Ec303 Hw 3 Solutions

EC 303 HW 3 Solutions

E9.1 Data from 2004 (1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

ln(AHE)

ln(AHE)

ln(AHE)

ln(AHE)

ln(AHE)

0.147** (0.042)

0.146** (0.042)

0.190** (0.056)

0.117* (0.056)

0.160 (0.064)

−0.0021* * (0.0007)

−0.0021* * (0.0007)

−0.0027* * (0.0009)

−0.0017 (0.0009)

−0.0023 (0.0011)

Dependent Variable

Age

AHE

ln(AHE)

0.439** (0.030)

0.024** (0.002)

ln(AHE)

Age2

ln(Age)

0.725** (0.052)

Female × Age

−0.097 (0.084)

−0.123 (0.084)

Female × Age2

0.0015 (0.0014)

0.0019 (0.0014)

Bachelor × Age

0.064 (0.083)

0.091 (0.084)

Bachelor × Age2

−0.0009 (0.0014)

−0.0013 (0.0014)

Female

−3.158** (0.176)

−0.180** (0.010)

−0.180** (0.010)

−0.180** (0.010)

−0.210** (0.014)

1.358* (1.230)

−0.210** (0.014)

1.764 (1.239)

Bachelor

6.865** (0.185)

0.405** (0.010)

0.405** (0.010)

0.405** (0.010)

0.378** (0.014)

0.378** (0.014)

−0.769 (1.228)

−1.186 (1.239)

0.064** (0.021)

0.063** (0.021)

0.066** (0.021)

0.066** (0.021)

0.059 (0.613)

0.078 (0.612)

−0.633 (0.819)

0.604 (0.819)

−0.095 (0.945)

98.54 (0.00)

100.30 (0.00)

51.42 (0.00)

53.04 (0.00)

36.72 (0.00)

4.12 (0.02)

7.15 (0.00)

6.43 (0.00)

Female × Bachelor Intercept

1.884 (0.897)

1.856** (0.053)

0.128 (0.177)

F-statistic and p-values on joint hypotheses (a) F-statistic on terms involving Age (b) Interaction terms with Age and Age2 SER R

2

7.884

0.457

0.457

0.457

0.457

0.456

0.456

0.456

0.1897

0.1921

0.1924

0.1929

0.1937

0.1943

0.1950

0.1959

Significant at the *5% and **1% significance level.

Data from 1992

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

ln(AHE)

ln(AHE)

ln(AHE)

ln(AHE)

ln(AHE)

0.157** (0.041)

0.156** (0.041)

0.120* (0.057)

0.138* (0.054)

0.104 (0.065)

−0.0022* * (0.0006)

−0.0022* * (0.0007)

−0.0015 (0.0010)

−0.0020* (0.0009)

−0.0013* (0.0011)

Dependent Variable

Age

AHE

ln(AHE)

0.461** (0.028)

0.027** (0.002)

ln(AHE)

Age2

ln(Age)

0.786** (0.052)

Female × Age

0.088 (0.083)

0.077 (0.083)

Female × Age2

−0.0017 (0.0013)

−0.0016 (0.0014)

Bachelor × Age

0.037 (0.084)

0.046 (0.083)

Bachelor × Age2

−0.0004 (0.0014)

−0.0006 (0.0014)

Female

−2.698** (0.152)

−0.167** (0.010)

−0.167** (0.010)

−0.167** (0.010)

−0.200** (0.013)

−1.273** (1.212)

−0.200** (0.013)

−1.102 (1.213)

Bachelor

5.903** (0.169)

0.377** (0.010)

0.377** (0.010)

0.377** (0.010)

0.340** (0.014)

0.340** (0.014)

−0.365** (1.227)

−0.504 (1.226)

0.085** (0.020)

0.079** (0.020)

0.086** (0.020)

0.080** (0.02)

−0.136 (0.608)

−0.119 (0.608)

0.306 (0.828)

0.209 (0.780)

0.617 (0.959)

115.93 (0.00)

118.89 (0.00)

62.51 (0.00)

65.17 (0.00)

45.71 (0.00)

9.04 (0.00)

4.80 (0.01)

7.26 (0.00)

Female × Bachelor Intercept

0.815 (0.815)

1.776** (0.054)

−0.099 (0.178)

F-statistic and p-values on joint hypotheses (a) F-statistic on terms involving Age (b) Interaction terms with Age and Age2 SER R

2

6.716

0.437

0.437

0.437

0.437

0.436

0.436

0.436

0.1946

0.1832

0.1836

0.1841

0.1858

0.1875

0.1866

0.1883

Significant at the *5% and **1% significance level.

