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- 1. For the model Yi = β1 + ui the OLS estimator of β1 is b1 = 1 n ∑ i Yi. (a) Demonstrate that b1 may be written as b1 = β1+c ∑ i ui, where c is some constant that you have to figure out. (b) Prove that var(b1) = σ2u n . Recall that var(b1) = E[b1 − E(b1)]2. You can take as given that E[b1] = β1.

1. For the model Yi = β1 + ui the OLS estimator of β1 is b1 = 1 n

∑ i Yi.

(a) Demonstrate that b1 may be written as b1 = β1+c ∑

i ui, where c is some constant

that you have to figure out.

(b) Prove that var(b1) = σ2u n . Recall that var(b1) = E[b1 − E(b1)]2. You can take as

given that E[b1] = β1.

2. Dougherty 4e edition exercise 2.9.

3. Suppose that the true regression model is Yi = β2Xi + ui. An unbiased and linear

estimator of β2 is b̂2 = Ȳ /X̄. Use the Gauss-Markov theorem to explain whether this

estimator is more or less efficient than the ordinary least squares estimator

bOLS2 = ∑

iXiYi/ ∑

i X 2 i which is also linear and unbiased. I am not asking for a

mathematical proof.

4. (a) Go to

http://global.oup.com/uk/orc/busecon/economics/dougherty4e/01student/datasets/

Click on data set Educational Attainment and Earnings Functions (EAEF), pick

the data format you want and then download “subset 22.” Use these data to

answer problem 1.7 in Dougherty. You must supply a print out of the regression

output produced by the software you are using (e.g. gretl, stata, R, etc).

(b) Still considering the model and data from part (a) suppose that someone per-

formed a two-sided hypothesis test (H0 : β2 = β 0 2 vs H1 : β2 ̸= β02) and got a test

statistic equal to 1.4. Calculate the p-value of that test and represent the p-value

in a graph.

(c) Still considering the model and data from part (a) suppose that someone per-

formed a one-sided hypothesis test (H0 : β2 = β 0 2 vs H1 : β2 > β

0 2) and got a test

statistic equal to 1.7. Calculate the p-value of that test and represent the p-value

in a graph.

(d) Still considering the model and data from part (a) suppose that someone per-

formed a one-sided hypothesis test (H0 : β2 = β 0 2 vs H1 : β2 < β

0 2) and got a test

statistic equal to -2.3. Calculate the p-value of that test and represent the p-value

in a graph.

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5. This question relates to problem 1.6 in Dougherty. Refer to appendix B (textbook

pages 548-550) for a definition of the variables appearing in the model. Estimating the

regression model Si = β1 + β2AVSABCi + ui yields the following

Ŝ = 6.56932 (0.48988)

+ 0.139146 (0.0093936)

ASVABC

T = 540 R2 = 0.29 s2u = 2.0958 2

(standard errors in parentheses)

(a) What is the fraction of the variation in S that is explained by the regression

model?

(b) Calculate the predicted value of schooling (S) when ASVABC = 50. Show your

work.

(c) Calculate a 90% confidence interval for β2. Show your work.

(d) Someone claims that years of schooling increases by one year when ASVABC

increases by 10 units. Set up the null and alternative hypotheses you would use

to test (two-sided) this person’s claim. Explain your answer. You do not actually

need to carry out the test.

(e) Test H0 : β2 = 0.125 against H1 : β2 ̸= 0.125 using a level of significant of 5%. To do so perform the following: (1) draw a t distribution where you represent the

rejection and non-rejection regions for H0; (2) calculate the relevant test statistic;

(3) decide whether you reject H0 or not; (4) interpret the outcome of your test

(i.e. what is the outcome of the test telling us).

(f) Given the specific model you are working with, explain why it might be justifiable

to perform a one-sided test H0 : β2 = 0 against H1 : β2 > 0 instead of a two-sided

test (H0 : β2 = 0 against H1 : β2 ̸= 0).

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