EC420 Michigan State University Econometric Method & Scaling Data Essay

Economics 420 Introduction to Econometrics Professor Woodbury — Fall Semester 2019 Model Specification 1. What is “model specification”? 2. Scaling data 3. Models with quadratics 4. What if X is a “dummy” (binary) variable? 5. Dummy variables for multiple categories … more to come 1 1. What is “model specification”? Model specification refers to: • the variables included in a model • how each of those variables is “specified” — linearly, as a log, as a quadratic, as a dummy variable, interacted with one or more other variables The earnings function is a canonical example: • how should earnings be specified? • which variables should be included? • how should they appear in the equation? 2 2. Scaling data (units of measurement) (Wooldridge, sections 2.4, 6.1) Yi = β0 + β1X1i + β2X2i + … + βkXki + ui, • It is often convenient to change the units of measurement to make the estimates from a model easier to interpret • For example, you may want to estimate the relationship between spending per student and math pass rate (MEAP93.dta) • But spending per student in measured in dollars, and the estimated coefficient may be very small • Rescaling the variable may make the results clearer 3 Main results to remember • When the dependent variable changes units of measurement: o Y • c (where c is a constant) means that the intercept and all the slope coefficients will be multiplied by c o Y / c (where c is a constant) means that the intercept and all the slope coefficients will be divided by c • So multiplying the dependent variable by a constant means the intercept and all the coefficients get multiplied by that constant • Dividing the dependent variable by a constant means the intercept and all the coefficients get divided by that constant 4 Example: CEOSAL1.dta (regress salary on roe) salary roe int float %9.0g %9.0g 1990 salary, thousands $ return on equity, 88-90 avg . sum salary roe Variable | Obs Mean Std. Dev. Min Max ————-+——————————————————–salary | 209 1281.12 1372.345 223 14822 roe | 209 17.18421 8.518509 .5 56.3 . reg salary roe Source | SS df MS ————-+———————————Model | 5166419.04 1 5166419.04 Residual | 386566563 207 1867471.32 ————-+———————————Total | 391732982 208 1883331.64 Number of obs F(1, 207) Prob > F R-squared Adj R-squared Root MSE = = = = = = 209 2.77 0.0978 0.0132 0.0084 1366.6 —————————————————————————–salary | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————roe | 18.50119 11.12325 1.66 0.098 -3.428196 40.43057 _cons | 963.1913 213.2403 4.52 0.000 542.7902 1383.592 —————————————————————————— 5 • What if we rescale salary to dollars (from $1,000s)? . gen salarydol = salary * 1000 . sum salary salarydol roe Variable | Obs Mean Std. Dev. Min Max ————-+——————————————————–salary | 209 1281.12 1372.345 223 14822 salarydol | 209 1281120 1372345 223000 1.48e+07 roe | 209 17.18421 8.518509 .5 56.3 . reg salarydol roe Source | SS df MS ————-+———————————Model | 5.1664e+12 1 5.1664e+12 Residual | 3.8657e+14 207 1.8675e+12 ————-+———————————Total | 3.9173e+14 208 1.8833e+12 Number of obs F(1, 207) Prob > F R-squared Adj R-squared Root MSE = = = = = = 209 2.77 0.0978 0.0132 0.0084 1.4e+06 —————————————————————————–salarydol | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————roe | 18501.19 11123.25 1.66 0.098 -3428.196 40430.57 _cons | 963191.3 213240.3 4.52 0.000 542790.2 1383592 —————————————————————————— 6 • When an independent variable changes units of measurement: o X • c (where c is a constant) means the coefficient on that X will be divided by c o X / c (where c is a constant) means the coefficient on that X will be multiplied by c • So multiplying an independent variable by a constant means the coefficient on that variable gets divided by the same constant • Dividing an independent variable by a constant means the coefficient on that variable gets multiplied the same constant 7 Example: MEAP93.dta (regress math10 on expend) math10 expend float int %9.0g %9.0g perc studs passing MEAP math expend. per stud, $ . sum math10 expend Variable | Obs Mean Std. Dev. Min Max ————-+——————————————————–math10 | 408 24.10686 10.49361 1.9 66.7 expend | 408 4376.578 