(a) (1) Omitted variables: There is the potential for omitted variable bias when a variable is excluded from the regression that (i) has an effect on ln(AHE) and (ii) is correlated with a variable that is included in the regression. There are several candidates. The most important is a worker’s Ability. Higher ability workers will, on average, have higher earnings and are more likely to go to college. Leaving Ability out of the regression may lead to omitted variable bias, particularly for the estimated effect of education on earnings. Also omitted from the regression is Occupation. Two workers with the same education (a BA for example) may have different occupations (accountant versus 3rd grade teacher) and have different earnings. To the extent that occupation choice is correlated with gender, this will lead to omitted variable bias. Occupation choice could also be correlated with Age. Because the data are a cross section, older workers entered the labor force before younger workers (35 year-olds in the sample were born in 1969, while 25 year-olds were born in 1979), and their occupation reflects, in part, the state of the labor market when they entered the labor force. (2) Misspecification of the functional form: This was investigated carefully in exercise 8.1. There does appear to be a nonlinear effect of Age on earnings, which is adequately captured by the polynomial regression with interaction terms. (3) Errors-in-variables: Age is included in the regression as a “proxy” for experience. Workers with more experience are expected to earn more because their productivity increases with experience. But Age is an imperfect measure of experience. (One worker might start his career at age 22, while another might start at age 25. Or, one worker might take a year off to start a family, while another might not). There is also potential measurement error in AHE as these data are collected by retrospective survey in which workers in March 2005 are asked about their average earnings in 2004. (4) Sample selection: The data are full-time workers only, so there is potential for sampleselection bias. (5) Simultaneous causality: This is unlikely to be a problem. It is unlikely that AHE affects Age or gender. (6) Inconsistency of OLS standard errors: Heteroskedastic robust standard errors were used in the analysis, so that heteroskedasticity is not a concern. The data are collected, at least approximately, using i.i.d. sampling, so that correlation across the errors is unlikely to be a problem. (b) Results for 1988 are shown in the table above. Using results from (8), several conclusions were reached in E8.1(l) using the data from 2004. These are summarized in the table below, and are followed by a similar table for the 1998 data. Results using (8) from the 2004 Data

Gender, Education

Predicted Value of ln(AHE) at Age

Predicted Increase in ln(AHE) (Percent per year)

25

32

35

25 to 32

32 to 35

Females, High School

2.32

2.41

2.44

1.2%

0.8%

Males, High School

2.46

2.65

2.67

2.8%

0.5%

Females, BA

2.68

2.89

2.93

3.0%

1.3%

Males, BA

2.74

3.06

3.09

4.6%

1.0%

Results using (8) from the 1998 Data

Gender, Education

Predicted Value of ln(AHE) at Age

Predicted Increase in ln(AHE) (Percent per year)

25

32

35

25 to 32

32 to 35

Females, High School

2.28

2.42

2.39

2.0

−0.9

Males, High School

2.42

2.65

2.70

3.2

1.9

Females, BA

2.64

2.86

2.86

3.3

−0.2

Males, BA

2.70

3.01

3.09

4.4

2.6

Based on the 2004 data E81.1(l) concluded: Earnings for those with a college education are higher than those with a high school degree, and earnings of the college educated increase more rapidly early in their careers (age 25–32). Earnings for men are higher than those of women, and earnings of men increase more rapidly early in their careers (age 25–32). For all categories of workers (men/women, high school/college) earnings increase more rapidly from age 25–32 than from 32–35. All of these conclusions continue to hold for the 1998 data (although the precise values for the differences change somewhat.)

E10.1

(1)

(2)

(3)

(4)

−0.443** (0.048)

−0.368** (0.035)

−0.0461* (0.019)

−0.0280 (0.017)

0.00161** (0.00018)

−0.00007 (0.00009)

0.0000760 (0.000090)

density

0.0267 (0.014)

−0.172** (0.085)

−0.0916 (0.076)

avginc

0.00121 (0.0073)

pop

0.0427** (0.0031)

0.0115 (0.0087)

−0.00475 (0.0079)

pb1064

0.0809** (0.020)

0.104** (0.018)

0.0292 (0.023)

pw1064

0.0312** (0.0097)

0.0409** (0.0051)

0.00925 (0.0079)

pm1029

0.00887 (0.012)

−0.0503** (0.0064)

0.0733** (0.016)

6.135** (0.019)

2.982** (0.61)

3.866** (0.38)

3.766** (0.47)

State Effects

No

No

Yes

Yes

Time Effects

No

No

No

Yes

shall incar_rate

Intercept

−0.00920 (0.0059)

