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A11113
hw2.pdf
Econ 103: Homework 2
Manu Navjeevan
August 15, 2022
Single Linear Regression Theory Review
1. Recall that we define our parameters of interest β0 and β1 as the parameters governing the “line of
best fit” between Y and X:
β0,β1 = arg min b0,b1
E[(Y − b0 − b1X)2]. (1)
Once we define these parameters we define the regression error term � = Y − β0 − β1X which then generates the linear model
Y = β0 + β1X + �.
(a) Using the first order conditions for β0 and β1 (set the derivatives of the right hand side of (1)
with respect to b0 and b1 equal to zero at) show why E[�] = E[�X] = 0.
(b) Using the definition of β0 and β1 as line of best fit parameters, give an intuitive explanation for
why E[�] = 0.
Hypothesis Testing and Confidence Intervals
In the following questions, whenver running a hypothesis test, please state the null and alternative hypotheses,
show some work, and state the conclusion of the test.
1. In an estimated simple regression model based on n = 64, the estimated slope parameter, β̂1, is 0.310
and the standard error of β̂1 is 0.082.
(a) What is σ̂2β1 ? Recall σβ1 is the terms such that, approximately for large n,
√ n(β̂1 −β1) ∼ N(0,σβ1 ).
(b) Test the hypothesis that the slope is zero against the alternative that it is not at the 1% level of
significance (α = 0.01).
(c) Test the hypothesis that the slope is negative against the alternative that it is positive at the 1%
level of significance (α = 0.01).
(d) Test the hypothesis that the slope is positive against the alternative that it is negative at the 5%
level of significance. What is the p-value?
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(e) Generate a 99% confidence interval for β1. How can we use this interval to run the hypothesis
test in part (b)?
2. Consider a simple regression of log-income (income is measured thousands of dollars), Y , against years
of education, X. After collecting a sample of size n = 50 we estimate the following regression equation.
Ŷ = β̂0 + 0.0180X.
(a) Using the following information to solve for β̂0 as well as the estimated variance V̂ar(β̂0), which
is the square of the standard error.
• The standard error of β̂0 is 2.174
• The test statistic, t∗, associated with the hypothesis test for
H0 : β0 = 0 vs. H1 : β0 6= 0,
is equal to 1.257.
(b) Use the following information to solve for the standard error β̂1 as well as the estimated variance
V̂ar(β̂1), which is the square of the standard error.
• The test statistic, t∗, associated with the hypothesis test for
H0 : β1 = 0 vs. H1 : β1 6= 0,
is equal to 5.754
(c) Given that Y is a logged variable, Y = log(income), how do we interpret β̂1?
(d) Suppose that we are interested in the average value of log-income for someone with 16 years of
education. We want to use the model above to test the hypothesis that the average value of
log-income for someone with 16 years of education is less than or equal to 1.85. That is we want
to test
H0 : λ = β0 + 16β1 ≤ 1.85 vs. H1 : λ = β0 + 16β1 > 1.85.
Use the fact that Ĉov(β̂0, β̂1) = 2.84 to test this hypothesis at level α = 0.1.
(e) Use the above to generate a 90% confidence interval for λ.
3. (Challenge) Suppose we find that β̂1 > 0. If we reject the null hypothesis that β1 = 0 in favor of an
alternative hypothesis that β1 6= 0 at level α, up to what level can we be sure that would we reject the null hypothesis that β1 ≤ 0 against an alternative that β1 > 0? (Please give some explanation here as well as your answer, which will be some multiple of α).
R2 and Goodness of Fit
1. Consider the following estimated regression equation.
Ŷ = 6.83 + 0.869X.
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Write the estimated regression equation that would result if
(a) All values of X were divided by 20 before estimation.
(b) All values of Y were divided by 20 before estimation.
(c) All values of X and Y were divided by 20 before estimation.
2. Given the quantities in the questions below, calculate and interpret R2:
(a) ∑n i=1(Yi − Ȳ )
2 = 631.63 and ∑n i=1 �̂
2 i = 182.85.
(b) ∑n i=1 Y
2 i = 5930.94, Ȳ = 16.035, n = 20, and SSR = 666.72.
3. Suppose R2 = 0.7911, SST = 552.36, and n = 20. Find σ̂2� .
