ECON 4706 B Winter 2022 Solution Set 2

ECON 4706 B Winter 2022 Simon Power Solution Set 2: Total Marks = 130 1. [40 marks] Consider the Ramanathan US Wage Data (available in the file wage.dta): Data Set 7-2 from Ramu Ramanathan, Introductory Econometrics with Applications (4th Edition), 1998. Data on salaries and employment characteristics of 48 employees in a certain company (compiled by Susan Wong). WAGE = wage rate per month EDUC = years of education beyond eighth grade when hired EXPER = number of years at the company AGE = age of employees GENDER = 1 for males, 0 for females RACE = 1 for white, 0 for non-white CLERICAL = 1 for clerical workers, 0 for others MAINT = 1 for maintenance workers, 0 for others CRAFTS = 1 for crafts workers, 0 for others There is also a control group of professional workers. a) Is there any evidence of gender discrimination (against women) in the raw wage data? Explain. NOTE: The phrase “Is there any evidence …” imp lies that you should do an appropriate hypothesis test. Also, the phrase “the raw wage data” implies that the wage data is not being adjusted or controlled for any other variable(s). To determine whether there is any evidence of gender discrimination (against women) in the raw wage data, we need to estimate the model ???? 𝑖 = 𝛽 1 + 𝛽 2 ?????? 𝑖 + ? 𝑖 reg wage gender Source | SS df MS Number of obs = 48 -------------+---------------------------------- F(1, 46) = 10.54 Model | 3706980.8 1 3706980.8 Prob > F = 0.0022 Residual | 16175989.1 46 351651.937 R-squared = 0.1864 -------------+---------------------------------- Adj R-squared = 0.1688 Total | 19882969.9 47 423041.913 Root MSE = 593 ------------------------------------------------------------------------------ wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- gender | 557.7413 171.7826 3.25 0.002 211.9609 903.5217 _cons | 1529.182 126.4286 12.10 0.000 1274.694 1783.669 ------------------------------------------------------------------------------ and then test the null hypothesis ? 0 : 𝛽 2 = 0 against the alternative hypothesis ? 1 : 𝛽 2 > 0

2 In this case, we can either obtain the appropriate t- statistic directly from the “reg” output or calculate it from the formula ? = 𝛽 ̂ 2 − 0 ?. 𝑒. (𝛽 ̂ 2 ) = 557.7413 − 0 171.7826 ≈ 3.25 Comparing this t-statistic of 3.25 with the critical value of 1.679 (46 degrees of freedom in Table A.2), we conclude that we should reject the null hypothesis at the 5% significance level. This suggests that there is evidence of gender discrimination (against women) in the raw wage data. (5 marks) b) Is there any evidence of racial discrimination (against non-whites) in the raw wage data? Explain. To determine whether there is any evidence of racial discrimination (against non-whites) in the raw wage data, we need to estimate the model ???? 𝑖 = 𝛽 1 + 𝛽 2 ???? 𝑖 + ? 𝑖 reg wage race Source | SS df MS Number of obs = 48 -------------+---------------------------------- F(1, 46) = 1.82 Model | 756475.339 1 756475.339 Prob > F = 0.1840 Residual | 19126494.6 46 415793.36 R-squared = 0.0380 -------------+---------------------------------- Adj R-squared = 0.0171 Total | 19882969.9 47 423041.913 Root MSE = 644.82 ------------------------------------------------------------------------------ wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- race | 259.3111 192.2483 1.35 0.184 -127.6647 646.2869 _cons | 1669.222 151.9856 10.98 0.000 1363.291 1975.153 ------------------------------------------------------------------------------ and then test the null hypothesis ? 0 : 𝛽 2 = 0 against the alternative hypothesis ? 1 : 𝛽 2 > 0 In this case, we can either obtain the appropriate t- statistic directly from the “reg” output or calculate it from the formula ? = 𝛽 ̂ 2 − 0 ?. 𝑒. (𝛽 ̂ 2 ) = 259.3111 − 0 192.2483 ≈ 1.35 Comparing this t-statistic of 1.35 with the critical value of 1.679 (46 degrees of freedom in Table A.2), we conclude that we should not reject the null hypothesis at the 5% significance level. This suggests that there is no evidence of racial discrimination (against non-whites) in the raw wage data. (5 marks)