0.000959 (0.0064)

F-Statistics and p-values testing exclusion of groups of variables State Effects Time Effects

210.38 (0.00)

309.29 (0.00) 13.90 (0.00)

R2

0.09

0.56

0.94

0.95

(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) Attitudes towards guns and crime. Quality of schools. Quality of police and other crimeprevention 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 95% confidence interval for β Shall is now −0.086 to −0.007 or −0.7% to −8.6%. The state effects are jointly statistically significant, so this regression seems better specified than (2). (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) shall

(1)

(2)

(3)

(4)

−0.773** (0.070)

−0.529** (0.051)

−0.008 (0.026)

0.027 (0.025)

F-Statistics and p-values testing exclusion of groups of variables State Effects

190.47 (0.00)

Time Effects

243.39 (0.00) 12.39 (0.00)

Dependent Variable = ln(mur) shall

−0.473** (0.049)

−0.313** (0.036)

−0.061* (0.027)

−0.015 (0.027)

F-Statistics and p-values testing exclusion of groups of variables State Effects Time Effects

88.22 (0.00)

106.69 (0.00) 9.73 (0.00)

The quantative 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 β Shall is +1% to −6.6%. This includes β Shall = 0. Thus, there is no statistically significant evidence that concealed weapons laws have any effect on crime rates. The interval is wide, however, and includes values as large as −6.6%. Thus, at a 5% level the hypothesis that β Shall = −0.066 (so that the laws reduce crime by 6.6%) cannot be rejected.

E10.2

Regressor

(1)

(2)

(3)

sb_useage

0.00407*** (0.0012)

−0.00577*** (0.0012)

−0.00372*** (0.0011)

speed65

0.000148 (0.00041)

−0.000425 (0.00033)

−0.000783* (0.00042)

speed70

0.00240*** (0.00047)

0.00123*** (0.00033)

0.000804** (0.00034)

ba08

−0.00192*** (0.00036)

−0.00138*** (0.00037)

−0.000822** (0.00035)

drinkage21

0.0000799 (0.00099)

0.000745 (0.00051)

−0.00113** (0.00054)

lninc

−0.0181*** (0.0011)

−0.0135*** (0.0014)

0.00626 (0.0039)

age

−0.00000722 (0.00016)

0.000979** (0.00038)

0.00132*** (0.00038)

State Effects

No

Yes

Yes

Year Effects

No

No

Yes

0.544

0.874

0.897

(a) The estimated coefficient on seat belt useage is positive and statistically significant. One the face of it, this suggests that seat belt useage leads to an increase in the fatality rate. (b) The results change. The coefficient on seat belt useage is now negative and the coefficient is statistically significant. The estimated value of β SB = −0.00577, so that a 10% increase in seat belt useage (so that sb_useage increases by 0.10) is estimated to lower the fatality rate by . 000577 fatalities per million traffic miles. States with more dangerous drving conditions (and a higher fatality rate) also have more people wearing seat belts. Thus (1) suffers from omitted variable bias. (c) The results change. The estimated value of β SB = −0.00372. (d) The time effects are statistically significant − the F-statistic = 10.91 with a p-value of 0.00. The results in (3) are the most reliable. (e) A 38% increase in seat belt useage from 0.52 to 0.90 is estimated to lower the fatality rate by 0.00372 × 0.38 = 0.0014 fatalities per million traffic miles. The average number of traffic miles per year per state in the sample is 41,447. For a state with the average number of traffic miles, the number of fatalities prevented is 0.0014 × 41,447 = 58 fatalities. (f) A regression yields á _ useage = 0.206 × primary + 0.109 × secondary + sb (0.021)

(0.011)

(speed65, speed70, ba08, drinkage21, logincome, age, time effects, state effects) where the coefficients on the other regressors are not reported to save space. The coefficients on primary and secondary are positive and significant. Primary enforcement is estimated to increase seat belt useage by 20.6% and secondary enforcement is estimated to increase seat belt useage by 10.9%. (g) This results in an estimated increase in seatbelt useage of 0.206−0.109 = 0.094 or 9.4% from (f). This is predicted to reduce the fatality rate by 0.00372 × 0.094 = 0.00035 fatalities per million traffic miles. The data set shows that there were 63,000 million traffic miles in 1997 in New Jersey, the last year for which data is available. Assuming the same number of traffic miles in 2000 yields 0.00035 × 63,000 = 22 lives saved.

