ECON4150 - Introductory Econometrics Seminar 4 Stock and Watson Chapter 8
empirical exercise E8.2: Data 2 In this exercise we use the data set CPS12.dta Each month the Bureau of Labor Statistics in the U.S. Department of Labor conducts the Current Population Survey (CPS), which provides data on labor force characteristics of the population, including the level of employment, unemployment, and earnings. Approximately 65,000 randomly selected U.S. households are surveyed each month. The file CPS12.dta contains the data for 2012 (from the March 2013 survey). These data are for full-time workers, defined as workers employed more than 35 hours per week for at least 48 weeks in the previous year. Data are provided for workers whose highest educational achievement is (1) a high school diploma, and (2) a bachelor s degree.
empirical exercise E8.2: Data 3 Series in Data Set: FEMALE: 1 if female; 0 if male YEAR: Year AHE : Average Hourly Earnings BACHELOR: 1 if worker has a bachelor s degree; 0 if worker has a high school degree
empirical exercise E8.2: Data 4 Tuesday March 3 14:58:52 2015 Page 1 (R) / / / / / / / / / / / / Statistics/Data Analysis 1. sum Variable Obs Mean Std. Dev. Min Max year 7440 2012 0 2012 2012 ahe 7440 19.80026 10.68632 2.136752 91.45602 bachelor 7440.531586.4990349 0 1 female 7440.424328.4942738 0 1 age 7440 29.64772 2.839661 25 34
Tuesday March 3 15:01:07 2015 Page 1 empirical exercise E8.2 part a) (R) / / / / / / / / / / / / Statistics/Data Analysis 5 1. regress ahe age female bachelor, robust Linear regression Number of obs = 7440 F( 3, 7436) = 539.54 Prob > F = 0.0000 R-squared = 0.1801 Root MSE = 9.6782 Robust ahe Coef. Std. Err. t P> t [95% Conf. Interval] age.510286.0395409 12.91 0.000.4327747.5877973 female -3.810305.2239148-17.02 0.000-4.249241-3.371368 bachelor 8.318628.2237329 37.18 0.000 7.880048 8.757208 _cons 1.866198 1.175373 1.59 0.112 -.4378656 4.170261 If age increases from 25 to 26, earnings are expected to increase by $0.51 per hour. If age increases from 33 to 34, earnings are expected to increase by $0.51 per hour.
empirical exercise E8.2 part b) (R) 6 / / / / / / / / / / / / Statistics/Data Analysis 1. gen lnahe=ln(ahe) 2. regress lnahe age female bachelor, robust Linear regression Number of obs = 7440 F( 3, 7436) = 623.31 Prob > F = 0.0000 R-squared = 0.1964 Root MSE =.47823 Robust lnahe Coef. Std. Err. t P> t [95% Conf. Interval] age.0255179.0019619 13.01 0.000.0216721.0293637 female -.1923376.0112614-17.08 0.000 -.2144132 -.170262 bachelor.4377833.0112003 39.09 0.000.4158275.4597391 _cons 1.941423.0590018 32.90 0.000 1.825763 2.057083 If age increases from 25 to 26, earnings are expected to increase by 2.55%. If age increases from 33 to 34, earnings are expected to increase by 2.55%. These values, in percentage terms, are the same because the regression is a linear function relating ln(ahe) and age.
Tuesday March 3 15:11:25 2015 Page 1 empirical exercise E8.2 part c) (R) / / / / / / / / / / / / Statistics/Data Analysis 7 1. gen lnage=ln(age) 2. regress lnahe lnage female bachelor, robust Linear regression Number of obs = 7440 F( 3, 7436) = 624.31 Prob > F = 0.0000 R-squared = 0.1966 Root MSE =.47817 Robust lnahe Coef. Std. Err. t P> t [95% Conf. Interval] lnage.7529407.0576153 13.07 0.000.6399984.8658831 female -.1923558.0112593-17.08 0.000 -.2144271 -.1702844 bachelor.4376637.0111993 39.08 0.000.4157099.4596175 _cons.1495318.1953385 0.77 0.444 -.2333871.5324506 ln(ahe) = 0.75 (ln(26) ln(25)) = 0.029 so if age increases from 25 to 26, earnings is expected to increase by 2.9%. ln(ahe) = 0.75 (ln(34) ln(33)) = 0.022 so if age increases from 25 to 26, earnings is expected to increase by 2.2%.
empirical exercise E8.2 part d) 1. gen age2=age^2 2. regress lnahe age age2 female bachelor, robust (R) / / / / / / / / / / / / Statistics/Data Analysis 8 Linear regression Number of obs = 7440 F( 4, 7435) = 469.24 Prob > F = 0.0000 R-squared = 0.1967 Root MSE =.47816 Robust lnahe Coef. Std. Err. t P> t [95% Conf. Interval] age.1040449.0457314 2.28 0.023.0143984.1936913 age2 -.0013284.0007728-1.72 0.086 -.0028433.0001864 female -.1923983.0112589-17.09 0.000 -.214469 -.1703276 bachelor.4374121.0112096 39.02 0.000.4154381.4593862 _cons.7918819.6712609 1.18 0.238 -.5239795 2.107743 ln(ahe) = ( 0.104 26 0.0013 26 2 ) ( 0.104 25 0.0013 25 2) = 0.037 so if age increases from 25 to 26, earnings is expected to increase by 3.7%. ln(ahe) = ( 0.104 34 0.0013 34 2 ) ( 0.104 33 0.0013 33 2) = 0.017 so if age increases from 25 to 26, earnings is expected to increase by 1.7%.
