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Econ 497 B1/B2 - Winter 2022

Midterm Exam 1

Question 1 (18 points)

Let the true population model be

yi = —1xi1 + —2xi2 + Ái ,

for i = 1,...,N. Notice that there is no constant term in the regression. Assume that the standard assumptions about this model apply.

Consider the sample regression model yi = —˜1xi1 + Á˜i for i = 1,...,N. Answer the following questions.

a. Specify the least squares function that is minimized by OLS.

b. Derive the expression for the OLS estimator —˜1 .

c. Suppose q xi1xi2 0. Compute the expected value of the OLS estimator —˜1 .

d. Derive an expression for the variance of the —˜1 .

Question 2 (12 points)

Consider the model

yi = —1 + —2xi2 + —3xi3 + —4xi4 + Ái ,

for i = 1,...,N and assume all the assumptions of the classical linear model hold.

You want to test the null hypothesis that —2 = —3 = —4 = 0. Answer the following questions.

a. Write the null hypothesis as the set of linear restrictions H0: R— = q .

R2 = 0.23. Using an appropriate form of the F-statistic, carry out a test of the null hypothesis H0: —2 = —3 = —4 = 0 at the 5% signiﬁcance level. Is the distribution

c. Suppose you now want to jointly test H0: —2 = —3 and —4 = 2. Write the null hypothesis as the set of linear restrictions H0: R— = q .

Question 3 (20 points)

For this exercise you need to use ﬁle WAGE2.csv available on eClass.

The data consists of 935 full-time workers where wage is monthly earnings, age is age (in years), and educ is years of education. We are interested in the eﬀect of education on wages and estimate following wage equation:

ln(wagei) = —1 + —2agei+ —3educi+ —4educ + Ái .

Use R to answer the following questions.

a. Estimate the model by OLS and interpret the meaning of —ˆ2 .

b. Carry out a two-sided test at the 5% signiﬁcance level of the null hypothesis that —2 = 0. What do you conclude?

c. What is the expected change in wage if education increases from 10 to 12 years? Are the marginal eﬀects increasing or decreasing?

d. Test for heteroskedasticity in the error term. Brieﬂy explain the your test and report your results. What do you conclude?

e. It would be nice to separate out the eﬀects of education, age, and on-the-job experience. Very few data sets collect information on on-the-job experience so labor economists frequently deﬁne a variable “potential experience” as age minus years of education minus 6. As an example, an 18 year old high school graduate is coded as zero. Explain what you would ﬁnd if you regressed log wage on age, education, potential experience, and an intercept.