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? 1 Page 2 (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 ...

1. 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

2. 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?

3. 1
Page 2
(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 V
̂ ar(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
V
̂ ar(β̂1), which is the square of the standard error.

4. • 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).

5. R2 and Goodness of Fit
1. Consider the following estimated regression equation.
Ŷ = 6.83 + 0.869X.
Page 3
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

6. 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� .
Multimedia Instructional Materials
Multimedia
Tool
Description
Cost
Age Level
Advantages
(use bullets)
Disadvantages
(use bullets)
Application in the Classroom
1.

7. ·
·
2.
·
·
3.
·
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4.
5.
·
·

8. Rationale
· In-text citations are also embedded within the MATRIX.
· Keep in mind that each paragraph should have an in-text
citation.
· The reference list should appear at the end of the paper, on a
separate page.
References
© 2017. Grand Canyon University. All Rights Reserved.
Staying current on technology is an essential aspect of being an
educator. Today’s students are digital natives, and they often
respond better to media than to traditional methods of teaching.
Having a strong technology repertoire is important.
Create a matrix detailing a variety of multimedia, technology,
games, apps, and other technological tools for teaching reading

9. and writing to struggling readers and writers. Include five
tools/media/apps and address the following, in 100-200 words
per tool:
App/technology tool description, app/technology location
(online, offline through software, through a game console, etc.),
and the cost.
Age level or academic level for which the technology is
appropriate.
Advantages of using the technology.
Drawbacks to using the technology.
Rationalize why struggling students may benefit from the
app/technology tool.
Additionally, write a 250-500 word overview of the contents of
the matrix, describing how you will implement technology in
your ELA classroom. Justify which of these technologies you
think will be most beneficial and describe how you might
convince an administrator to help you acquire the technology.
Support the matrix and summary with 3-5 resources.
While APA format is not required for the body of this
assignment, solid academic writing is expected, and in-text
citations and references should be presented using APA
documentation guidelines, which can be found in the APA Style
Guide, located in the Student Success Center
Econ 103: Homework 1
Manu Navjeevan
July 31, 2022

10. Econ 41 Review
1. Discrete Random Variables. Suppose that we are interested in
the number of cups of coffee drank by a
(randomly selected) student at UCLA. This quantity can be
represented as a random variable Y with
probability mass function:
pY (a) =
1
4
if a ∈ {0, 1, 2}
1
8
if a = 3
3
32
if a = 4
c if a = 5
0 otherwise
,

11. where c is an unknown constant.
(a) Explain why the number of cups of coffee drank in a day by
a randomly selected student at UCLA
is a random variable.
(b) What is the relevant outcome space of the random variable
Y ?
(c) Explain what the distribution of this random variable
represents. In other words distribution of
Y assigns a probability to any subset of the outcome space. How
do we interpret this probability?
(d) Solve for c. (Hint: Recall that PY (OY ) = 1 so that
∑
a∈ OY pY (a) must equal one).
(e) What is the probability that a randomly selected student at
UCLA drinks at least 3 cups of coffee
a day, PY (Y ≥ 3)?
(f) What is the expected number of cups of coffee drank per day
for a randomly selected student at
UCLA?
2. Continuous Random Variables. Suppose that we are
interested in the income of a randomly selected
Angeleno. The distribution of incomes (in tens of thousands of
dollars) for residents of Los Angeles

12. can be described as a random variable, X, with the following
pdf.
fX(a) =
where c is an unkown constant.
1
Page 2
(a) What is the outcome space of X, OX?
(b) Using the relationship
PX(l ≤ X ≤ m) =
∫ m
l
fX(a) da,
explain why the pdf must always be weakly positive, fX(a) ≥ 0,
for any a ∈ R.
(c) Because PX(OX) = 1 we must have that
∫ 10
0
fX(a) da = 1. Using this fact, solve for c.
(d) What is the expected value of X, E[X]?

13. (e) What is the variance of X, Var(X)?
3. Variance and Covariance. Let Y be a random variable
representing income (in tens of thousands of
dollars) and X be a random variable representing years of
education. Suppose that the marginal
distribution of X is described by its probability mass function
pX(x) =
0.05 if x ∈ {1, 2, . . . , 12}
0.09 if x ∈ {13, 14, 15, 16}
0.04 if x ∈ {17}
0 otherwise
.
The marginal distribution of Y is described by its probability
density function
fY (y) =
(a) What is the expectation of Y , E[Y ]? What is its variance,
Var(Y )?

