5. Let W follow a normal distribution with mean of μ and the variance of 1. Then, the pdf of W is

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1 Practice Final Exam Last Name:, First Name:. Please write LEGIBLY. Answer all questions on this exam in the space provided (you may use the back of any page if you need more space). Show all work but do not include irrelevant material. Good luck! 1. Explain differences between u (the error term) and û (the residual). 2. Consider the linear regression model in matrix form, y = Xβ + u. Under the Gauss-Markov assumptions, the variance-covariance matrix of u is σ 2 I where I is an identity matrix. That is, Var(u X) =σ 2 I where Var(u i X) =σ 2 : true or false? 3. Consider the regression of SAT scores on high school sizes: sat = β 0 + β 1 hsize + u where sat = the average SAT score of the graduating class and hsize = the size of the graduating class in hundreds. When estimated, we have ˆβ 1 =5,andthet statistic testing for H 0 : β 1 =0is3. Suppose that we measure the high school sizes in thousands, not in hundreds and sat is regressed upon the high school size variable in thousands. Then, the estimated slope coefficient is 50 and the reported t statistic for the null of no effect is 30: true or false? 4. Consider log(wage) = female+ where female = 1 if a person s gender is female. The exact percentage difference in the predicted wage between male and female is 100[exp(.297) 1] = 25.7: true or false? 5. Let W follow a normal distribution with mean of μ and the variance of 1. Then, the pdf of W is f(w ; μ) = 1 ) (W μ)2 exp ( 2π 2 In order to estimate μ, we collect a random sample of size n and plan to use the maximum likelihood estimation (MLE). To carry out MLE, we need a likelihood function. Write down the likelihood function for this problem. 1

2 6. Consider the model for gas mileage: MPG = β 1 + β 2 hp + β 3 vol + β 4 weight + u, where MPG = the average miles per gallon, hp = engine horsepower, vol = cubic feet of cab space, and weight = vehicle weight measured in hundreds of pounds. The observation units are cars and the data are collected by U.S. Environmental Protection Agency (EPA) in The data consist of 20 observations and, from this data, the followings are available: ˆβ = , Var( ˆβ) =ˆσ 2 (X X) 1 = At the job interview with U.S. EPA, you are asked to suggest a different model for explaining MPG. With the knowledge in nonlinear regressors, you suggest the following model: MPG = β 1 + β 2 hp + β 3 hp 2 + β 4 vol + β 5 weight + β 6 (vol weight)+ɛ, and when estimated, MPG = hp hp vol weight (vol weight). (a) Using the estimated model, sketch a graph of MPG versus hp that shows the approximate relationship between these two variables (we assume that all the other regressors are held constant).. (b) Using the formula for the turning point, hp (estimated coefficient on hp) = 2 (estimated coefficient on hp 2 ), calculate the value of hp. (c) Do you think this (hp from the above) is the level of horsepower that maximizes MPG? Explain! Note that in the sample data, the minimum of hp is 49 and the maximum of hp is 92. (d) Explain the rationale for including the variable (vol weight) in the regression. In particular, explain why anyone might want to include such a variable. (e) Consider the most popular car brand, FB by Gord. FB has a cab space of 84, and Gord has decided to decrease the weight of FB in an attempt to increase FB s mileage. Then, what is the marginal effect of weight on FB s MPG given that vol = 84? 2

3 7. Consider the following regression where typo i is the number of typing errors and hours i is the number of hours spent in practicing: typo i = β 0 + β 1 hours i + u i, Var(u i hours i )= σ2 hours i, where it is known that β 0 > 0andβ 1 < 0. The Gauss-Markov assumptions other than the constant error variance assumption hold. (a) Draw a graph illustrating the heteroskedasticity in the above regression. Be sure to label each axis. (b) Are the OLS estimates, ˆβ 0 and ˆβ 1 from the regression of typo on hours BLUE? If not, how would you transform the regression equation so that you would get BLUE estimates by running OLS on the transformed regression? Write down the transformed equation with homoskedasticity. 8. Consider the following regression: rdintens t = log(sales t )+0.050profmarg t (1.369) (0.216) (0.046) T =32, R 2 =0.883, DW =1.4, where the numbers in the parentheses are standard errors. The variable rdintens t is expenditures on research and development (R&D) as a percentage of sales, sales t are measured in millions of dollars and the variable profmarg t is profit as a percentage of sales. (a) Is there first-order autocorrelation in the above regression? Explain. If first-order autocorrelation is present, indicate whether it is the positive type or the negative type and provide specific statistical evidence to support your answer. 3

