ECO-2400-5
Problem Set 2
Due : 2 March
Instructions:(a) Soft copy needs to be uploaded on Moodle before March 2, 11:59 PM.
(b) Hard Copy should be submitted on March 3 between 1:30-2:30 PM in WS-820.
(c) Write name & Roll No on the top. You may submit hard copy through your friend.
Q1. The following simple linear regression model has been estimated with data (Yi , Xi ), i =
1, ..., 70 and the fit is the following:
a
a
Ŷi = 3.25 − 2.5Xi
(1.22) (0.6)
We know that both Yi and Xi are positive for all i = 1, ..., n, and that the sample
variance of X is s2X = 1.23. What will be the sign of the slope estimator in a regression
model without a constant using the same data? Justify the answer.(10)
Q2. Consider the following regression:lnYi = −0.6702 + 0.7256lnXi
se = (0.5624)
(0.1015)
t = (−1.1915)
(7.1440)
P = (0.2676)
(0.0001)
r2 = 0.8644. Where Xi denotes income and Yi denotes expenditure. Answer the following:
(i) Interpret slope coefficient.(5)
(ii) Is it elastic?(5)
(iii) Interpret intercept coefficient.(5)
(iv) Interpret r2 .(5)
Q3. Let βˆ0 and βˆ1 be the OLS intercept and slope estimators, respectively, and let ū
be the sample average of the errors (not the residuals!).
P
(xi − x̄)
(i) Show that βˆ1 can be written as βˆ1 = β1 + ni=1 wi ui where wi =
.(10)
(xi − x̄)2
P
(ii) use part (i) along with ni=1 wi = 0 to show that βˆ1 and ū are uncorrelated. (10)
(iii) Show that βˆ0 can be written as βˆ0 = β0 + ū − (βˆ1 − β1 )x̄. (10)
P
σ 2 ni=1 x2i
ˆ
(iv) Use parts (ii) and (iii) to show that V (β0 ) = Pn
.(10)
n i=1 (xi − x̄)2
Q4. Consider the standard simple regression model y = β0 + β1 x + ui under the GaussMarkov Assumptions SLR.1 through SLR.5. The usual OLS estimators βˆ0 and βˆ1 are
ECO-2400-5
Problem Set 2
Due : 2 March
unbiased for their respective population parameters. Let β˜1 be the estimator of β1 obtained by assuming the intercept is zero.
(i) Find E(β˜1 ) in terms of the xi , β0 and β1 . Verify that β˜1 is unbiased for β1 when
the population intercept (β0 ) is zero. Are there other cases where β˜1 is unbiased? (5)
(ii) Find the variance of β˜1 .(10)
(iii) Show that V (β˜1 ) ≤ V (βˆ1 )(10)
(iv) Comment on the tradeoff between bias and variance when choosing between β˜1 and
βˆ1 .(5)