ECON 310 – Econometrics I Assignment 1

ECON 310 – Econometrics I
Assignment 1
Mirza Troki´c
[email protected]
Department of Economics, Bilkent University, Ankara, Turkey
Abstract
The assignment is to be handed in on Wednesday, April 6, 2015, at the beginning of
class. Late assignments will not be accepted.
1. Suppose that the population model determining y is
y = β0 + β1 x 1 + β2 x 2 + β3 x 3 + u
and this model satisfies assumptions MLR.1 through MLR.4. However, we estimate
the model that omits x3 . Let β˜0 , β˜1 and β˜2 be the OLS estimators from regression
of y on x1 and x2 . Show that the expected value of β˜2 is
n
P
r˜i2 xi3
E(β˜2 ) = β2 + β3 i=1
n
P
2
r˜i2
i=1
where the r˜2i are the OLS residueals from the regression of x2 on x1 . [Hint: this
question is not same as the question in your text book, but it is quite similar.]
2. Consider the multiple regression model containing three independent variables,
under Assumptions MLR.1 through MLR.4:
y = β0 + β1 x 1 + β2 x 2 + β3 x 3 + u
you are interested in estimating a linear combination of the parameters on x1 and
x3 , which is θ = 2β1 + 12 β3 .
(a) Show that θˆ1 = 2βˆ1 + 12 βˆ3 is an unbiased estimator of θ where βˆ1 and βˆ3 are
OLS estimates from above model.
ˆ in terms of V ar(βˆ1 ), V ar(βˆ3 ) and Cov(βˆ1 , βˆ3 ).
(b) Find V ar(θ)
March 23, 2015
3. Consider the simple linear model y = β0 +β1 x1 +u. Let βˆ0 and βˆ1 OLS estimates of
β0 and β1 respectively. Now, we transform the variables in the model as y ∗ = 2y +3
and x∗ = 3x + 2 and run the regression model y ∗ = β0 + β1 x∗ + u to obtain the
estimates β˜0 and β˜1 . Write there estimates from transformed model in terms of
original regression estimates βˆ0 and βˆ1 . (Hint: A similar question was solved in
recitation hour. You need to use same algebraic operations).
4. Let yi = β0 + ui for i = 1, 2, ..., n and V ar(ui ) = σ2 for all i, Cov(ui , uj ) = 0 for
all i 6= j. You want to estimate β0 . As estimator of β0 following estimators are
n
given βˆ0 = y¯ and β˜0 = y1 +y
where y1 is first observation for y and yn is the last
2
observation.
(a) Which of these two estimators are unbiased?
(b) Compare the variances of two estimator and conclude which has smaller variance.
5. Solve following computer based questions Chapter 3: C.4 C.6.
6. Solve Following computer based questions from Chapter 2: C.2, C.3.
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