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PROBLEM SET 3

(Due: April 23)

1. Use the variables Sales and Ads in the data file on the I-Campus (sales.data). Sales

denotes the total amount of sales in dollars, and Ads is the total cost of advertisement

in dollars.

(a) Fit the following linear regression model to the data by calculating the OLS

estimates of β1 and β2. Do this by HAND with a calculator and show your

work. Compare your result with the output by Stata.

Salesi = β1 + β2Adsi + ei

(b) What factors might influence ei?

(c) Based on your estimates, what is the predicted amount of dollar that the company

would sell if it decided to do no advertising.

(d) Suppose that this company makes a profit of 75% of total amount of sales.

Should it be advertising? Why or why not?

2. We will run a regression to see if there exists linear relationship between interest

rates (CD rate) and unemployment rate with a recent Korean data set. Use the

umint kor.txt. (It is monthly data for the period June, 1999 - February 2018 inclusive.)

(a) Suppose unemployment and interest are related by the regression:

unemploymenti = β1 + β2interesti + errori

Regress unemployment on interest. What are the estimated intercept and

slope?

(b) Calculate the standard errors of the estimated intercept and slope.

(c) What is the R2 and the standard error of the regression? How can you interpret

your R2

?

(d) Based on the estimated regression, what would be the effect on unemployment

of a 1% increase in interest rates?

(e) Find a 95% confidence interval for the slope coefficient. Explain the meaning

of the confidence interval you obtained.

2

(f) Based on the estimated regression, what is the predicted level of unemployment

when interest = 12%.

(g) Test the hypothesis H0 : β2 = 0 vs. H1 : β2 6= 0 at the 5% significance level.

What is your p-value? Why is the p-value useful?

(h) Using the confidence interval in (e), test the hypotheses in part (g). Do you

get the same answer? Why or why not?

(i) Suppose that instead unemployment was related to interest in the following

way:

log(unemploymenti) = β1 + β2 log(interesti) + errori

.

How does the economic interpretation of β2 differ from that in part (a)?

3. Consider the simple regression model given by

Yi = β1 + β2Xi + ei.

Assume that you multiply all the Xi values by 10, but not the Yi values.

(a) What happens to the parameter values β1 and β2?

(b) What happens to the OLS estimators βb1 and βb2?

(c) What happens to the variances of OLS estimators βb1 and βb2? Has residual ebi

changed?

(d) Assume that you multiply all the Yi values by 10, but not the Xi values. Answer

the (a), (b), and (c) for this regression.

4. Consider the following two simple linear regression models

Yi = β1 + β2Xi + ei (1)

and

Yi = α2Xi + ui (2)

Assume that the regression errors ei (mean zero and variance σ2e) and ui (mean zero and variance σ2u) satisfy the classical assumptions. ebi and ubi denote the residuals

from OLS regression. Consider the OLS estimators (of the slope parameters β2, α2)

βb2 for (1) and αb2 for (2).

(a) Compare βb2 and αb2. Are they identical?

3

(b) Calculate the bias of αb2 and variance of αb2.

(c) We already know that Pn

i=1 Xiebi = 0 from the first order condition of OLS in

the simple linear regression (1). Is Pn

i=1 Xiubi = 0 true?

(d) We already know that Pn

i=1 ebi = 0 from the first order condition of OLS in the

simple linear regression (1). Is Pn

i=1 ubi = 0 true?

5. Show that in the simple regression the following result (introduced in the class)

holds:

is the coefficient of determination and rX,Y is the sample correlation coef-

ficient between Yi and Xi, i.e.,rX,Y =Pn

i=1(Xi − X)(Yi − Y )qPni=1(Xi − X)2qPni=1(Yi − Y )2

6. You first estimate a simple linear regression

Yi = β1 + β2Xi + ei,

and then estimate the regression model without intercept, that is, estimate the

following model

Yi = βXi + ei.

Which one give smaller RSS (Residual sum of squares)? Explain.

7. Consider a simple linear regression

Yi = β1 + β2Xi + ei

and assume that ei

iid∼ U(−√3σ, √3σ).

(Hint : If Xi ∼ U(a, b), then E(Xi) = 12

(a + b), V ar(Xi) = 112 (b − a)2))

(a) Obtain the OLS estimators βb1 and βb2.

(b) Calculate the bias and variance of βb1 and βb2.

(c) Some argue that βb1 and βb2 are still BLUE. Do you agree or disagree? Explain

briefly.

(d) Do βb1 and βb2 follow the normal distribution in small sample? Explain briefly.

(No formal proof required)

(e) What about in large sample? Explain briefly. (No formal proof required)

8. In the simple linear regression model Yi = β1 + β2Xi + ei

, suppose that E(ei) 6= 0.

Letting α1 = E(ei), show that the model can always be rewritten with the same

slope, but a new intercept and error, where the new error has a zero expected value.

9. Consider a simple linear regression relationship

Yi = β1 + β2Xi + ui

where ui

is an error term that satisfies the 5 assumptions for the regression. One

may assume that

is the unbiased OLS estimator for this model(xi = Xi−X¯ and yi = Yi−Y¯ ). However,

a researcher has decided to use

as an estimate for β2.

(a) Show that βˆ

2 is an unbiased estimator for β2.

(b) Show that β˜

2 is a biased estimator for β2.

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