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Jan 9, 2024

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Department of Economics UN3412 Columbia University Fall 2023 Problem Set 5 Introduction to Econometrics (Erden_ Section 1) ______________________________________________________________________________ Please make sure to select the page number for each question while you are uploading your solutions to Gradescope. Otherwise, it is tough to grade your answers, and you may lose points. Part I Question 1 (12p) True, False, Uncertain with Explanation: (a) (3p) If the key explanatory variable is constant over time, we cannot use fixed effects to estimate its effect on y (the dependent variable). (b) (3p) Using fixed effects is mechanically the same as allowing a different intercept for each cross-sectional unit. (c) (3p) In the fixed-effects regression model, you should exclude one of the binary variables for the entities when an intercept is present in the equation. (d) (3p) Time fixed effects regressions are useful in dealing with omitted variables if these omitted variables are constant over time but not across entities. Question 2 (12p) A researcher investigating the determinants of crime in the United Kingdom has data for 42 police regions over 22 years. She estimates by OLS the following regression 𝐿?𝑔(𝑐𝑟??) 𝑖𝑡 = ? 𝑖 + 𝜑 𝑡 + ? 1 ??𝑟?? 𝑖𝑡 + ? 2 ?𝑟?𝑦?ℎ 𝑖𝑡 + ? 3 log (??) 𝑖𝑡 + ? 𝑖𝑡 where cmrt is the crime rate per head of population, unrtm is the unemployment rate of males, proyth is the proportion of youths, and pp is the probability of punishment measured as (number of convictions)/(number of crimes reported). ? and 𝜑 are area and year fixed effects, coeffcient 𝜑 1 is not included. (a) (3p) What is the purpose of excluding 𝜑 1 ? What are the terms ? and 𝜑 likely to pick up? Discuss the advantages of using panel data for this type of investigation. (b) (3p) Estimation by OLS using heteroskedasticity-robust standard errors results in the following output, where the coeffcients of the fixed effects are not reported: 𝐿?𝑔(𝑐𝑟??) 𝑖𝑡 ̂ = 0.063??𝑟?? 𝑖𝑡 + 3.739?𝑟?𝑦?ℎ 𝑖𝑡 − 0.558log (??) 𝑖𝑡 ; 𝑅 2 = 0.904 (0.109) (0.179) (0.024) Comment on the results. In particular, what is the effect of a ten percent increase in the probability of punishment? (c) (3p) To test for the relevance of the area fixed effects, you restrict the regression by dropping all regional fixed effects and adding a single constant. The relevant F-statistic is 135.28. What are the degrees of freedom? What is the critical value from your F table ?

(d) (3p) Although the test rejects the hypothesis of eliminating the fixed effectsfrom the regression, you want to analyze what happens to the coefficients and their standard errors when the equation is re-estimated without fixed effects. In the resulting regression, ? ̂ 2 and ? ̂ 3 do not change by much, although their standard errors roughly double. However, ? ̂ 1 is now 1.340 with a standard error of 0.234. Why do you think that is? Part II Question 1 (24p) Data file fatality_extra.dta includes the following variables for 50 states and Washington DC from 1983 to 1997. Variables in fatality_extra.dta Variable Definition fips State ID (FIPS) Code year Year fatalityrate The number of traffic deaths in a given state in a given year, per 10,000 people living in that state in that year sb_usege Percent of drivers using seatbelt in a given state ba08 Blood alcohol level 0.08% law is applied. drinkage21 State has a drinking age limit of 21yr-old speed65 =1 for states with speed limit 65mph, zero otherwise. speed70 =1 for states with speed limit 70mph, zero otherwise. lnincome Logarithm of average income level in that state y83 through y97 Dummy variables for each year (a) (9p) Fill out Table 1 on the separate table file provided, report your do file here and answer the following questions: (b) (3p) Do seat belts change the fatality rate significantly? (c) (3p) What is wrong with regression 1? (d) (3p) Are state fixed effects significant? (e) (3p) Are time fixed effects significant? (f) (3p) Why do you need to use HAC errors?

Table 1 The Effect of Seatbelt Usage on Traffic Deaths: Regression Results Dependent variable: fatalityrate (1) (2) (3) (4) (5) Coefficient on seat belt usage ( ) ( ) ( ) ( ) ( ) State characteristic control variables a ? No Yes Yes Yes Yes State fixed effects? No No Yes Yes Yes Year fixed effects? No No No Yes Yes F -statistic testing the hypothesis that the state fixed effects are zero – – (p< ) (p< ) F -statistic testing the hypothesis that the year fixed effects are zero – – – (p< ) (p< ) HAC (clustered) SEs? No No No No Yes n Notes: All regressions include an intercept. Heteroskedasticity-robust standard errors appear in parentheses below estimated coefficients; p-values appear in parentheses beneath heteroskedasticity-robust F-statistics. a Regressions with “state characteristic control variables” include the following regressors: ba08 drinkage21 speed65 speed70 lnincome. Question 2 (24p) The data file RENTAL.dta include rental prices and other variables for college towns in 1980 and in 1990. The idea is to see whether a stronger presence of students affects rental rates. The unobserved effects model is log(rent it ) = β 0 + δ 0 y90 t + β 0 log(pop it ) + β 2 log(avginc it ) + β 3 pctstu it + a i + u it Variables needed are explained in below Variables in RENTAL.dta Variable Definition pop City population avginc Average income pctstu Student population as a percentage of city population (during the school year) y90 =1 for 1990, zero otherwise.

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