ps5_sol_fall2023

Department of Economics UN3412 Columbia University Fall 2023 SOLUTIONS to 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 True, False, Uncertain with Explanation: (a) If the key explanatory variable is constant over time, we cannot use fixed effects to estimate its effect on y (the dependent variable). TRUE. In this case, the key explanatory variable would be perfectly collinear with the fixed effects. Intuitively, the fixed effect estimator transforms the model by demeaning for each individual. Therefore, we identify the effects of X on Y using the variation within individuals. If X is constant over time, there is no variation within individuals. (b) Using fixed effects is mechanically the same as allowing a different intercept for each cross- sectional unit. TRUE. One way to estimate a fixed effects model is to include one dummy variable for each entity (if we include a constant, then we have to drop one dummy variable). This implies that we will have one intercept for each individual. (c) 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. TRUE. As mentioned in the previous item, we would have perfect multi-collinearity if we include a constant and one dummy variable for each individual. Therefore, we need to exclude the dummy variable for one of the individuals. If we do not include the constant, then we can include all dummy variables. (d) Time fixed effects regressions are useful in dealing with omitted variables if these omitted variables are constant over time but not across entities. FALSE. It is the opposite. Time fixed effects control for omitted variables that can vary over time, but are constant across entities. Question 2 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??) 𝑖? = ? 𝑖 + 𝜑 ? + ? 1 ??𝑟?? 𝑖? + ? 2 ?𝑟?𝑦?ℎ 𝑖? + ? 3 log (??) 𝑖? + ? 𝑖? 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) 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. We need to exclude 𝜑 1 because otherwise the complete set of police region dummies would be multicollinear with the constant. ? picks up the individual fixed effects, which is constant over time but different across police regions. 𝜑 picks up the time fixed effects, which is constant across police regions but different over time. By using panel data, we can control for unobserved heterogeneity that may cause omitted variable bias with cross section or time series data only . (b) 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??) 𝑖? ̂ = 0.063??𝑟?? 𝑖? + 3.739?𝑟?𝑦?ℎ 𝑖? − 0.558log (??) 𝑖? ; 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? Controlling for other variables, crime rate increases by 6.3% when the unemployment rate increases by 1 percentage point. But this is not statistically significant. Crime rate increases by 3.739% when young population increases by 1 percentage point (note that proportion of young population is in ratio which needs to be scaled up by 100 to be converted as percentage.) Finally, crime rate decreases by 0.58% in response to a 1% increase in the punishment probability (note that this is not a response to 1 percentage point increase of punishment probability, which would be true if we used pp instead of log(pp)). Therefore, a 10% increase in the probability of punishment would reduce crime by 5.88%. The coefficient on probability of punishment is significant at 5%, as the t-statistic is much greater than 1.96. (c) 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 ? We are testing if the 41 dummies (remember, we exclude one) are all equal to zero. Therefore, the q = 41. The other degree of freedom is ? − 𝑘 ?𝑛?????𝑖???? − 1 , but since this is generally a big number, we can approximate to ∞ . Thus, the F-statistic will have an 𝐹 41 distribution under the null. The 5% critical value is around 1.38. Since the F-statistic is 135.28, we reject that the fixed effects are all zero. (d) 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? We can definitely suspect that there is omitted variable bias here. That is, the omitted region fixed effects are highly correlated with unemployment rate, but not so much with other regressors. A possible explanation is that one of omitted variables is black/hispanic population. If it is a fact that black/hispanic population has higher unemployment rate and are correlated with higher crime rate. Then, omitting this variable will bias the coefficient on unemployment rate upward. Another important consequence of including fixed effects is that