E11.1 (1)

(2)

(3)

Linear Probabili ty

Linear Probability

Probit

−0.078** (0.009)

−0.047** (0.009)

−0.159** (0.029)

Age

0.0097** (0.0018)

0.035** (0.007)

Age2

−0.00013** (0.00002)

−0.00047* * (0.00008)

Hsdrop

0.323** (0.019)

1.142** (0.072)

Hsgrad

0.233** (0.013)

0.883** (0.060)

Colsome

0.164** (0.013)

0.677** (0.061)

Colgrad

0.045** (0.012)

0.235** (0.065)

Black

−0.028 (0.016)

−0.084 (0.053)

Hispanic

−0.105** (0.014)

−0.338** (0.048)

Female

−0.033** (0.009)

−0.112** (0.028)

−0.014 (0.041)

−1.735** (0.053)

Smkban

Intercept

F-statistic and p-values on joint hypotheses Education indicators Significant at the 5% * or 1% ** level.

140.09 (0.00)

464.90 (0.00)

(a) Estimated probability of smoking (mean of smoker)

All Workers No Smoking Ban Smoking Ban

pˆ 0.242 0.290 0.212

SE ( pˆ ) 0.004 0.007 0.005

(b) From model (1), the difference in −0.078 we a standard error of 0.009. The resulting t-statistic is −8.66, so the coefficient is statistically significant. (c) From model (2) the estimated difference is −0.047, smaller than the effect in model (1). Evidently (1) suffers from omitted variable bias in (1). That is, smkban may be correlated with the education/race/gender indicators or with age. For example, workers with a college degree are more likely to work in an office with a smoking ban than high-school dropouts, and college graduates are less likely to smoke than high-school dropouts. (d) The t-statistic is −5.27, so the coefficient is statistically significant at the 1% level. (e) The F-statistic has a p-value of 0.00, so the coefficients are significant. The omitted education status is “Masters degree or higher”. Thus the coefficients show the increase in probability relative to someone with a postgraduate degree. For example, the coefficient on Colgrad is 0.045, so the probability of smoking for a college graduate is 0.045 (4.5%) higher than for someone with a postgraduate degree. Similarly, the coefficient on HSdrop is 0.323, so the probability of smoking for a college graduate is 0.323 (32.3%) higher than for someone with a postgraduate degree. Because the coefficients are all positive and get smaller as educational attainment increases, the probability of smoking falls as educational attainment increases. (f) The coefficient on Age2 is statistically significant. This suggests a nonlinear relationship between age and the probability of smoking. The figure below shows the estimated probability for a white, non-Hispanic male college graduate with no workplace smoking ban.

E11.2 (a) (b) (c) (d)

See the table above. The t-statistic is −5.47, very similar to the value for the linear probability model. The F-statistic is significant at the 1% level, as in the linear probability model. To calculate the probabilities, take the estimation results from the probit model to calculate zö, and calculate the cumulative standard normal distribution at zö, i.e., Prob( smoke) = Φ ( zö). The probability of Mr. A smoking without the workplace ban is 0.464 and the probability of smoking with the workplace bans is 0.401. Therefore the workplace bans would reduce the probability of smoking by 0.063 (6.3%). (e) To calculate the probabilities, take the estimation results from the probit model to calculate zö, and calculate the cumulative standard normal distribution at zö, i.e., Prob( smoke) = Φ ( zö). The probability of Ms. B smoking without the workplace ban is 0.143 and the probability of smoking with the workplace ban is 0.110. Therefore the workplace bans would reduce the probability of smoking by .033 (3.3%). (f) For Mr. A, the probability of smoking without the workplace ban is 0.449 and the probability of smoking with the workplace ban is 0.402. Therefore the workplace ban would have a considerable impact on the probability that Mr. A would smoke. For Ms. B, the probability of smoking without the workplace ban is 0.145 and the probability of smoking with the workplace ban is 0.098. In both cases the probability of smoking declines by 0.047 or 4.7%. (Notice that this is given by the coefficient on smkban, −0.047, in the linear probability model.) (g) The linear probability model assumes that the marginal impact of workplace smoking bans on the probability of an individual smoking is not dependent on the other characteristics of the individual. On the other hand, the probit model’s predicted marginal impact of workplace smoking bans on the probability of smoking depends on individual characteristics. Therefore, in the linear probability model, the marginal impact of workplace smoking bans is the same for Mr. A and Mr. B, although their profiles would suggest that Mr. A has a higher probability of smoking based on his characteristics. Looking at the probit model’s results, the marginal impact of workplace smoking bans on the odds of smoking are different for Mr. A and Ms. B, because their different characteristics are incorporated into the impact of the laws on the probability of smoking. In this sense the probit model is likely more appropriate. Are the impacts of workplace smoking bans “large” in a real-world sense? Most people might believe the impacts are large. For example, in (d) the reduction on the probability is 6.3%. Applied to a large number of people, this translates into a 6.3% reduction in the number of people smoking. (h) An important concern is two-way causality. Do companies that impose a smoking ban have fewer smokers to begin with? Do smokers seek employment with employers that do not have a smoking ban? Do states with smoking bans already have more or fewer smokers than states without smoking bans?