4 9. Consider the following normal regression model, which is used to determine how marital status and exercise affect the health level of adults in the US: health i = β 0 + β 1 exercise i + β 2 married i + β 3 (exercise i married i )+u i, where health i = the health status of person i (this is a number from 0 to 100 which measures how healthy person i is; 0 indicates worst health and 100 indicates best health); exercise i =the average number of hours per week of exercise obtained by person i ; married i = an intercept dummy variable that takes on a value of 1 if person i has been married for a majority of his or her adult life, and 0 if not. You may assume that this model satisfies all ideal conditions. (a) What is the base group of this model? Write out the expression for the slope of the regression line for the included group of this model. (b) Now suppose the above model was estimated using the procedure of OLS using a random sample of 27 observations. These results are shown below (number in parentheses are estimated standard errors): ĥealth i =35+5exercise i +15married i +3(exercise i married i ) (17) (2) (6.2) (1) i. Sketch a graph of health status versus exercise that illustrates the above regression results. Be sure to draw two regression lines, one for married individuals and one for non-married individuals, and clearly label which is which. In addition, be sure to indicate the exact numerical values of the intercepts and slopes for these two regression lines. ii. Based on the above results, is there any evidence (at α =.01) to support the idea that the marginal effect of exercise on health status is greater for people who have been married for a majority of their adult life? As evidence for you answer, be sure to do the following: (1) write out the null and alternative hypotheses of the test (in terms of betas), (2) write out the form of the test statistic and calculate the statistic, (3) note the degrees of freedom and critical value of the test, and (4) explain your conclusion about the test. 4

5 10. We are interested in the women s labor force participation decision. Consider the following regression model inlf = β 0 + β 1 nwifeinc + β 2 educ + β 3 exper + β 4 expersq + β 5 age + β 6 kidslt6+β 7 kidsge6+u, where inlf = 1 if a woman is in the labor force and 0 otherwise, nwifeinc = households income excluding a woman s income measured in thousands of dollars, educ = years of education, exper = years of experience, expersq = exper 2, age = age in years, kidslt6 =thenumberofkidslessthan six-years old, and kidsge6 = the number of kids six-years old or older. (a) The above regression model is called the linear probability model. Why is it? (b) By running OLS using the data on married women aged between 30 and 60, we have obtained the following estimated regression equation: inlf = nwifeinc educ exper 0.001expersq 0.016age 0.262kidslt kidsge6. i. Suppose Rachel s husband gets a raise of $50,000. Holding other factors constant, what happens to Rachel s labor force participation probability? Is it percent change or percentage point change? 11. Consider the following simple linear regression model, y i = β 0 + β 1 x i + u i. The OLS estimator for β 1 is ˆβ 1 = n i=1 (x i x)(y i ȳ) n i=1 (x i x) 2 Show that, under the Gauss-Markov assumptions, Var( ˆβ 1 X) = σ 2 n i=1 (x i x) 2 where Var(u i X) =σ 2 for all i. 5

6 12. Consider the Stata output on the handout. The goal is to see the effect of a college student s PC ownership on the student s college GPA. The variables used in the regression model are: colgp A = a student s college GPA, hsgp A = a student s high school GPA, ACT = a student s ACT test score, skipped = average lectures missed per week, and PC = 1 if a student owns a computer and 0 otherwise. The residuals (uhat), the fitted values (yhat) and some powers of the residuals and the fitted values are generated. The regression equation that we are interested in is colgp A = β 0 + β 1 hsgp A + β 2 ACT + β 3 skipped + β 4 PC + u (a) Using the first regression output, interpret the estimated coefficient on PC.IsPC statistically significant at α =.05? (b) We are worried about a possible heteroskedasticity in u so that we perform the White test for heteroskedasticity. Given the Stata output with α =.05, what do you conclude? (c) Given the White test result, what is the reliable standard error for the estimate on PC? Find the standard error for the estimate on PC that we can use for t test on H 0 : β 4 =0. (d) Someone points out that there may be a nonlinear relationship between colgp A and some regressors. Given the Stata output, can we conclude that we do not need to worry about such nonlinear relationship? How do we know? 13. If your (future) boss is asking you what is Econometrics, what would you tell him or her in less than 30 words? 6

7 The Stata output is generated using GPA1.DTA.. regress colgpa hsgpa ACT skipped PC Source SS df MS Number of obs = 141 F( 4, 136) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = colgpa Coef. Std. Err. t P> t [95% Conf. Interval] hsgpa ACT skipped PC _cons predict yhat, xb. predict uhat, resid. gen uhatsq = uhat^2. gen yhatsq = yhat^2. gen yhatcubed = yhat^3. regress uhatsq yhat yhatsq Source SS df MS Number of obs = 141 F( 2, 138) = 3.58 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = uhatsq Coef. Std. Err. t P> t [95% Conf. Interval] yhat yhatsq _cons test yhat yhatsq ( 1) yhat = 0 ( 2) yhatsq = 0 F( 2, 138) = 3.58 Prob > F = regress colgpa hsgpa ACT skipped PC, robust Linear regression Number of obs = 141 F( 4, 136) = Prob > F = R-squared = Root MSE = Robust colgpa Coef. Std. Err. t P> t [95% Conf. Interval] hsgpa ACT skipped PC _cons

8 . regress colgpa hsgpa ACT skipped PC yhatsq yhatcubed Source SS df MS Number of obs = 141 F( 6, 134) = 8.48 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = colgpa Coef. Std. Err. t P> t [95% Conf. Interval] hsgpa ACT skipped PC yhatsq yhatcubed _cons test yhatsq yhatcubed ( 1) yhatsq = 0 ( 2) yhatcubed = 0 F( 2, 134) = 1.46 Prob > F =