression may
be negative.