1. What are the first-order conditions for the minimization of least squares in
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1. What are the first-order conditions for the minimization of least squares in the simple linear model? Write out these two equations in terms of the coefficient estimates and in terms of the estimated error term.
2. Use these two equations to solve for , and . Interpret these coefficients in terms of sample covariances and variances. What is the interpretation of betahat1?
3. Which of the following are random variables: , , , yi, , If they are not random variables, what are they? Which of previous were population or “true” values? What is the difference between a sample and population or “true” value?
4. What is meant by the term unbiased estimate?
5. Show that is an unbiased estimator of . Be sure to be clear on what assumptions from your answer to 10 that you use. Does this mean that every estimate is beta1? Explain.
6. Show that the variance of = . Be sure to be clear on what assumptions you use.
7. The regression below is a regression of the wages of full-time workers on the workers sex2, which equals 0 for males and 1 for females, and their age, and on their age squared. Interpret the coefficient on sex2? How do you calculate a t-statistic? How do you use it? Also, use the data from this regression to calculate the F-statistic for the null hypothesis that age does not matter for the earnings of full time workers.
8. Suppose you are adding a new bathroom to your house. You are interested in how this addition will affect the value of your home. Your contractor brings you this regression.
Where the units are current dollars. You are immediately concerned because the regression does not include the number of bedrooms. What do you believe is the sign of the covariance between the number of baths and the number of bedrooms in a house, and what do you believe is the relationship between the number of bedrooms on the price of a home? Is the above regression over or understating the increase in the value of your home after you add the new bathroom? Explain briefly.
9. What is heteroskedasticity? What OLS estimator properties are lost as a result of heteroscedasticity?
10. How can you test for heteroscedasticity? What is the difference between the Breusch-Pagan test and White’s test? What is the heteroskedasticity test that conserves on degrees of freedom?
11. How does heteroskedasticity affect hypothesis testing? How did White suggest remedying this problem?
12. How can you use a Lagrange multiplier test to test for heteroskedasticity? What is the distribution of a Lagrange multiplier test?
13. What is the difference between large sample and small sample properties? What is meant by saying that an estimator is consistent?
14. A student asked me “Does measurement error mess up my coefficient estimates?” As a good economist, I answered “It depends.” Explain my answer. Use your answer to explain attenuation bias.
15. Consider the regression where crime is the crime in city i in a given year and police is the number of police in city i in the same year. Explain why policei and are likely to be correlated. What is the consequence of this correlation for the OLS estimate of ? Explain.
16. The economist Steve Levitt argues for using the number of firefighters in city i in the year as an instrument for police. What must be true of about the variable firefighters for it to be valid instrument?
17. Use the above questions to describe the two stage least squares estimation method.
18. Suppose the demand for bananas in country i depends on the price of bananas the particular country and the income in that country. The supply of bananas depends on the price of bananas and the average rainfall in banana producing countries. You are interested in how the quantity of bananas demanded and supplied reacts to the price of bananas. Explain what is wrong with estimating the regression quantity of bananasi = b0 + b 1price of bananasi + ui. Explain how you would estimate the demand and supply curves for bananas.
19. What are the two conditions a valid instrument must satisfy? How can you assess whether or not these conditions are satisfied in your problem?
20. What is a weak instrument
21. What is a Hausman test? What role does it serve? Explain how you would carry out this test in the problem discussed in questions 11 and 12.
22. How can you run a test of overidentifying restrictions? If you reject the null in this test, what is the implication for moving forward on your research question?
23. How does two stage least squares estimation address the causation problem in economics. Use the Acemoglu, Gallego, and Robinson paper to formulate your answer.