775.7897 3332 7419 . reg math10 expend Source | SS df MS ————-+———————————Model | 1477.19666 1 1477.19666 Residual | 43339.9838 406 106.748729 ————-+———————————Total | 44817.1805 407 110.115923 Number of obs F(1, 406) Prob > F R-squared Adj R-squared Root MSE = = = = = = 408 13.84 0.0002 0.0330 0.0306 10.332 —————————————————————————–math10 | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————expend | .0024557 .0006601 3.72 0.000 .001158 .0037534 _cons | 13.35923 2.934111 4.55 0.000 7.591287 19.12718 —————————————————————————— 8 • But it may be easier to think about expenditures per students in $1,000s — divide expend by 1,000 . gen expendthou = expend / 1000 . sum math10 expend expendthou Variable | Obs Mean Std. Dev. Min Max ————-+——————————————————–math10 | 408 24.10686 10.49361 1.9 66.7 expend | 408 4376.578 775.7897 3332 7419 expendthou | 408 4.376578 .7757897 3.332 7.419 . reg math10 expendthou Source | SS df MS ————-+———————————Model | 1477.19667 1 1477.19667 Residual | 43339.9838 406 106.748729 ————-+———————————Total | 44817.1805 407 110.115923 Number of obs F(1, 406) Prob > F R-squared Adj R-squared Root MSE = = = = = = 408 13.84 0.0002 0.0330 0.0306 10.332 —————————————————————————–math10 | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————expendthou | 2.455715 .6601462 3.72 0.000 1.157983 3.753446 _cons | 13.35923 2.934111 4.55 0.000 7.591287 19.12718 —————————————————————————–9 3. Models with quadratics (Wooldridge, section 6.2) Consider the following model: E(Y) = β0 + β1X + β2X2 What is the relationship between X and Y in this model? ΔE(Y ) = β1 + 2β 2 X ΔX • What does this tell you? • The relationship depends on the level of X 10 ΔE(Y ) = β1 + 2β 2 X ΔX • Suppose X = 0, then ΔE(Y ) = β1 ΔX • Otherwise the value of X matters (as long as β2 ≠ 0) • A common approach for obtaining a summary of ΔE(Y)/ΔX is to plug in the mean value of X into the expression at the top of this slide and calculate ΔE(Y)/ΔX for that value 11 Example • The so-called Mincer earnings equation combines the log transformation with a quadratic specification of experience • In the following earnings equation, look at the relationship between log(earnings) and experience, and calculate the level of experience for which log(earnings) is a maximum: log(earnings) = β0 + β1educ + β2exper + β3exper2 + u • Typically in an earnings equation, we have: β2 > 0 and β3 < 0 so the experience-earnings profile starts with a positive (but decreasing) slope that reaches a maximum, then falls 12 • To find the turning point, use a little calculus: ∂log(earnings) = β 2 + 2β3exper ∂exper • Set this expression to 0 and solve for exper: β 2 + 2β3exper = 0 −β2 exper = 2β3 • So if β2 = 0.041 and β3 = –0.00071 (as in WAGE1.dta), earnings max out after 28.9 years of experience 13 Questions What is the return to experience when exper = 0? What is the return to experience when exper = 1? What is the return to experience when exper = 5? What is the return to experience when exper = 10? What is the return to experience when exper = 20? What is the return to experience when exper = 30? 14 Solution for the return to experience when exper = 10 Apply the formula: ΔE(Y ) = β1 + 2β 2 X ΔX In this case, ∂log(earnings) = β 2 + 2β3exper ∂exper = 0.041 + 2(–0.00071)•exper = 0.041 + (–0.00142)•10 = 0.041 – 0.0142 = 0.0268 So, for someone with 10 years of experience, the expected return to an additional year of experience is ~2.7 percent 15 Another example • A study of the relationship between alcohol and health in Lancet takes little care in matching results with conclusions (courtesy of Professor Imberman) • https://urldefense.proofpoint.com/v2/url? u=https-3A__mobile.twitter.com_scottimberman_status_10330438 28717092865&d=DwICAg&c=nE__W8dFEshTxStwXtp0A&r=5gNfshNzZWFcbhoDspnU-A&m=wSiArarzsSlua1mFTRSmOmIF3zVBRsAbtvY52kJtUs&s=t7O1zf7xvyRnDzCegXb hRwc0lvHvh5ULadg23oKcjJ0&e= • The figure shows the weighted relative risk of death or disability from alcohol, by standard drinks consumed per day 16 rve, we tandard l health ly with weighted e effects disease otective cancers, on. In a 4·5 4·0 3·5 Relative risk men and tandard daily for relative diabetes her outtive risk umption 3·0 2·5 2·0 1·5 