Regression Analysis of Taxicab Drivers' Hours Worked and Wages: Farber (2008) - Prof. Melv, Exams of Introduction to Econometrics

Download Regression Analysis of Taxicab Drivers' Hours Worked and Wages: Farber (2008) - Prof. Melv and more Exams Introduction to Econometrics in PDF only on Docsity! 1. This problem has 13 parts, labeled A-M. The following dataset is taken from the article “Reference-Dependent Preferences and Labor Supply: The Case of New York City Taxi Drivers,” by Henry S. Farber from the June 2008 American Economic Review. The dataset contains information from a sample of 538 taxicab driver tripsheets which record information on the amount of time and the total fare for each trip during a shift. The statistics below show the amount of time working in hours (the variable hours), the average hourly wage in dollars (wage1), the amount of precipitation during the shirt (precip), and the high temperature during the shift (maxtemp). summarize hours wage1 precip maxtemp Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- hours | 538 6.837887 2.493785 .1333333 13.56667 wage1 | 538 23.80357 4.52074 13.04132 57.89189 precip | 538 .1455204 .5541368 0 6.46 maxtemp | 538 56.9684 17.43701 22 101 The corr command in Stata is used to get the correlation matrix for the variables in the dataset. The results of using this command on the sample described above are shown below. corr hours wage1 precip maxtemp (obs=538) | hours wage1 precip maxtemp -------------+------------------------------------ hours | 1.0000 wage1 | -0.2430 1.0000 precip | -0.0452 0.0188 1.0000 maxtemp | -0.0641 0.0121 -0.1510 1.0000 Name:_____________________ 2 A. (5 points) In this dataset, what is the standard error of the sample mean for the high temperature during the shift? Name:_____________________ 5 We are interested in the parameters of the population regression function ii10i uWage1*ββ Hours ++= (1) Using OLS to estimate regression (1) with the taxicab tripsheet dataset yields the following ANOVA table Source | SS df MS -------------+------------------------------ Model | 197.186473 1 197.186473 Residual | 3142.39766 536 5.86268219 -------------+------------------------------ Total | 3339.58413 537 6.21896486 D. (5 points) Using the information provided on page 2 and/or above, what is the OLS estimate of 1β for regression model (1)? Name:_____________________ 6 E. (5 points) Using the information provided on page 2 and/or on the previous page, compute Var( 1β̂ ) when estimating regression model (1). Name:_____________________ 7 F. (10 points) Test the null hypothesis 0β:H 10 = at the 0.05α = level of significance for regression model (1). Be certain to state your decision rule along with the null and alternative hypotheses. Name:_____________________ 10 J. (5 points) Using the information provided on page 2 and/or on page 6, compute the r2 for the estimated model. Name:_____________________ 11 At the end of the exam, regression results A-B contain additional regression specifications that are estimated using this dataset of taxicab tripsheets. ii3i2i10i uMaxtemp*βPrecip*βWage1*ββ Hours ++++= (2) K. (10 points) Test the overall significance of regression model (2) at the 0.05α = level of significance. Be sure to state the null hypothesis, the alternative hypothesis, and your decision rule. Name:_____________________ 12 L. (10 points) Using an F-test, test the null hypothesis 0β:H 30 = at the 0.05α = level of significance. Be sure to state your decision rule along with the null