4 wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- _cons | 1831.292 93.87957 19.51 0.000 1642.43 2020.153 ------------------------------------------------------------------------------ Comparing this F-statistic of 18.40 with the critical value of 2.84 (3 and 40 degrees of freedom, closest to the correct 3 and 44 degrees of freedom, in Table A.3), we conclude that we should reject the null hypothesis at the 5% significance level. This suggests that there is evidence of a difference in wage across occupational categories in the raw wage data. (5 marks) d) Estimate a basic overall linear regression model for wage that controls for both age and number of years of education beyond eighth grade when hired, and takes gender, race, and occupational category into account. (Do NOT include any interaction terms or any non-linear terms.) For future reference, call this model “the basic model”. Copy and paste the results into your assignment. reg wage age educ gender race clerical maint crafts Source | SS df MS Number of obs = 48 -------------+---------------------------------- F(7, 40) = 12.41 Model | 13615270.3 7 1945038.62 Prob > F = 0.0000 Residual | 6267699.58 40 156692.49 R-squared = 0.6848 -------------+---------------------------------- Adj R-squared = 0.6296 Total | 19882969.9 47 423041.913 Root MSE = 395.84 ------------------------------------------------------------------------------ wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | -1.5585 5.954216 -0.26 0.795 -13.59242 10.47542 educ | 34.90691 31.16176 1.12 0.269 -28.07336 97.88718 gender | 376.6267 176.9503 2.13 0.040 18.99671 734.2566 race | 249.327 146.7225 1.70 0.097 -47.2103 545.8643 clerical | -1108.715 185.7291 -5.97 0.000 -1484.087 -733.3424 maint | -962.76 224.0082 -4.30 0.000 -1415.498 -510.0225 crafts | -623.39 203.7489 -3.06 0.004 -1035.182 -211.5981 _cons | 2086.752 374.2348 5.58 0.000 1330.396 2843.109 ------------------------------------------------------------------------------ (5 marks) e) Is there any evidence of gender discrimination (against women) in the basic model? Explain. To determine whether there is any evidence of gender discrimination (against women) in the basic model, we need to test the null hypothesis ? 0 : 𝛽 4 = 0 against the alternative hypothesis ? 1 : 𝛽 4 > 0 . Comparing the t- statistic of 2.13, taken from the “reg” output, w ith the critical value of 1.684 (40 degrees of freedom in Table A.2), we conclude that we should reject the null hypothesis at the 5% significance level. This suggests that there is evidence of gender discrimination (against women) in the basic model. (5 marks) f) Is there any evidence of racial discrimination (against non-whites) in the basic model

5 To determine whether there is any evidence of racial discrimination (against non-whites) in the basic model, we need to test the null hypothesis ? 0 : 𝛽 5 = 0 against the alternative hypothesis ? 1 : 𝛽 5 > 0 . Comparing the t- statistic of 1.70, taken from the “reg” output, with the critical value of 1.684 (40 degrees of freedom in Table A.2), we conclude that we should reject the null hypothesis at the 5% significance level. This suggests that there is evidence of racial discrimination (against non- whites) in the basic model. (5 marks) NOTE: You may NOT use the STATA “test” command or the STATA “lincom” command for parts g) and h). g) Is there any evidence of a difference in wage across occupational categories in the basic model? Copy and paste any additional STATA output that you may require into your assignment. To determine whether there is any evidence of a difference in wage across occupational categories in the basic model, we need to test the null hypothesis ? 0 : 𝛽 6 = 𝛽 7 = 𝛽 8 = 0 against the alternative hypothesis ? 1 : ??ℎ𝑒?𝑤𝑖?𝑒 . In this case, we can calculate the appropriate F-statistic from the formula ? = (???? − ????) ? ⁄ ???? (? − ?) ⁄ = (12935853.3 − 6267699.58) 3 ⁄ 6267699.58 (48 − 8) ⁄ ≈ 14.19 where, as usual, ???? denotes the ??? from the restricted model and ???? denotes the ??? from the original or unrestricted model. NOTE: To obtain ???? , I just used the ??? from the “reg” output for the unrestricted or basic model above. And, to obtain ???? , I just used the ??? from the following “reg” output for the restricted model . reg wage age educ gender race Source | SS df MS Number of obs = 48 -------------+---------------------------------- F(4, 43) = 5.77 Model | 6947116.64 4 1736779.16 Prob > F = 0.0008 Residual | 12935853.3 43 300833.797 R-squared = 0.3494 -------------+---------------------------------- Adj R-squared = 0.2889 Total | 19882969.9 47 423041.913 Root MSE = 548.48 ------------------------------------------------------------------------------ wage | Coefficient Std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- age | .9081983 8.135893 0.11 0.912 -15.49939 17.31579 educ | 99.43782 38.52316 2.58 0.013 21.74847 177.1272 gender | 547.4983 164.3424 3.33 0.002 216.0703 878.9264 race | 96.88581 188.3884 0.51 0.610 -283.0356 476.8073 _cons | 816.0308 429.7004 1.90 0.064 -50.54264 1682.604