empirical exercise E8.2 part e) 9 The regressions differ in their choice of one of the regressors. They can be compared on the basis of the adjusted R 2 Part b): display e(r2_a).19605996 Part c): display e(r2_a).19623914 Regression in part c) has a marginally higher adjusted R 2, but the differences are very small.
empirical exercise E8.2 part f) 10 The regression in (d) adds the variable age 2 to regression (b). The coefficient on age 2 is not statistically significant at the 5% level (t = 1.72<1.96) and the estimated coefficient is very close to zero. This suggests that (b) is preferred to (d) (note: conclusion would be different at 10% level)
empirical exercise E8.2 part g) 11 The regressions differ in their choice of the regressors (ln(age) in (c) and age and age 2 in (d)). They can be compared on the basis of the adjusted R 2. Part c): display e(r2_a).19623914 Part d) display e(r2_a).19627256 The regression in (d) has a (marginally) higher adjusted R 2, so it is slightly preferred.
empirical exercise E8.2 part h) 12 2.8 2.75 ln(ahe) 2.7 2.65 2.6 2.55 24 26 28 30 32 34 age part b) log-linear part d) log-quadratic part c) log-log The regression functions for women with a college degree will look just like these, but they will be shifted by the amount of the coefficient on the binary regressors female and bachelor.
Tuesday March 3 16:14:25 2015 Page 1 empirical exercise E8.2 part i) (R) / / / / / / / / / / / / Statistics/Data Analysis 13 1. gen interaction=female*bachelor 2. regress lnahe age age2 female bachelor interaction, robust Linear regression Number of obs = 7440 F( 5, 7434) = 382.92 Prob > F = 0.0000 R-squared = 0.1984 Root MSE =.4777 Robust lnahe Coef. Std. Err. t P> t [95% Conf. Interval] age.1043224.04568 2.28 0.022.0147767.1938682 age2 -.0013316.0007719-1.73 0.085 -.0028447.0001815 female -.2423732.0166376-14.57 0.000 -.2749877 -.2097587 bachelor.4004463.0148482 26.97 0.000.3713396.4295531 interaction.0898571.0225592 3.98 0.000.0456346.1340796 _cons.8037407.6706449 1.20 0.231 -.5109132 2.118395 The coefficient on the interaction term shows the difference between men and women in the association between having a bachelor degree and log hourly earnings.
empirical exercise E8.2 part i) 14 Predicted values of ln(ahe): Alexis: 0.104 30 0.0013 30 2 0.24 1 + 0.40 1 + 0.090 1 x + 0.80 = 3.00 Jane: 0.104 30 0.0013 30 2 0.24 1 + 0.40 0 + 0.090 0 1 + 0.80 = 2.51 Alexis-Jane= 3.00 2.51 = 0.49 Bob: 0.104 30 0.0013 30 2 0.24 0 + 0.40 1 + 0.090 1 0 + 0.80 = 3.15 Jim: 0.104 30 0.0013 30 2 0.24 0 + 0.40 0 + 0.090 0 0 + 0.80 = 2.75 Bob-Jim= 3.15 2.75 = 0.4
empirical exercise E8.2 part j) Tuesday March 3 16:41:27 2015 Page 1 15 (R) / / / / / / / / / / / / Statistics/Data Analysis 1. gen age_female=age*female 2. regress lnahe age female age_female bachelor, robust Linear regression Number of obs = 7440 F( 4, 7435) = 468.12 Prob > F = 0.0000 R-squared = 0.1968 Root MSE =.47814 Robust lnahe Coef. Std. Err. t P> t [95% Conf. Interval] age.0287246.0025862 11.11 0.000.0236548.0337944 female.0328354.1180552 0.28 0.781 -.1985862.2642569 age_female -.0075971.0039688-1.91 0.056 -.015377.0001828 bachelor.4375294.0111983 39.07 0.000.4155775.4594813 _cons 1.846351.0773751 23.86 0.000 1.694674 1.998028
empirical exercise E8.2 part k) Tuesday March 3 16:42:32 2015 Page 1 16 (R) / / / / / / / / / / / / Statistics/Data Analysis 1. gen age_bachelor=age*bachelor 2. regress lnahe age female age_bachelor bachelor, robust Linear regression Number of obs = 7440 F( 4, 7435) = 467.46 Prob > F = 0.0000 R-squared = 0.1965 Root MSE =.47824 Robust lnahe Coef. Std. Err. t P> t [95% Conf. Interval] age.0238619.002771 8.61 0.000.01843.0292939 female -.1922201.0112624-17.07 0.000 -.2142976 -.1701425 age_bachelor.0031672.0039178 0.81 0.419 -.0045128.0108472 bachelor.3438603.1166697 2.95 0.003.1151547.5725658 _cons 1.990507.0826508 24.08 0.000 1.828488 2.152526
empirical exercise E8.2 part l) 17 The estimated regressions suggest that earnings increase as workers age from 25 35, the range of age studied in this sample. Can we say more...?