E12.1 1.

This table shows the OLS and 2SLS estimates. Values for the intercept and coefficients on Seas are not shown. Regressor ln(Price) Ice Seas and intercept

OLS −0.639 (0.073) 0.448 (0.135) Not Shown

First Stage F-statistic

2SLS −0.867 (0.134) 0.423 (0.135) Not Shown 183.0

(a) See column the table above. The estimated elasticity is −0.639 with a standard error of 0.073. (b) A positive demand “error” will shift the demand curve to the right. This will increase the equilibrium quantity and price in the market. Thus ln(Price) is positively correlated with the regression error in the demand model. This means that the OLS coefficient will be positively biased. (c) Cartel shifts the supply curve. As the cartel strengthens, the supply curve shifts in, reducing supply and increasing price and profits for the cartel’s members. Thus, Cartel is relevant. For Cartel to be a valid instrument it must be exogenous, that is, it must be unrelated to the factors affecting demand that are omitted from the demand specification (i.e., those factors that make up the error in the demand model.) This seems plausible. (d) The first stage F-statistic is 183.0. Cartel is not a weak instrument. (e) See the table. The estimated elasticity is −0.867 with a standard error of 0.134. Notice that the estimate is “more negative” than the OLS estimate, which is consistent with the OLS estimator having a positive bias. (f) In the standard model of monopoly, a monopolist should increase price if the demand elasticity is less than 1. (The increase in price will reduce quantity but increase revenue and profits.) Here, the elasticity is less than 1.

.do file clear set memory 35m infile using "/Users/blah/Desktop/default-441bab20263c8.dct" rename R0000100 ID rename R0017300 HighGrade rename R0153000 Rot1A rename R0153100 Rot1B rename R0153200 Rot2A rename R0153300 Rot2B rename R0153400 Rot3A rename R0153500 Rot3B rename R0153600 Rot4A rename R0153700 Rot4B rename R0214700 Race rename R0214800 Sex

tab HighGrade replace HighGrade= . if HighGrade==-4 replace HighGrade= . if HighGrade==-3 replace HighGrade= . if HighGrade==-2 replace HighGrade= . if HighGrade==95 tab Rot1A replace Rot1A= . if Rot1A==-3 replace Rot1A= . if Rot1A==-2 replace Rot1A= . if Rot1A==-1 tab Rot1B replace Rot1B= . if Rot1B==-3 replace Rot1B= . if Rot1B==-2 replace Rot1B= . if Rot1B==-1 tab Rot2A replace Rot2A= . if Rot2A==-3 replace Rot2A= . if Rot2A==-2 replace Rot2A= . if Rot2A==-1 tab Rot2B replace Rot2B= . if Rot2B==-3 replace Rot2B= . if Rot2B==-2 replace Rot2B= . if Rot2B==-1 tab Rot3A replace Rot3A= . if Rot3A==-3 replace Rot3A= . if Rot3A==-2 replace Rot3A= . if Rot3A==-1 tab Rot3B replace Rot3B= . if Rot3B==-3 replace Rot3B= . if Rot3B==-2 replace Rot3B= . if Rot3B==-1 tab Rot4A replace Rot4A= . if Rot4A==-3 replace Rot4A= . if Rot4A==-2 replace Rot4A= . if Rot4A==-1 tab Rot4B replace Rot4B= . if Rot4B==-3 replace Rot4B= . if Rot4B==-2 replace Rot4B= . if Rot4B==-1 tab Race tab Sex tab Race, gen(racedum) rename racedum1 Hispanics rename racedum2 Blacks rename racedum3 NonBlackHispanic tab HighGrade gen HSLevel=1 if HighGrade<=8 replace HSLevel=2 if HighGrade<8 & HighGrade<=12 replace HSLevel=3 if HighGrade>12 & HighGrade ~=. gen RotScore=Rot1A+Rot1B+Rot2A+Rot2B+Rot3A+Rot3B+Rot4A+Rot4B drop Rot1A Rot1B Rot2A Rot2B Rot3A Rot3B Rot4A Rot4B gen lnRotScore=ln(RotScore) egen RotHSLevel=mean(RotScore), by(HSLevel) gen Rot60=pctile(RotScore), p(60) egen RaceSexMed=median(RotScore), by(Race Sex)