ECON 310 { Econometrics I Assignment 1 - Mirza...
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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 = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + u and this model satisfies assumptions MLR.1 through MLR.4. However, we estimate the model that omits x 3 . Let ˜ β 0 , ˜ β 1 and ˜ β 2 be the OLS estimators from regression of y on x 1 and x 2 . Show that the expected value of ˜ β 2 is E( ˜ β 2 )= β 2 + β 3 n ∑ i=1 ˜ r i2 x i3 n ∑ i=1 ˜ r 2 i2 where the ˜ r 2i are the OLS residueals from the regression of x 2 on x 1 . [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 = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + u you are interested in estimating a linear combination of the parameters on x 1 and x 3 , which is θ =2β 1 + 1 2 β 3 . (a) Show that ˆ θ 1 =2 ˆ β 1 + 1 2 ˆ β 3 is an unbiased estimator of θ where ˆ β 1 and ˆ β 3 are OLS estimates from above model. (b) Find V ar( ˆ θ) in terms of V ar( ˆ β 1 ), V ar( ˆ β 3 ) and Cov( ˆ β 1 , ˆ β 3 ). March 23, 2015
Transcript of ECON 310 { Econometrics I Assignment 1 - Mirza...
ECON 310 – Econometrics I
Assignment 1
Mirza [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 ofclass. Late assignments will not be accepted.
1. Suppose that the population model determining y is
y = β0 + β1x1 + β2x2 + β3x3 + u
and this model satisfies assumptions MLR.1 through MLR.4. However, we estimatethe model that omits x3. Let β0, β1 and β2 be the OLS estimators from regressionof y on x1 and x2. Show that the expected value of β2 is
E(β2) = β2 + β3
n∑i=1
ri2xi3
n∑i=1
r2i2
where the r2i are the OLS residueals from the regression of x2 on x1. [Hint: thisquestion 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 + β1x1 + β2x2 + β3x3 + u
you are interested in estimating a linear combination of the parameters on x1 andx3, which is θ = 2β1 +
12β3.
(a) Show that θ1 = 2β1 +12 β3 is an unbiased estimator of θ where β1 and β3 are
OLS estimates from above model.(b) Find V ar(θ) in terms of V ar(β1), V ar(β3) and Cov(β1, β3).
March 23, 2015
3. Consider the simple linear model y = β0+β1x1+u. Let β0 and β1 OLS estimates ofβ0 and β1 respectively. Now, we transform the variables in the model as y∗ = 2y+3and x∗ = 3x + 2 and run the regression model y∗ = β0 + β1x
∗ + u to obtain theestimates β0 and β1. Write there estimates from transformed model in terms oforiginal regression estimates β0 and β1. (Hint: A similar question was solved inrecitation 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 forall i 6= j. You want to estimate β0. As estimator of β0 following estimators aregiven β0 = y and β0 = y1+yn
2 where y1 is first observation for y and yn is the lastobservation.
(a) Which of these two estimators are unbiased?(b) Compare the variances of two estimator and conclude which has smaller vari-
ance.
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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