1·0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Standard drinks daily Figure 5: Weighted relative risk of alcohol for all attributable causes, by standard drinks consumed per day 17 • “In 2016, alcohol use led to 2.8 million deaths and was the leading risk factor for premature death and disability among people aged 15–49 years, with nearly 9% of all attributable DALYs (deaths and disability-adjusted life-years) for men and more than 2% for women.” • Two problems: first, the study draws conclusions that could only be made if the study were causal — it takes little care in matching results with conclusions 18 • Second, it ignores the nonlinearity in the effect of alcohol consumption on health • “… the level of consumption that minimizes health loss is zero.” • The dotted line is a reference line for a relative risk of 1 (DALYs = deaths and disability-adjusted life-years) • Comment (Aaron E. Carroll @aaronecarroll): “Can someone please explain to me how a figure that shows that at one drink per day you have a relative risk of 1 can lead to the conclusion that there is no safe level of alcohol consumption?” 19 4. What if X is a “dummy” (binary) variable? (Wooldridge, sections 7.1, 7.2) • Dummy variables (aka zero-one, binary, or indicator variables) are useful when you need to model something that is qualitative or categorical X = 1 if female, 0 if male X = 1 if treated (experimental drug), 0 if not X = 1 if small class size, 0 if not 20 • Other examples Race or ethnicity (Caucasian, African-American, Asian, Hispanic) o Marital status (married, never married, separated, divorced, widowed) o Industry or occupation • Things are easy when there are only two categories (male or female), but can be a little more involved when there are many (the other examples—for later) • So far, β1 has been called a “slope,” but that doesn’t make sense if X is binary • How do we interpret regression with a binary regressor? o 21 Back to a simple regression: Yi = β0 + β1Xi + ui • If X is binary (Xi = 0 or 1), then when Xi = 0: Yi = β0 + ui when Xi = 1: Yi = β0 + β1 + ui • so: when Xi = 0, the mean of Yi is β0 when Xi = 1, the mean of Yi is β0 + β1 • or (equivalently): E(Yi | Xi = 0) = β0 E(Yi | Xi = 1) = β0 + β1 22 So we have this result: • When Yi = β0 + β1Xi + ui and X is binary: E(Yi) = β0 when Xi = 0 E(Yi) = β0 + β1 when Xi = 1 • So β1 = E(Yi | Xi = 1) – E(Yi | Xi = 0) = population difference between mean Y for group 1 and mean Y for group 2 or the expected difference between the two group means 23 • SE( β̂1 ) has the usual interpretation o t-statistics, p-values, and confidence intervals constructed as usual Examples • Differences in wages by gender, race, and marital status — see Examples 7.1, 7.6, 7.10, 7.11 in Wooldridge • Experimental effects (difference-in-means analysis — reemployment bonus experiment — see below) 24 Wooldridge Example 7.1—Gender wage gaps use WAGE1.DTA reg wage female Source | SS df MS ————-+—————————–Model | 828.220467 1 828.220467 Residual | 6332.19382 524 12.0843394 ————-+—————————–Total | 7160.41429 525 13.6388844 Number of obs F( 1, 524) Prob > F R-squared Adj R-squared Root MSE = = = = = = 526 68.54 0.0000 0.1157 0.1140 3.4763 —————————————————————————–wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————female | -2.51183 .3034092 -8.28 0.000 -3.107878 -1.915782 _cons | 7.099489 .2100082 33.81 0.000 6.686928 7.51205 —————————————————————————— for difference of means, use the Stata command: ttest wage, by (female) 25 reg wage female educ exper tenure Source | SS df MS ————-+—————————–Model | 2603.10658 4 650.776644 Residual | 4557.30771 521 8.7472317 ————-+—————————–Total | 7160.41429 525 13.6388844 Number of obs F( 4, 521) Prob > F R-squared Adj R-squared Root MSE = = = = = = 526 74.40 0.0000 0.3635 0.3587 2.9576 —————————————————————————–wage | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————female | -1.810852 .2648252 -6.84 0.000 -2.331109 -1.290596 educ | .5715048 .0493373 11.58 0.000 .4745803 .6684293 exper | .0253959 .0115694 2.20 0.029 .0026674 .0481243 tenure | .1410051 .0211617 6.66 0.000 .0994323 .1825778 _cons | -1.567939 .7245511 -2.16 0.031 -2.991339 -.144538 —————————————————————————— 26 Another