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Theoretical Problems

1. Suppose y = β0 + β1x1 + β2x2 + u and MLR.1 to MLR.6 are satisfified. When we use OLS to estimate this model with n=100 observations, estimates (and non-robust estimated standard errors) are:

βˆ0 = 1 (???) βˆ1= 5 (3)βˆ2= 6 (???)

where some standard errors are left as question marks for now. Defifine ¯ x1 as the sample average of x1, and ¯ x2 as the sample average of x2, and defifine xij as observation i for variable xj . Suppose that:

1n nXi=1(xi1− ¯x1)2= 41n nXi=1(xi2− ¯x2)2= 251n nXi=1(xi1 − ¯x1)(xi2 − ¯x2)=6

a. Test the null hypothesis that β1 = −1 against the alternative hypothesis that β1 6=−1. What is the test statistic? What is the critical value for this test at the 5% level? Do you reject the null at the 5% level? (For this problem set, find thecritical value by looking in the back of your book, using degrees of freedom closest to the number in this problem. For example, if there were 55 degrees of freedom, you should use the critical value for 60 degrees of freedom.)

b. Test the null hypothesis that β1 = −1 against the alternative hypothesis that β1 < −1. What is the test statistic? What is the critical value for this test at the 5% level? Do you reject the null at the 5% level?

c. Test the null hypothesis that β1 = −1 against the alternative hypothesis that β1 > −1. What is the test statistic? What is the critical value for this test at the 5% level? Do you reject the null at the 5% level?

d. Find the standard error of βˆ2. Hint: recall that the variance of an OLS estimator is estimated as: Vˆ ?βˆj? = σˆ2SSTj ?1− Rj2?where all terms on the right hand side are defifined in the slides and the textbook.

e. Calculate the sum of squared residuals from using OLS to estimate Equation.

f. Find the R2 when we use OLS to estimate Equation 1. Hint: use the fact that yi = βˆ0 +βˆ1xi1 +βˆ2xi2 + ui. Also, defifine ˜xij = (xij − ¯x

g. which simplififies notation.2=Varcxy=?arc? 21=(or C Xi , ? 2)(g)

h. Test the joint hypothesis that β1 = β2 = 0. What is the test statistic? What is the critical value for this test at the 5% level? Do you reject the null at the 5% level?

i. Test the joint hypothesis that β1 = 1 and β2 = 0. What is the test statistic? What is the critical value for this test at the 5% level? Do you reject the null at the 5% level?

3 An company is considering opening a new retail location. To help them decide where to locate, they use OLS to estimate the following model of store profifits:

y = β0 + β1x1 + β2x2 + β3x1x2 + β4x2 2 + u

We will assume MLR.1 to MLR.4 are satisfified, and:-y = ln( annual profifits in millions of dollars ) x1= an indicator variable, taking a value of 1 if the store stays open late and 0 otherwise x2= the store’s distance from headquarters, in kilometers.

The company estimates:

βˆ0= 6βˆ1= 0.2βˆ2= 0.4βˆ3=−0.2βˆ4=−0.1

a. Suppose a store is located 10 km from headquarters, and is not open late. By what percentage do you expect profifits would change if they do open late? (Note: do not use an approximation here, because this is a large di↵erence.)

b. Suppose the company wants a store that does open late. They are considering placing that store 8 km from headquarters. By what percentage do you expect profifits would change if, instead, they place it 1 meter further away? (Note: do use an approximation here, because this is a small di↵erence. You are being asked about 1 meter further away, not 1 km.)

d. Given these estimates, what choice of staying open late and distance from headquarters maximizes expected log profifits?

e. An economist in the company creates a complicated model of the fifirm’s choices, and a key parameter is γ = β1 ? β2. Can you give an unbiased estimate of γ? Can you give a consistent estimate of γ? For each question: if yes, give that estimate and explain how you derived it. If not, explain why not.

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