7 crafts | 572.247 662.3621 0.86 0.393 -769.826 1914.32 educcler | -194.6684 72.22824 -2.70 0.011 -341.0168 -48.32012 educmain | -149.1496 157.2879 -0.95 0.349 -467.8451 169.5459 educcraf | -144.9551 81.00859 -1.79 0.082 -309.0941 19.18393 _cons | 993.7299 582.7079 1.71 0.097 -186.9485 2174.408 ------------------------------------------------------------------------------ And, to obtain ???? , I just used the ??? from the “reg” output for the basic model above. Comparing this F-statistic of 2.43 with the critical value of 2.87 (3 and 35 degrees of freedom, closest to the correct 3 and 37, in Table A.3), we conclude that we should not reject the null hypothesis at the 5% significance level. This suggests that a set of interaction terms between number of years of education beyond eighth grade when hired and occupational category should not be added to the basic model. (5 marks) 2. [20 marks] Consider the following extension of the semi-log wage equation model considered on pp. 206-208: ?????? 𝑖 = 𝛽 1 + 𝛽 2 ? 𝑖 + 𝛽 3 ??? 𝑖 + 𝛽 4 ???? 𝑖 + ? 𝑖 𝑖 = 1, 2, … , ? NOTE: Be sure carefully to read the definitions of the variables given in Appendix B (see pp. 565-569). Using the EAWE11.dta dataset: a) Estimate this model using OLS and then copy and paste the results into your assignment. . gen LGEARN = ln(EARNINGS) . reg LGEARN S EXP MALE Source | SS df MS Number of obs = 500 -------------+---------------------------------- F(3, 496) = 43.47 Model | 28.3950753 3 9.46502509 Prob > F = 0.0000 Residual | 107.997755 496 .217737409 R-squared = 0.2082 -------------+---------------------------------- Adj R-squared = 0.2034 Total | 136.39283 499 .273332325 Root MSE = .46662 ------------------------------------------------------------------------------ LGEARN | Coefficient Std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- S | .1018833 .0091293 11.16 0.000 .0839464 .1198202 EXP | .0505384 .008711 5.80 0.000 .0334234 .0676534 MALE | .1631964 .0422056 3.87 0.000 .0802725 .2461202 _cons | .8880866 .1758592 5.05 0.000 .5425658 1.233607 ------------------------------------------------------------------------------ (5 marks) b) Carefully interpret the values of the two estimated slope parameters associated with the two continuous explanatory variables using the “more accurate” method discussed on p. 207.

8 The more accurate method of interpreting the values of the two estimated slope parameters associated with the two continuous explanatory variables, suggests that i) the change in EARNINGS resulting from a one-year increase in the number of years of schooling completed (holding the number of years of work experience and gender constant) would be approximately 10.7%, based on 𝑒 𝛽 ̂ 2 = 𝑒 0.1018833 ≈ 1.107 and ii) the change in EARNINGS resulting from a one-year increase in the number of years of work experience (holding the number of years of schooling completed and gender constant) would be approximately 5.2%, based on 𝑒 𝛽 ̂ 3 = 𝑒 0.0505384 ≈ 1.052 (5 marks) c) Use the estimated slope parameter associated with the dummy variable to obtain both an approximate measure AND a more accurate measure of the proportional difference in male earnings as compared to female earnings using the method outlined in Box 5.1 on p. 235. NOTE: The re is an important typo in Box 5.1. The sentence that begins “If 𝛿 is small, 𝑒 is approximately equal to (1 + 𝛿) , implying …” should begin as follows “If 𝛿 is small, 𝑒 𝛿 is approximately equal to (1 + 𝛿) , implying …” . The estimated slope parameter associated with the (dummy variable) MALE implies that the proportional change in EARNINGS, holding other things constant, for being male, rather than being female, is approximately 𝛽 ̂ 4 = 0.1632 , or in percentage terms, a percentage change in EARNINGS of approximately 16.32%. And that, using the more accurate method of evaluating the estimated slope parameter, namely, using the formula (𝑒 𝛿 − 1) , would imply that the proportional change in EARNINGS, holding other things constant, for being male, rather than being female, would be approximately 0.1773 (based on (𝑒 𝛽 ̂ 4 − 1) = (𝑒 0.1631964 − 1) ≈ 0.1773 ), or in percentage terms, a percentage change in EARNINGS of approximately 17.73%. (10 marks) 3. [60 marks] Consider the following variation on the wage equation model discussed in the textbook: ???????? 𝑖 = 𝛽 1 + 𝛽 2 ? 𝑖 + 𝛽 3 ?????? 𝑖 + 𝛽 4 ????? 𝑖 + 𝛽 5 ???? 𝑖 + ? 𝑖 𝑖 = 1, 2, … , ? NOTE: Be sure carefully to read the definitions of the variables given in Appendix B (see pp. 565-569).

