Econometrics question

HW1 (Due Next Wednesday May 17, at the beginning of class)

May 9, 2017

0.1 HW1

Please explain and show all workings.

Q1. Chapter 2 from the text book, fifth Edition :

a) Exercise 2.2.

b) Exercise 2.5.

c) Exercise 2.6.

d) Exercise 2.9 (do only part a and b)

e) Exercise 2.15 ( do only part a and b). This is a data question, please download the data set from

blackboard and conduct the empirical analysis in STATA.

Q2. Consider the simple linear regression yi = β1 + β2Xi + ui for i = 1, 2, . . . ,n, where β1 and β2

are population parameters and the ui are IID (0,σ2).

(a) Derive OLS estimates (β̂1, β̂2) for the parameters (β1, β2), respectively. Note that this is done by

minimizing the residuals sum of squares with respect to β̂1and β̂2.

(b) Show that β̂1and β̂2 are unbiased estimators for the population parameters (β1, β2), respectively.

(c) derive the variances of β̂1and β̂2.

(d) Suppose β̂ ∗ 2 = β̂2 +

1 n , where n is the sample size. Is β̂

∗ 2 unbiased, show it? Note that β̂2 is derived

in part c.

(e) Is β̂ ∗ 2 asymptotically unbaised, show it?

1

Q3. Consider the simple linear regression yi = β1 + β2Xi + ui for i = 1, 2, . . . ,n. The ui are IID

(0,σ2) and verify the following numerical properties of OLS

a. Σni=1ûi = 0

b. Σni=1ûiXi = 0

c. Σni=1ûiŶi = 0

d. Σni=1Ŷi = Σ n i=1Yi

Q4. Suppose β2 = 0 ⇒yi = β1 + ui,i = 1, 2, . . . ,n. The ui are IID (0,σ2).

a. derive the estimate of β1 and called it β̂1

b. derive the variance of β̂1

2