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# assignment1_f13 - solutions.pdf

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School
York University
Department
Economics
Course
ECON 3210
Professor
Idd
Semester
Fall

Description
YORK UNIVERSITY Faculty of Liberal Arts and Professional Studies Department of Economics Use of Economic Data – AP/ECON3210.03A Assignment #1: Questions and Solutions th Date: September 30 , 2th3 Due Date: October 7 , 2013 Course Director: Ida Ferrara Weight of Assignment: 5% of the final grade. NOTES: (1) assignments will have to be submitted either electronically within MOODLE or in hard copy at the beginning of class on the due date; (2) students failing to submit the assignment on the due date will receive a grade of zero for the assignment; (3) the data files for the two questions are provided as dat files if STATA is used and as xlsx files if EXCEL is used (either software is acceptable); (4) computer outputs have to be included in the submission and as part of the submission (e.g., for electronic submission, everything should be in one file). 1. (2.15/3.11) How much does education affect wage rates? The date file cps4_small.dat contains 1000 observations on hourly wage rates, education, and other variables from the 2008 Current Population Survey (CPS). (a) Obtain the summary statistics and histograms for the variables 𝑊𝐴𝐺𝐸 and 𝐸𝐷𝑈𝐶. Discuss the data characteristics. (2.15a) (b) Estimate the linear regression 𝑊𝐴𝐺𝐸 = 𝛽1+𝛽 2𝐷𝑈𝐶 + 𝑒 and discuss the results. (2.15b) (c) Test the statistical significance of the estimated slope of the relationship at the 5% level. Use a one-tail test. (3.11b) (d) Calculate the least squares residuals and plot them against 𝐸𝐷𝑈𝐶. Are any patterns evident? If the CLRM assumptions hold, should any pattern be evident? (2.15c) (e) Estimate separate regressions for males, females, blacks, and whites. Compare the results. (2.15d) 2 (f) Estimate the quadratic regression 𝑊𝐴𝐺𝐸 = 𝛽 +1 𝐸2𝑈𝐶 + 𝑒 and discuss the results. Estimate the marginal effect of another year of education on wage for a person with 12 years of education, and for a person with 14 years of education. Compare these values to the estimated marginal effect of education from the linear regression in part (b). (2.15e) (g) Plot the fitted linear model from part (b) and the fitted values from the quadratic model from part (f) in the same graph with the data on 𝑊𝐴𝐺𝐸 and 𝐸𝐷𝑈𝐶. Which model appears to fit the data better? (2.15f) (h) Construct a histogram of 𝑙𝑛 𝑊𝐴𝐺𝐸 . Compare the shape of the histogram to that for 𝑊𝐴𝐺𝐸 from part (a). Which appears more symmetric and bell-shaped? (2.15g) (i) Estimate the log-linear regression 𝑙𝑛 𝑊𝐴𝐺𝐸 = 𝛾 +𝛾 𝐸𝐷𝑈𝐶 + 𝑒. Estimate the marginal effect of 1 2 another year of education on wage for a person with 12 years of education, and for a person with 14 years of education. Compare these values to the estimated marginal effects of education from the linear regression in part (b) and the quadratic regression in part (f). (2.15h) 2. (2.10/3.7) The capital asset pricing model (CAPM) is an important model in the field of finance. It explains variations in the rate of return on a security as a function of the rate of return on a portfolio consistingof allpubliclytraded stocks, whichiscalled the market portfolio.Generally, therateof return on any investment is measured relative to its opportunity cost, which is the return on a risk-free asset. The resulting difference is called the risk premium, since it is the reward or punishment for making a risky investment. The CAPM says that the risk premium on security j is proportional to the risk premium on the market portfolio, that is, 𝑟 −𝑗𝑟 =𝑓𝛽 (𝑟 𝑗 𝑟𝑚), wh𝑓re 𝑟 and 𝑟 𝑗re the𝑓returns to security j and the risk-free rate, respectively𝑚 𝑟 is the return on the market portfolio, and𝑗𝛽 is the jth security’s “beta” value. A stock’s beta is important to investors since it reveals the stock’s volatility. It measures the sensitivity of security j’s return to variation in the whole stock market. As such, values of beta less than 1 indicate that the stock is “defensive” since its variation is less than the market’s. A beta greater than 1 indicates an “aggressive” stock. Investors usually want an estimate of a stock’s beta before purchasing it. The CAPM model shown above is the “economic model” in this case. The “econometric model” is obtained by including an intercept in the model (even though theory says it should be zero) and an error term, 𝑟𝑗− 𝑟 𝑓 𝛼 + 𝑗 (𝑟 𝑗 𝑟𝑚) + 𝑓. (a) Explain why the econometric model above is a simple regression model. (2.10a) (b) In the data file capm4.dat are data on the monthly returns of six firms (Microsoft, GE, GM, IBM, Disney, and Mobil-Exxon), the rate of return on the market portfolio (MKT), and the rate of return on the risk-free asset (RISKFREE). The 132 observations cover January 1998 to December 2008. Estimate the CAPM model for each firm and comment on their estimated beta values. Which firm appears most aggressive? Which firm appears most defensive? (2.10b) (c) Finance theory says that the intercept parameter 𝛼𝑗should be zero. Does this seem correct given your estimates? For the Microsoft stock, plot the fitted regression line along with the data scatter. (2.10c) (d) Estimate the model for each firm