10 years with current employer as reported at the 2011 interview), and URBAN (whether living in an urban area or not as reported at the 2011 interview) constant. (10 marks) c) Are the signs of the estimated slope parameters in line with your prior expectations? Explain. The positive sign of the estimated slope parameters associated with both S and TENURE are in line with my prior expectations. They are both positive and this makes sense, the higher the number of years of education completed, other things being equal, the more productive the respondent should be, and the longer that the respondent has held a job, other things being equal, the higher that earnings tend to be, because of increases in productivity due to learning by doing and/or additional pay for “seniority”. The negative sign of the estimated slope parameter associated with URBAN is not in line with my prior expectations. In general, I would expect that, other things being equal, pay would be higher in an urban setting, due to increased competition for workers. The negative sign of the estimated slope parameter associated with JOBS is in line with my prior expectations. This makes sense, the higher the number of jobs that an individual has had, other things being equal, the more an employer might begin to worry about things like reliability. (10 marks) d) Are the estimated slope parameters individually statistically significant? Explain. At the 5% significance level, the estimated slope parameters associated with the S and TENURE explanatory variables are individually statistically significant (because their p-values are less than 0.05), while the estimated slope parameters associated with the URBAN and JOBS explanatory variables are not individually statistically significant (because their p-values are greater than 0.05). (5 marks) NOTE: You may NOT use the STATA “test” command or the STATA “lincom” command for part e). e) Conduct a suitable (joint) F-test to test whether the two explanatory variables URBAN and JOBS belong in this model. (Be sure to include a clear statement of the appropriate null and alternative hypotheses, the formula for the test statistic, and all of the necessary calculations for your test in your answer.) What do you conclude? Explain your reasoning and copy and paste any additional STATA output that you may require into your assignment. To determine whether the two explanatory variables URBAN and JOBS belong in this model, we need to test the null hypothesis ? 0 : 𝛽 4 = 𝛽 5 = 0 against the alternative hypothesis ? 1 : ??ℎ𝑒?𝑤𝑖?𝑒 . In this case, we can calculate the appropriate F-statistic from the formula

11 ? = (???? − ????) ? ⁄ ???? (? − ?) ⁄ = (45895.6608 − 45556.6257) 2 ⁄ 45556.6257 (485 − 5) ⁄ ≈ 1.79 where, as usual, ???? denotes the ??? from the restricted model and ???? denotes the ??? from the original or unrestricted model. NOTE: To obtain ???? , I just used the ??? from the “reg” output for the unrestricted or basic model above. And, to obtain ???? , used the ??? from the following “reg” output for the restricted model . reg EARNINGS S TENURE if JOBS < . Source | SS df MS Number of obs = 485 -------------+---------------------------------- F(2, 482) = 36.11 Model | 6875.91536 2 3437.95768 Prob > F = 0.0000 Residual | 45895.6608 482 95.2192134 R-squared = 0.1303 -------------+---------------------------------- Adj R-squared = 0.1267 Total | 52771.5762 484 109.032182 Root MSE = 9.758 ------------------------------------------------------------------------------ EARNINGS | Coefficient Std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- S | 1.267805 .1639093 7.73 0.000 .9457397 1.58987 TENURE | .5464383 .160837 3.40 0.001 .23041 .8624667 _cons | -1.776567 2.482654 -0.72 0.475 -6.654729 3.101595 ------------------------------------------------------------------------------ NOTE: For the estimation of the restricted model, I had to explicitly tell STATA to drop the 15 observations for which there were missing observations for the JOBS variable, by using the qualifier “if JOBS <.” on the “reg” command . This ensured that we were comparing apples and apples, that is, two regressions with the same number of observations. Comparing this F-statistic of 1.79 with the critical value of 3.01 (2 and 500 degrees of freedom, closest to the correct 2 and 480 degrees of freedom, in Table A.3), we conclude that we should not reject the null hypothesis at the 5% significance level. This suggests that the URBAN and JOBS variables probably do not belong in this model. (10 marks) NOTE: You may NOT use the STATA “test” command or the STATA “lincom” command or the STATA “ovtest” command for part f). f) Perform a RESET test, exactly as described on p. 222, on this model. (Be sure to include a clear statement of the appropriate null and alternative hypotheses, the formula for the test statistic, and the necessary calculations for your test in your answer.) What do you conclude? Explain your reasoning and copy and paste any additional STATA output that you may require into your assignment. We need to test the null hypothesis that the parameter associated with the squared predictions of the dependent variable from the original model is equal to zero in the appropriately augmented model, that is, we need to test the null hypothesis