and alternative hypotheses. (Note: you will not be given credit for using another approach to test this null hypothesis.) Name:_____________________ 15 REGRESSION RESULT A reg hours wage1 precip Source | SS df MS Number of obs = 538 -------------+------------------------------ Model | 202.714362 2 101.357181 Residual | 3136.86977 535 5.86330798 -------------+------------------------------ Total | 3339.58413 537 6.21896486 ------------------------------------------------------------------------------ hours | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- wage1 | -.1336207 .0231181 -5.78 0.000 -.179034 -.0882074 precip | -.1831269 .1886009 -0.97 0.332 -.5536161 .1873623 _cons | 10.04518 .5602729 17.93 0.000 8.94458 11.14579 ------------------------------------------------------------------------------ estat vce e(V) | wage1 precip _cons -------------+------------------------------------ wage1 | .00053444 precip | -.0000819 .03557031 _cons | -.01270977 -.00322672 .31390573 Name:_____________________ 16 REGRESSION RESULT B reg hours wage1 precip maxtemp Source | SS df MS Number of obs = 538 -------------+------------------------------ Model | 218.185325 3 72.7284417 Residual | 3121.3988 534 5.84531611 -------------+------------------------------ Total | 3339.58413 537 6.21896486 ------------------------------------------------------------------------------ hours | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- wage1 | -.1330532 .0230852 -5.76 0.000 -.1784022 -.0877043 precip | -.2300056 .1905032 -1.21 0.228 -.6042332 .1442221 maxtemp | -.0098482 .0060534 -1.63 0.104 -.0217397 .0020433 _cons | 10.59954 .6550193 16.18 0.000 9.312806 11.88627 ------------------------------------------------------------------------------ estat vce e(V) | wage1 precip maxtemp _cons -------------+------------------------------------------------ wage1 | .00053293 precip | -.0000917 .03629148 maxtemp | -2.111e-06 .00017443 .00003664 _cons | -.01255192 -.01303546 -.00206269 .42905025 Name:_____________________ 17 Features of Probability Distribution Functions Expected Value: ∑ = == k 1j jjX )f(xxμE(X) Variance: ∑ = −=−== k 1j j 2 Xj 2 X 2 X )f(x)μ(x)μE(XσVar(X) Covariance: ( )( )[ ] [ ] ∑∑ = = −−=−= −−= k 1j m 1l ljYX,YlXjYX YX )y,(x)fμ)(yμ(xμμXYE μYμXEY)Cov(X, Correlation Coefficient: YXσσ Y)Cov(X,ρ = Conditional Expectation: ∑ = k 1j jX|Yj x)|(yfy = x)|E(Y Variance of Sum/Difference: Var(X±Y) = Var(X) + Var(Y) ± 2Cov(X,Y) Sample Characteristics Sample Mean: ∑ = = n 1i i n X X Sample Variance: ∑ = = n 1i 2 i2 X 1-n )X-(X S Sample Covariance: ∑ = = n 1i ii 1-n )Y-)(YX-(X Y)Cov(X, Sample Sample Correlation Coefficient: YXSS Y)Cov(X, Sampler = Probability Distributions Normal Distribution: If )σ,N(μ~X 2XX , then A. N(0,1)~Z where X X 2 X X σ μ-X σ μ-XZ == B. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ n σ ,μN~X 2 X X and /nσ μ-X Z 2 X X= C. n/S μ-X /nS μ-X t X X 2 X X 1-n == Chi-Squared: 2(k) 2 k 2 2 2 1 2 i χZZZZ =+++=∑ L F-Dist.: If 2mχ ~U and 2nχ ~V then nm,F~V/n U/m Central Limit Theorem: Even if Xi is not normally distributed, n/σ μ-X Z 2 X X= and /nS μ-X t 2 X X 1-n = as ∞→n . Hypothesis testing Confidence interval for population mean: [ ] [ ]( ) α1nStXμnStXP Xα/2XXα/2 −=+≤≤− Bias of an estimator: ( ) ( )θ-θ̂Eθ̂bias = Mean squared error of an estimator: ( ) ( ) ( ) 22 )θ̂bias( )θ̂Var( θ-θ̂Eθ̂MSE +== F-test of H0: 2Y 2 X σσ = against HA: 2 Y 2 X σσ ≠ 2 Y 2 X 1n1,m S S F =−− Testing YX0 μμ:H = : ( ) ( )nσmσ )μμ()YX( )YXVar( )μμ()YX(z 2 Y 2 X YX1YX + −−− = − −−− =