Table 4 The Effect of Seatbelt Usage on Traffic Deaths: Regression Results Dependent variable: fatalityrate (1) (2) (3) (4) (5) Coefficient on seat belt usage -0.0119 (0.0012) 0.0041 (0.0012) -0.0056 (0.0013) -0.0034 (0.0011) -0.0034 (0.0014) 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 – – 37.88 (p<0.001) 47.72 (p<0.001) F -statistic testing the hypothesis that the year fixed effects are zero – – – 10.60 (p<0.001) 9.15 (p<0.001) HAC (clustered) SEs? No No No No Yes n 556 556 556 556 556 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. Do file for question 1: xtset fips year reg fatalityrate sb_usage,r regress fatalityrate sb_usage ba08 drinkage21 speed65 speed70 lnincome, r xtreg fatalityrate sb_usage ba08 drinkage21 speed65 speed70 lnincome, fe vce(cluster fips) xtreg fatalityrate sb_usage ba08 drinkage21 speed65 speed70 lnincome i.year, fe testparm i.year xtreg fatalityrate sb_usage ba08 drinkage21 speed65 speed70 lnincome i.year, fe vce (cluster fips) testparm i.year Question 2 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. (a) Estimate the equation by pooled OLS and report the results in standard form. What do you make of the estimate on the 1990 dummy variable? Using pooled OLS we obtain log(rent it ) = -0.569 + 0.262 y90 t + 0.041 log(pop it ) + 0.571 log(avginc it ) + 0.0050 pctstu it (0.535) (0.035) (0.023) (0.053) (0.0010) n = 128, R 2 = 0.861 The positive and very significant coefficient on d90 simply means that, other things in the equation fixed, nominal rents grew by over 26% over the 10 year period. (b) Interpret the sample coefficient of pctstu The coefficient on pctstu means that a one percentage point increase in pctstu increases rent by half a percent (.5%). The t statistic of five shows that, at least based on the usual analysis, pctstu is very statistically significant. (c) Are the standard errors you report in part (a) valid? Explain. The standard errors from part (i) are not valid, unless we think ai does not really appear in the equation. If ai is in the error term, the errors across the two time periods for each city are positively correlated, and this invalidates the usual OLS standard errors and t statistics. (d) Now, difference the equation and estimate by OLS. Compare your estimate of β 3 with that of part (a). Does the relative size of the student population appear to affect rental prices? The equation estimated in differences is Δlog( rent ) = .386 + .072 Δlog( pop ) + .310 Δ log( avginc ) + .0112 Δ pctstu (.037) (.088) (.066) (.0041) n = 64, R 2 = .322. Interestingly, the effect of pctstu is over twice as large as we estimated in the pooled OLS equation. Now, a one percentage point increase in pctstu is estimated to increase rental rates

Question 3 U.S. airlines were deregulated in 1975, allowing them to charge whatever prices they wished and to choose routes for their flights more freely than previously. One anticipated gain from deregulations was cost reduction, to be derived in part by allowing airlines to reduce excess capacity. Baltagi, Griffin and Vadali estimate that airlines did, indeed, reduce excess capacity following deregulations 1 . Their analysis combined data on variable costs and factor shares to efficiently estimate excess capacity for 23 airlines in the years 1971-1986. Data file deregulate.dta contain the following variables: Variable Description airline A number indicating the airline in the observation. pf The price of fuel pl The price of labor pm The price of materials reg =1 if the observation is from the regulated period =0 otherwise stage Average length of the airline’s flights that year vc Variable cost (fuel+labor+materials) y An index of annual passenger miles flown by the airline year The year of the observation (a) Regress the log of costs on the regulation dummy, year and the natural logs of three price variables and of stage (i) using OLS (ii) using firm-specific fixed effects without cluster (iii) with cluster (b) What is the interpretation of regulation dummy’s coefficient in these regression? (c) What is the interpretation of year’s coefficient in these regression? (d) Briefly explain why we can conclude that the estimated standard errors reported for OLS are probably incorrect as well as the ones in fixed effects regression without cluster errors? (e) What does the fixed effects regression imply about the effect of deregulation on airlines’ variable cost? (f) How do you counter the objection that technical change would have reduced airline costs even without the deregulation? (g) Add the squares of the logged regressors to the fixed effects regression in (a). What does this regression suggests about the conclusions in (e)? (h) Are the added terms in regression (g), taken together, jointly statistically significant? Show the needed test results. (i) Some have argued that deregulation enables airlines to better plan their flight. This could mean that more efficient flight lengths were chosen after deregulation. How does this affect the interpretations in (e) and (g), and how would you take this consideration into account? 1 Badi H. Baltagi, James M. Griffin, and Sharada R. Vadali, “Excess Capacity: A Permanent Characteristic of U.S. Airlines,” Journal of Applied Economtrics 13, no.5 (1998): 645-657