13 g) Using the Box-Cox test discussed in class, determine whether a semi-log version of this model should be preferred to the original model. Explain your reasoning and copy and paste any additional STATA output that you may require into your assignment. First, we need to calculate the geometric mean of EARNINGS . means EARNINGS if JOBS < . Variable | Type Obs Mean [95% conf. interval] -------------+--------------------------------------------------------------- EARNINGS | Arithmetic 485 18.63367 17.70204 19.5653 | Geometric 485 16.33589 15.60169 17.10463 | Harmonic 485 14.19561 13.4058 15.08432 ----------------------------------------------------------------------------- NOTE: We have to be careful here, to calculate (and to use below) the geometric mean of EARNINGS for ONLY those observations that are included in the two auxiliary regressions. Next, we need to generate the new variables and run the two auxiliary regressions . gen EARNINGSstar = EARNINGS/16.33589 . gen lnEARNINGSstar = ln(EARNINGSstar) . reg EARNINGSstar S TENURE URBAN JOBS Source | SS df MS Number of obs = 485 -------------+---------------------------------- F(4, 480) = 19.00 Model | 27.036331 4 6.75908274 Prob > F = 0.0000 Residual | 170.712747 480 .355651556 R-squared = 0.1367 -------------+---------------------------------- Adj R-squared = 0.1295 Total | 197.749078 484 .408572475 Root MSE = .59637 ------------------------------------------------------------------------------ EARNINGSstar | Coefficient Std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- S | .0810544 .0102074 7.94 0.000 .0609976 .1011112 TENURE | .0247514 .0110629 2.24 0.026 .0030138 .046489 URBAN | -.0556144 .0657864 -0.85 0.398 -.1848793 .0736505 JOBS | -.0173799 .0103597 -1.68 0.094 -.0377358 .002976 _cons | .0159256 .1668365 0.10 0.924 -.3118945 .3437456 ------------------------------------------------------------------------------ . reg lnEARNINGSstar S TENURE URBAN JOBS Source | SS df MS Number of obs = 485 -------------+---------------------------------- F(4, 480) = 24.73 Model | 21.9707027 4 5.49267569 Prob > F = 0.0000 Residual | 106.600979 480 .222085373 R-squared = 0.1709 -------------+---------------------------------- Adj R-squared = 0.1640 Total | 128.571682 484 .26564397 Root MSE = .47126 ------------------------------------------------------------------------------ lnEARNINGS~r | Coefficient Std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- S | .0691218 .0080661 8.57 0.000 .0532725 .0849711 TENURE | .0290109 .0087421 3.32 0.001 .0118334 .0461884

14 URBAN | -.0805523 .0519857 -1.55 0.122 -.1826999 .0215953 JOBS | -.0149607 .0081864 -1.83 0.068 -.0310463 .0011249 _cons | -.961362 .1318374 -7.29 0.000 -1.220412 -.7023123 ------------------------------------------------------------------------------ Finally, we need to calculate the Box-Cox statistic ?? = ? 2 ?? ( ??? ?𝑎?𝑔𝑒? ??? ??𝑎??𝑒? ) = 485 2 ?? ( 170.712747 106.600979 ) ≈ 114.191 Comparing this Box-Cox statistic of 114.191 with the critical value of 3.841 (chi-squared, 1 degree of freedom in Table A.4), we obtain a significant result, which suggests that one model is significantly better than the other, which in turn suggests that the semi-log model version of the model should be preferred to the original (linear) model. (10 marks) 4. [10 marks] Write down the STATA command that would be required to estimate the following nonlinear regression model: ? 𝑖 = 𝛽 1 1+𝑒 (−𝛽 2 −𝛽 3 𝑋 𝑖 ) + ? 𝑖 𝑖 = 1, 2, … , ? Answer: The STATA command that would be used is: . nl (Y = {beta1}/(1 + exp(-{beta2}-{beta3}*X))) (10 marks)