Intermediate Econometrics 2024 Spring Assignment 2
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Intermediate Econometrics 2024 Spring Assignment 2
National School of Development
Due on April 9th 12:00 a.m., 2024
1 Basic Concepts
Please explain the following concepts:
Q1. What is omitted variable bias? When does it happen?
Q2. “Partialling out” Interpretation of Multiple Regression.
Q3. p-values for t Tests.
2 Theoretical Deduction
2.1
Consider the regression model
yi = β1 xi1 + β2 xi2 + ui
for i = 1, ...n. Notice that there is no constant term in the regression.
Q4. Specify the least square function that is minimized by OLS.
Q5. Compute the partial derivatives of the objective function with respect to β1 and β2 .
Q6. Suppose that 
Derive an expression for 
as a function of the data (yi , xi1, xi2), i =
1, ... n
Q7. Suppose that the model includes an intercept:
yi = β0 + β1 xi1 + β2 xi2 + ui
Show that the least square estimators satisfy 
Q8. Suppose that the model contains an intercept. Also suppose that
Show that
How does this compare to the OLS estimator of β1 from the regression that omits x2 ?
2.2
Consider a simple linear regression
yi = β0 + β1 xi + ui
with n units i = 1, . . . , n. Denote the OLS estimator of β1 by 
. Make the Homoskedasticity (HMK) Assumption: var(ui
|xi) = σ2. In this problem, we assume xi are fixed data points. That is to say, you don’t need to treat them as random variables.
Q9. Check the formula in Slides Chap. 5, p. 17. Prove that
Q10. Give an unbiased estimator for σ2 = var(ui ) = var(ui |xi ).
Q11. You have two ways to test H0 : β1 = 0. The first way is to use the t-statistic
Q12. Define
Prove that
Q13. The second way is to use the F-statistic with form
Here q = 1 and k = 1. Use the sample set 
SSRr and SSRur.
Q14. Prove that
So t2 = F.
3 Application and Stata
Are rent rates influenced by the student population in a college town? Let rent be the average monthly rent paid on rental units in a college town in the United States. Let pop denote the total city popu- lation, avginc the average city income, and pctstu the student population as a percentage of the total population. One model to test for a relationship is
log( rent ) = β0 + β1 log( pop ) + β2 log( avginc ) + β3 pctstu + u.
Q15. State the null hypothesis that size of the student body relative to the population has no ceteris paribus effect on monthly rents. State the alternative that there is an effect.
Q16. The equation estimated using 1990 data from RENTAL for 64 college towns is
What is wrong with the statement: ”A 10% increase in population is associated with about a 6.6% increase in rent”?
Q17. Test the hypothesis stated in Q16 at the 1% level.
Q18. Write down the Stata code to
• Show the number of observations, mean, and standard deviation of pctstu.
• Regress model in Q16.
• Test the null hypothesis in Q15.
2024-04-06