example: Difference-in-means analysis using dummy variables • Dummy variables give you a simple way to do difference-inmeans analysis — especially useful when we have additional regressors • Problem: Unemployment insurance (UI) may induce recipients to search less hard for a job and take longer to become reemployed (moral hazard effect) • This is a problem both for the worker (longer spell of unemployment, lower earnings, possibly lower earnings in the long-term) and for the UI system (more benefits paid) 27 • One policy to ameliorate this problem is the reemployment bonus (or job search incentive) — a lump sum paid to a UI claimant who finds and job within a specified number of weeks, and who holds the job for a minimum number of months following reemployment • First reemployment bonus: Illinois in the 1980s o o Maximum potential duration of benefits in Illinois: 26 weeks Terms of the reemployment bonus: $500 cash if you find a job within 11 weeks and hold the job for 4 months 28 • An alternative policy (also tried in Illinois but not thereafter) is a hiring incentive bonus — each UI claimant told to inform prospective employers, “Hire me within 11 weeks, and employ me for at least 4 months, and you will receive a $500 payment from the State of Illinois.” • How do you evaluate policies like these? • Random assignment! 29 • In Illinois, new UI claimants randomly assigned to one of three groups o o o Control group (treated like a normal UI claimants) Offered a job search incentive bonus Offered a hiring incentive bonus • How do you know whether one (or both) of the programs worked? • Compare mean outcome(s) of controls and those who received the treatment 30 (Illinois bonus extract (FSC ineligibles)) 2 . drop if hie==1 (1581 observations deleted) Two ways of calculating experimental effects 3 . ttest wkpaid, by jsie comparison of means • Straight option by incorrectly specified r(198); o In Stata, ttest outcome, by (treatment) 4 . ttest wkpaid, by (jsie) Two-sample t test with equal variances Group Obs Mean 0 1 1725 1735 combined 3460 diff Std. Err. Std. Dev. [95% Conf. Interval] 16.86145 15.83516 .2384533 .2478865 9.903709 10.3253 16.39376 15.34897 17.32914 16.32135 16.34682 .1721957 10.12886 16.00921 16.68444 1.026291 .3440003 .3518265 1.700755 diff = mean(0) – mean(1) Ho: diff = 0 Ha: diff < 0 Pr(T < t) = 0.9986 5 . t = degrees of freedom = Ha: diff != 0 Pr(|T| > |t|) = 0.0029 2.9834 3458 Ha: diff > 0 Pr(T > t) = 0.0014 31 wkpaid Coef. jsie _cons -1.026291 16.86145 Std. Err. .3440003 .2435962 • Dummy variable regression o t -2.98 69.22 P>|t| 0.003 0.000 [95% Conf. Interval] -1.700755 16.38384 -.3518265 17.33906 reg outcome treatment, robust 7 . reg wkpaid jsie, robust Linear regression Number of obs F( 1, 3458) Prob > F R-squared Root MSE wkpaid Coef. jsie _cons -1.026291 16.86145 Robust Std. Err. .3439589 .2384531 t -2.98 70.71 P>|t| 0.003 0.000 = = = = = 3460 8.90 0.0029 0.0026 10.117 [95% Conf. Interval] -1.700674 16.39393 -.3519077 17.32897 8 . 32 • Adding control variables (age, education, etc.) to the preceding regression would reduce the sampling variation in β1-hat, so the standard error of β1-hat should fall • Why? Remember: 2 σ var( β̂ j ) = σ β̂2 = j SSTj (1− R 2j ) and adding control variables reduces σ2 33 Other points about the Illinois data • Treatment effects for other outcomes (including pre- and post-program earnings) could be estimated • Important to check for balance in the treatment groups (are the variable means similar in the control and treatment groups?) • How many received bonuses? 34 5. Dummy variables for multiple categories See Wooldridge, section 7.3 (especially “Incorporating ordinal information by using dummy variables” and Example 7.7) A very short review • Dummy variables (zero-one, binary, or indicator variables) are useful when you need to model something that is qualitative or categorical X = 1 if female, 0 if male X = 1 if treated (experimental drug), 0 if not X = 1 if small class size, 0 if not 35 • When Yi = β0 + β1Xi + ui and X is binary: β0 = mean of Y for X = 0 β0 + β1 = mean of Y for X = 1 β1 = difference in group means = E(Yi | Xi = 1) – E(Yi | Xi = 0) 36 • Other examples Race or ethnicity (Caucasian, African-American, Asian, Hispanic) o Marital status (married, never married, separated, divorced, widowed) o Industry or occupation • Things are easy when there are only two categories (male or female), but can be a little more involved when there are many (the other examples) o 37 Looks (or “beauty”) and earnings • The data in BEAUTY.dta provide a good example • These data come from Daniel Hamermesh and Jeff Biddle, “Beauty and the Labor Market,” American Economic Review 84 (1994): 1174-1194 • The variable looks in this dataset is ordinal • The five categories are: homely (1), quite plain (2), average (3), good looking (4), and strikingly beautiful/handsome (5) • We will look at three possible specifications of looks 38 • Here is the distribution of looks in this sample . tab looks from 1 to 5 | Freq. Percent Cum. ————+———————————-1 | 13 1.03 1.03 2 | 142 11.27 12.30 3 | 722 57.30 69.60 4 | 364 28.89 98.49 5 | 19 1.51 100.00 ————+———————————-Total | 1,260 100.00 • Would you want to include looks as is (1–5)? log(wage) = β0 + β1educ + β2exper + β3exper2 + β4female + β5looks + u (1) In this model, looks is the original ordinal variable 39 • Or is another approach better? For example, log(wage) = β0 + β1educ + β2exper + β3exper2 + β4female + β5looks1 + β6looks2 + β7looks3 + β8looks4 + β9looks5 + u (2) In this model looks1 = 1 if looks = 1 (0 otherwise) looks2 = 1 if looks = 2 (0 otherwise) looks3 = 1 if looks = 3 (0 otherwise) looks4 = 1 if looks = 4 (0 otherwise) looks5 = 1 if looks = 5 (0 otherwise) So looks1 … looks5 are five mutually exclusive and exhaustive dummy variables capturing all the information in looks 40 • Here is a third possible specification log(wage) = β0 + β1educ + β2exper + β3exper2 + β4female + β5belavg + β6avg + β7abvavg + u (3) This is Biddle and Hamermesh’s preferred specification, where belavg = 1 if looks = 1 or 2 (0 otherwise), avg = 1 if looks = 3 (0 otherwise) abvavg = 1 if looks = 4 or 5 (0 otherwise) • Estimates of the three models follow 41 • Here is the model with the original variable (looks) included — equation (1) • Why is this not a good way to specify the model? . reg lwage educ exper expersq looks Source | SS df MS ————-+—————————–Model | 103.347555 4 25.8368886 Residual | 341.632418 1255 .272217066 ————-+—————————–Total | 444.979972 1259 .353439215 Number of obs F( 4, 1255) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1260 94.91 0.0000 0.2323 0.2298 .52174 —————————————————————————–lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————educ | .0683689 .0058127 11.76 0.000 .0569653 .0797725 exper | .0487326 .0047884 10.18 0.000 .0393386 .0581267 expersq | -.0006985 .0001077 -6.49 0.000 -.0009097 -.0004872 looks | .0620681 .0219232 2.83 0.005 .019058 .1050782 _cons | .0462774 .105333 0.44 0.660 -.1603708 .2529255 —————————————————————————— 42 • Here is the model with all five looks categories included as dummies — equation (2) • Notice that Stata drops one category (Why?) . reg lwage educ exper expersq looks1 looks2 looks3 looks4 looks5 note: looks5 omitted because of collinearity Source | SS df MS ————-+—————————–Model | 106.116516 7 15.1595023 Residual | 338.863456 1252 .270657712 ————-+—————————–Total | 444.979972 1259 .353439215 Number of obs F( 7, 1252) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1260 56.01 0.0000 0.2385 0.2342 .52025 —————————————————————————–lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————educ | .0682993 .0058026 11.77 0.000 .0569154 .0796833 exper | .0484757 .0047766 10.15 0.000 .0391048 .0578467 expersq | -.0006987 .0001074 -6.50 0.000 -.0009094 -.000488 looks1 | -.4653829 .1889268 -2.46 0.014 -.836031 -.0947348 looks2 | -.3025159 .1275879 -2.37 0.018 -.5528256 -.0522061 looks3 | -.1358712 .1213901 -1.12 0.263 -.3740217 .1022793 looks4 | -.1556256 .1225351 -1.27 0.204 -.3960224 .0847711 looks5 | 0 (omitted) _cons | .4113932 .145361 2.83 0.005 .1262153 .6965712 43 It usually makes sense to omit a category (rather than let Stata do it for you) — then you choose the reference category . reg lwage educ exper expersq looks1 looks2 Source | SS df MS ————-+—————————–Model | 106.116516 7 15.1595023 Residual | 338.863456 1252 .270657712 ————-+—————————–Total | 444.979972 1259 .353439215 looks4 looks5 Number of obs F( 7, 1252) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1260 56.01 0.0000 0.2385 0.2342 .52025 —————————————————————————–lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————educ | .0682993 .0058026 11.77 0.000 .0569154 .0796833 exper | .0484757 .0047766 10.15 0.000 .0391048 .0578467 expersq | -.0006987 .0001074 -6.50 0.000 -.0009094 -.000488 looks1 | -.3295117 .1464298 -2.25 0.025 -.6167865 -.042237 looks2 | -.1666447 .0478185 -3.48 0.001 -.2604578 -.0728315 looks4 | -.0197544 .0339561 -0.58 0.561 -.0863716 .0468627 looks5 | .1358712 .1213901 1.12 0.263 -.1022793 .3740217 _cons | .2755221 .0845305 3.26 0.001 .109685 .4413591 —————————————————————————— 44 Finally, here is the model with looks specified using belavg and abvavg — equation (3) . reg lwage educ exper expersq belavg abvavg Source | SS df MS ————-+—————————–Model | 105.369105 5 21.0738209 Residual | 339.610868 1254 .270822063 ————-+—————————–Total | 444.979972 1259 .353439215 Number of obs F( 5, 1254) Prob > F R-squared Adj R-squared Root MSE = = = = = = 1260 77.81 0.0000 0.2368 0.2338 .52041 —————————————————————————–lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval] ————-+—————————————————————educ | .0687694 .0057921 11.87 0.000 .0574061 .0801327 exper | .0483835 .0047764 10.13 0.000 .0390129 .0577542 expersq | -.0006996 .0001074 -6.51 0.000 -.0009104 -.0004889 belavg | -.1800631 .0461625 -3.90 0.000 -.2706273 -.0894988 abvavg | -.012913 .0334551 -0.39 0.700 -.0785471 .0527212 _cons | .2719752 .0844519 3.22 0.001 .1062927 .4376577 —————————————————————————— 45 Points to note • What are the problems with (1) — using an ordinal variable (looks) as independent variable? o What restrictive assumption are you imposing? • Is (2) better? o Can you estimate all five coefficients? No o It leads you into a “dummy variable trap” o You need to interpret the coefficient estimates with respect to the reference group — the omitted category • Why did Biddle and Hamermesh prefer (3)? o Look at the distribution of looks (see output following the Stata “tab” command, repeated below) 46 . tab looks from 1 to 5 | Freq. Percent Cum. ————+———————————-1 | 13 1.03 1.03 2 | 142 11.27 12.30 3 | 722 57.30 69.60 4 | 364 28.89 98.49 5 | 19 1.51 100.00 ————+———————————-Total | 1,260 100.00 47 Key point again • In specifications (2) and (3), one category necessarily drops out due to the perfect collinearity (the “dummy variable trap” • The coefficient estimates are interpreted with respect to the reference group — the omitted category • For example, in specification (3) belavg = –0.180 (0.046) o Interpretation: Other things equal, those with below average looks have expected earnings that are ~18% less than those with average looks • Also in specification (3), abvavg = –0.013 (0.033) o Interpretation? 48 Economics 420 (sections 2 and 3) Professor Woodbury Fall Semester 2019 Problem Set #4 (Due:Thursday, November 14) Instructions • • • Use Stata and a word processor for this assignment. Read the question and answer what is asked. For each question that asks you to use Stata, copy and paste the Stata output into a wordprocessing document, then type your answer. Staple all pages together at the upper-left corner before you turn in your homework. • • • Assignments turned in unstapled will be returned with a grade of zero. Only stapling is acceptable—paper clips and other methods of binding are not acceptable. If we cannot discern the meaning of your work, your response will be scored as incorrect. This problem covers three topics: models with a quadratic specification, regression with dummy variables and interaction terms; and hypothesis tests involving multiple parameters. Use the Stata file BEAUTY.dta to answer all the questions. BEAUTY.dta contains the following variables used by Hamermesh and Biddle (American Economic Review 1994): wage educ exper female black union goodhlth looks hourly wage years of schooling years of workforce experience =1 if female, 0 otherwise =1 if black, 0 otherwise =1 if a union member, 0 otherwise =1 if in good health, 0 otherwise physical attractiveness score ranging from 1 to 5 You will also need to generate a few additional variables from those already in the dataset. Part I: Simple dummy variables and the quadratic specification (25 points total) 1. (5 points) Estimate the simple linear regression model: ln(wage) = β0 + β1female + u Interpret the OLS estimates of the intercept and the coefficient on female. Hint: See the section on “Difference-in-means analysis using dummy variables” in . 