under the assumption that 𝛼 =𝑗0. Do the estimates of the beta values change much? (2.10d) (e) Test at the 5% level of significance the hypothesis that each stock’s beta value is 1 against the alternative that it is not equal to 1. What is the economic interpretation of a beta equal to 1? (3.7a) (f) Test at the 5% level of significance the hypothesis that Mobil-Exxon’s beta value is greater than or equal to 1. What is the economic interpretation of a beta less than 1? (3.7b) (g) Test at the 5% level of significance the hypothesis that Microsoft’s beta value is less than or equal to 1. What is the economic interpretation of a beta greater than 1? (3.7c) (h) Construct a 95% confidence interval for Microsoft’s beta value. Assume that you are a stockbroker. Explain this result to an investor who has come to you for advice. (3.7d) (i) Test at the 5% level of significance the hypothesis that the intercept term in the CAPM model for each stock is zero against the alternative that it is not. What do you conclude? (3.7e) Chapter 2, Exercise Solutions, Principles of Econometrics, 4e 50 EXERCISE 2.15 (a) Figure xr2.15(a) Histogram and statistics for EDUC Most people had 12 years of education, implying that they finished their education at the end of high school. There are a few observations at less than 12, representing those who did not complete high school. The spike at 16 years describes those who completed a 4- year college degree, while those at 18 and 21 years represent a master’s degree, and further education such as a PhD, respectivel y. Spikes at 13 and 14 years are people who had one or two years at college. 140 Series: WAGE 120 Sample 1 1000 Observations 1000 100 Mean 20.61566 80 Median 17.30000 Maximu76.39000 60 Minimum.970000 Std. Dev. 12.83472 Skewness 1.583909 40 Kurtosis 5.921362 Jarque-Bera 773.7260 20 Probability 0.000000 0 0 10 20 30 40 50 60 70 Figure xr2.15(a) Histogram and statistics for WAGE The observations for WAGE are skewed to the right indicating that most of the observations lie between the hourly wages of 5 to 40, and that there is a smaller proportion of observations with an hourly wage greater th an 40. Half of the sample earns an hourly wage of more than 17.30 dollars per hour, w ith the average being 20.62 dollars per hour. The maximum earned in this sample is 76.39 dollars per hour and the least earned in this sample is 1.97 dollars per hour. (b) The estimated equation is  WAGE  6.7103 1.9803EDUC The coefficient 1.9803 represents the estimated increase in the expected hourly wage rate for an extra year of education. The coefficient −6.7103 represents the estimated wage rate of a worker with no years of education. It s hould not be considered meaningful as it is not possible to have a negative hourly wage rate. Chapter 2, Exercise Solutions, Principles of Econometrics, 4e51 Exercise 2.15 (continued) (c) The residuals are plotted against education in Figure xr2.15(c). There is a pattern evident; as EDUC increases, the magnitude of the residuals also increases, suggesting that the error variance is larger for larger values ofa violation of assumption SR3. If the assumptions SR1-SR5 hold, there should not be any patterns evident in the residuals. 60 50 40 30 20 RESID 0 -10 -20 -30 0 4 8 12 16 20 24 EDUC Figure xr2.15(c) Residuals against education (d) The estimated equations are  femIafle: WAGE  14.1681 2.3575EDUC malef: WAGE  3.0544 1.8753EDUC blaf:k WAGE  15.0859 2.4491EDUC whif: WAGE  6.5507 1.9919EDUC The white equation is obtained from those workers who are neither black nor Asian. From the results we can see that an extra y ear of education increases the wage rate of a black worker more than it does for a wworker. And an extra year of education increases the wage rate of a female worker more than it does for a male worker. (e) The estimated quadratic equation is WAGE  .08283 0.073489EDUC 2 The marginal effect is therefore:   dGE  slope   2 073489 EDUC dEDUC For a person with 12 years of educate estimated marginal effect of an additional year of education on expected wage is: dGE  slop    2 073489 12 1.7637 dEDUC That is, an additional year of education for a person with 12 years of education is expected to increase wage by \$1.76. Chapter 2, Exercise Solutions, Principles of Econometrics, 4e 52 Exercise 2.15(e) (continued) For a person with 14 years of education, the marginal effect of an additional year of education is: dWE  slope    2  73489 14 2.0577 dEDUC An additional year of education for a personwith 14 years of education is expected to increase wage by \$2.06. The linear model in (b) suggested that an additional year of education is expected to increase wage by \$1.98 regardless of the numbe r of years of education attained. That is, the rate of change is constant. The quadratic model suggests that the effect of an additional year of education on wage increases with the level of education already attained. (f) Figure xr2.15(f) Quadratic and linear equations for wage on education The quadratic model appears to fit the data slightly better than the linear equation. (g) The histogram of ln(AGE) in the figure below is more symmetrical and bell-shaped than the histogram ofAGE given in part (a). .8 .6 Density .2 0 1 2 3 4 5 lwage
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