14. (b) What is the expectation of X, E[X]? What is its variance,
Var(X)?
(c) Using E[Y X] = 60 compute the covariance between Y and
X, Cov(X,Y ).
(d) Calculate the correlation coefficient between X and Y .
ρY X =
Cov(X,Y )
σXσY
.
(e) What does this covariance tell us about the relationship
between education levels and income? Is
there a positive or negative association?
(f) Should we interpret this result as a causal relationship
between education and income? What are
some reasons we may want to refrain from this interpretation?
(g) (Challenge) A common inequality used in econometrics is
the Cauchy-Schwarz inequality. It
states that, for any random variables X and Y , and any
functions g(·) and h(·),
∣ ∣ E[g(X)h(Y )]∣ ∣ ≤ √E[g2(X)]√E[h2(Y )].
Use this inequality to show why the correlation coefficient is
bounded between negative one and

15. Page 3
one, −1 ≤ ρXY ≤ 1. (Hint: Try g(x) = x−µX and h(y) = y −µY ).
Introduction to Single Linear Regression
1. Useful Equalities. Recall that in deriving the form of β̂1 we
used the following equalities
1
n
n∑
i=1
(Yi − Ȳ )(Xi − X
̄ ) =
1
n
n∑
i=1
YiXi − Ȳ X
̄ and
1
n
n∑
i=1
(Xi − X
̄ )2 =
1
n

16. n∑
i=1
X2i − (X
̄ )
2.
Show either one of these equalities (only have to show one or
the other).
2. Assumptions for Inference. Suppose we are interested in the
relationship between the size of the average
American’s social circle, X, and whether or not they are
unemployed, Y . To investigate this relationship
we want to estimate the following regression equation1
Y = β0 + β1X + �, E[�] = E[�X] = 0.
To estimate the regression coefficient parameters we collect a
sample of size n, {Yi,Xi}ni=1. Recall
that for valid asymptotic inference on our estimates β̂0 and β̂1
we require the following assumptions:
Random Sampling, Homoskedasticity, and Rank condition.
• Random Sampling: Assume that {Y,Xi} are independently and
identically distributed from the
population of interest, (Yi,Xi)
i.i.d∼ (Y,X).
• Homoskedasticity: Assume that Var(�|X = x) = σ2� for all
possible values of x.

17. • Rank Condition: There must be at least two distinct values of
X that appear in the population.
(a) Suppose we collect our sample by only randomly surveying
people on UCLA campus. Which
assumption would be violated?
(b) Suppose we collect our sample and find that everyone
appears to have exactly one friend. Which
assumption would be violated? Why is this a problem when
computing the line of best fit through
our sample?
(c) Suppose random sampling, homoskedasticity, and the rank
condition are all satisfied, but n = 10.
Why might inferences based on the approximation
β̂1 −β1
σ̂β1/
√
n
∼ N(0, 1)
not be valid?
3. Hypothesis Testing. Suppose now that we are interested in
investigating the relationship between the
size of someone’s social circle, X, and their income (in tens of
thousands of dollars), Y . We want to
estimate the following linear regression model

18. Y = β0 + β1X + �, E[�] = E[�X] = 0.
1Recall that this regression specification corresponds to finding
the line of best fit parameters β0,β1 = arg minb0,b1 E[(Y −
b0 − b1X)2] and defining � = Y − β0 − β1X
Page 4
To do so we collect a random sample of size n = 64,
{Yi,Xi}64i=1 and find that
1
n
∑n
i=1(Xi−X
̄ )
2 = 100,
1
n
∑n
i=1(Yi − Ȳ )(Xi − X
̄ ) = 225, Ȳ = 5.5, and X
̄ = 1.5.
(a) Using this information find and interpret β̂1 and β̂0.
(b) After finding β̂1 and β̂1 describe how you would construct
the estimated residuals �̂i.
(c) We find that 1
n
∑n
i=1 �̂

19. 2
i = 36. Use this and the result that, for n large,
β̂1 −β1
σ̂β1/
√
n
∼ N(0, 1),
to compute the (approximate) probability that, if the true value
was given β1 = 0, we would see
a value of |β̂1| equal to or larger than the one that we observed.
(d) Use this result to test, at level α = 0.1, the hypotheses
H0 : β1 = 0 vs. H1 : β1 6= 0
(e) Conduct this test in another fashion by constructing the test
statistic t∗ and comparing to either
z0.95 = 1.64 or z0.9 = 1.24 (indicate which value you are
comparing the test statistic to).
(f) Construct a 90% confidence interval for β1. How could we
use this to conduct the hypothesis test
in part (d)?
(g) Suppose that we find we made an error in our calculation
and actually 1
n
∑n
i=1(Xi − X
̄ )

20. 2 = 1. If
all other values stayed the same, how would this change the
result of the hypothesis test in part
(d)?