a. OLS results: . reg lvc reg lpl lpf lpm lstage year, r Linear regression Number of obs = 268 F( 6, 261) = 176.21 Prob > F = 0.0000 R-squared = 0.6939 Root MSE = .63537 ------------------------------------------------------------------------------ | Robust lvc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- reg | -.1044246 .1419933 -0.74 0.463 -.3840228 .1751736 lpl | .9137027 .439856 2.08 0.039 .0475846 1.779821 lpf | -.4192051 .2361399 -1.78 0.077 -.8841869 .0457766 lpm | 1.673205 .8965462 1.87 0.063 -.0921797 3.438589 lstage | 1.31977 .0511289 25.81 0.000 1.219092 1.420448 year | -.0688188 .0515875 -1.33 0.183 -.1703994 .0327618 _cons | -5.619306 3.28694 -1.71 0.089 -12.0916 .8529905 ------------------------------------------------------------------------------ Fixed Effects WITHOUT CLUSTER: . xtreg lvc reg lpl lpf lpm lstage year, fe Fixed-effects (within) regression Number of obs = 268 Group variable: airline Number of groups = 23 R-sq: within = 0.9324 Obs per group: min = 5 between = 0.4710 avg = 11.7 overall = 0.6358 max = 16 F(6,239) = 549.18 corr(u_i, Xb) = 0.1966 Prob > F = 0.0000 ------------------------------------------------------------------------------ lvc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- reg | -.0103893 .0404275 -0.26 0.797 -.090029 .0692504 lpl | .12939 .1013104 1.28 0.203 -.0701854 .3289654 lpf | .0880113 .0603803 1.46 0.146 -.0309343 .2069569 lpm | .3837664 .2618333 1.47 0.144 -.1320294 .8995621 lstage | .8636402 .0610974 14.14 0.000 .7432821 .9839984 year | .0458234 .0130707 3.51 0.001 .0200749 .071572 _cons | -4.573632 .9142783 -5.00 0.000 -6.374705 -2.772559 -------------+---------------------------------------------------------------- sigma_u | .74149963 sigma_e | .15044171 rho | .96046374 (fraction of variance due to u_i) ------------------------------------------------------------------------------ F test that all u_i=0: F(22, 239) = 200.74 Prob > F = 0.0000

with cluster because there is autocorrelation over time within the same airline since errors overtime would be correlated for each airline (but not necessarily across airlines) e. The estimated coefficients for reg is -.0103, indicating that on average, the variable costs during regulated period are 1.03 percent lower than the costs during deregulated period. So the airlines’ variable costs became higher as U.S. airlines were deregulated. f. If tech change is a smooth change then it will be captured by “year” variable. But if tech change happens as suddenly as regulation then it cannot be accounted by “year” variable hence it will cause omitted variable bias. g. . xtreg lvc reg lpl lpf lpm lstage lpl2 lpf2 lpm2 lstage2 year, fe vce(cluster > airline) Fixed-effects (within) regression Number of obs = 268 Group variable: airline Number of groups = 23 R-sq: within = 0.9385 Obs per group: min = 5 between = 0.4484 avg = 11.7 overall = 0.6246 max = 16 F(10,22) = 158.01 corr(u_i, Xb) = 0.2217 Prob > F = 0.0000 (Std. Err. adjusted for 23 clusters in airline) ------------------------------------------------------------------------------ | Robust lvc | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- reg | .0523131 .0262211 2.00 0.059 -.0020662 .1066923 lpl | .1602116 .1623721 0.99 0.335 -.1765276 .4969508 lpf | .0102717 .2519246 0.04 0.968 -.5121878 .5327313 lpm | -5.049218 7.213431 -0.70 0.491 -20.00896 9.910523 lstage | 1.432682 1.354852 1.06 0.302 -1.377109 4.242472 lpl2 | -.4498118 .1330767 -3.38 0.003 -.7257959 -.1738277 lpf2 | -.0179439 .0672491 -0.27 0.792 -.1574101 .1215222 lpm2 | .5261826 .7548597 0.70 0.493 -1.039301 2.091666 lstage2 | -.0526781 .1142455 -0.46 0.649 -.2896087 .1842525 year | .0465786 .0304395 1.53 0.140 -.0165491 .1097063 _cons | 7.859137 16.95165 0.46 0.647 -27.29643 43.01471 -------------+---------------------------------------------------------------- sigma_u | .76036594 sigma_e | .1447302 rho | .96503636 (fraction of variance due to u_i) ------------------------------------------------------------------------------ If we add the squares of the logged regressors the fixed effects regression in (a), the estimated coefficients for reg is 0.052. This indicates that the variable costs during regulated period are approximately 5.2 percent higher than the costs during deregulated period. In other words, it suggests that deregulation did contribute to the reduction on airlines’ variable cost.