2. (10 points) Now estimate the following modified Mincer wage equation: ln(wage) = β0 + β1educ + β2exper + β3exper2 + β4female + β5black + β6union + β7goodhlth +u Is the estimated on expersq statistically significant at the 5% level? How do you know? After how many years of experience does a worker’s ln(wage) reach a maximum? Show how you obtained your answer. 3. (5 points) Interpret the coefficient on the “union” dummy variable in the Mincer equation you just estimated. Be sure to consider statistical significance. 4. (5 points) Interpret the coefficient on the “good health” dummy variable in the Mincer equation you just estimated. Be sure to consider statistical significance. Part 2: Interaction terms (25 points total) 1. (10 points) Consider the following simple regression models of log-earnings: ln(wage) = α0 + α1educ + uf (for men only) ln(wage) = β0 + β1educ + um (for women only) Estimate the two regressions and interpret the parameters on education (α1 and β1).Are these coefficients are statistically significant at a level of 0.01? What is the difference between α1 and β1, and how do you interpret it? Hint:To estimate the models, type in Stata: reg lwage educ if female==0 reg lwage educ if female==1 2. (15 points) Now consider the following regression model: ln(wage) = γ0 + γ1educ + γ2female + γ3educ•female + ub (for both women and men) Estimate this regression and interpret fully the four estimated parameters (γ0, γ1, γ2, and γ3). Which of these estimated coefficients are statistically significant at a significance level of 0.01? Does your estimate γ3 make sense in light of what you found in question 1 about the difference between α1 and β1? Explain. Hints:To estimate this model, first generate the interaction between educ and female, which is the product of variables educ and female.To do this, type: gen educ•female = educ*female Then estimate the regression: reg lwage educ female educ•female Part 3: Dummy variables for multiple-categories (30 points total) BEAUTY.dta includes a variable “looks” indicating a persons’s score on physical attractiveness, as ranked by an interviewer.Attractiveness was coded in five categories: 1 = homely 2 = quite plain 3 = average 4 = good looking 5=strikingly beautiful/handsome The dataset also includes two additional variables that we discussed in class: belavg = 1 if looks = 1 or looks = 2, 0 otherwise abvavg = 1 if looks = 4 or looks = 5, 0 otherwise 2 1. (5 points) Create three interaction terms, one for female interacted with belavg, one for female interacted with looks = 3, and a third for female interacted with abvavg. Hint:All you need to do for this question is type in Stata: belavg*female gen abvavg•female = abvavg*female gen avg = (looks==3) gen belavg•female = gen avg•female = avg*female (Note: Just create these dummies; you don’t need to show anything on your homework.) 2. (10 points) Estimate the following model for ln(wage) and report the estimates: ln(wage) = β0 + β1educ + β2exper + β3exper2 + β4belavg + β5abvavg + β6belavg•female + β7avg•female + β8abvavg•female + u Interpret the estimates of β4, and β5. Hint:The omitted category is men of “average” looks, so the coefficients on belavg and abvavg pertain to men of below and above average looks. 3. (10 points) Interpret the estimates of β6, β7, and β8. Hint:Again, the omitted category is men of “average” looks, so for example, the coefficient on belavg•females tells you something about females of “below average” looks relative to men of average looks. 4. (5 points) What is the wage gap between women of “above average” looks and women of “average” looks? Is the difference statistically significant? Hint: Compare the coefficient on abvavg•female with the coefficient on avg•female.To test for the difference, type: lincom abvavg•female – avg•female