h. Yes they are, see below . test lpl2 lpf2 lpm2 lstage2 ( 1) lpl2 = 0 ( 2) lpf2 = 0 ( 3) lpm2 = 0 ( 4) lstage2 = 0 F( 4, 22) = 6.00 Prob > F = 0.0020 i. Regression (e) and (g) give opposite results about regulation, in (e) we are not controlling for efficiency of the flight length variable but in (g) by adding the stage squared term we may be better addressing the efficiency of the flight length so we are seeing the true impact of the regulation. Do file for question 5: use deregulate.dta, clear sum vc gen lvc=log(vc) gen lpl=log(pl) gen lpf=log(pf) gen lpm=log(pm) gen lstage=log(stage) xtset airline year reg lvc reg year lpl lpf lpm lstage, r xtreg lvc reg year lpl lpf lpm lstage, fe xtreg lvc reg year lpl lpf lpm lstage, fe vce(cluster airline) gen lpl2=lpl^2 gen lpf2=lpf^2 gen lpm2=lpm^2 gen lstage2=lstage^2 xtreg lvc reg year lpl lpf lpm lstage lpl2 lpf2 lpm2 lstage2, fe Following questions will not be graded, they are for you to practice and will be discussed at the recitation:

(a) (i) The coefficient is − 0.368, which suggests that shall-issue laws reduce violent crime by 36%. This is a large effect. (ii) The coefficient in (1) is − 0.443; in (2) it is − 0.369. Both are highly statistically significant. Adding the control variables results in a small drop in the coefficient. (iii) There are several examples. Here are two: Attitudes towards guns and crime, and quality of police and other crime-prevention programs. (b) In (3) the coefficient on shall falls to − 0.046, a large reduction in the coefficient from (2). Evidently there was important omitted variable bias in (2). The estimate is not statistically significantly different from zero. (c) The coefficient falls further to − 0.028. The coefficient is insignificantly different from zero. The time effects are jointly statistically significant, so this regression seems better specified than (3). (d) This table shows the coefficient on shall in the regression specifications (1) – (4). To save space, coefficients for variables other than shall are not reported. Dependent Variable = ln( rob ) (1) (2) (3) (4) shall − 0.773 ** (0.225) − 0.529** (0.161) − 0.008 (0.055) 0.027 (0.052) F -Statistics and p- values testing exclusion of groups of variables Time Effects 25.9 (0.00) Dependent Variable = ln( mur ) shall − 0.473 ** (0.149) − 0.313** (0.099) − 0.061 (0.037) − 0.015 (0.038) F -Statistics and p- values testing exclusion of groups of variables Time Effects 19.61 (0.00) The quantitative results are similar to the results using violent crimes: there is a large estimated effect of concealed weapons laws in specifications (1) and (2). This effect is spurious and is due to omitted variable bias as specification (3) and (4) show. (e) There is potential two- way causality between this year’s incarceration rate and the number of crimes. Because this year’s incarceration rate is much like last year’s rate, there is a potential two-way causality problem. There are similar two-way causality issues relating crime and shall . (f) The most credible results are given by regression (4). The 95% confidence interval for  Shall is − 11.0% to + 5.3%. This includes  Shall = 0. Thus, there is no statistically significant evidence that concealed weapons laws have any effect on crime rates. Question 5: SW Exercise 10.5 Let D 2 i = 1 if i = 2 and 0 otherwise; D 3 i = 1 if i = 3 and 0 otherwise … Dn i = 1 if i = n and 0 otherwise. Let B 2 t = 1 if t = 2 and 0 otherwise; B 3 t = 1 if t = 3 and 0 otherwise … BT t = 1 if t = T and 0 otherwise. Then  0 =  1 +  1 ;  i =  i −  1 and  t =  t −  1 .