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Econometrics Nova School of Business and Economics

Exercise Book

Econometrics

1. Simple Linear Regression Model Exercise 1.1 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

Consider the savings function sav = β0 + β1 inc + u, u =

√

inc.e,

where e is a random variable with E(e)=0 and Var(e)=σe2. Assume that e is independent of inc. (a) Show that E(u|inc)=0, so that the key zero conditional mean assumption (SLR.4) is satisfied. [Hint: If e is independent of inc, then E(e|inc)= E(e)]. (b) Show that Var(u|inc)=σe2 inc, so that homoskedasticity (SLR.5) is violated. In particular, the variance of sav increases with inc. [Hint: Var(e|inc)= Var(e), if e and inc are independent.] (c) Provide a discussion that supports the assumption that the variance of savings increases with family income.

Exercise 1.2 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

The following equation relates housing price (price) to the distance from a recently built garbage incinerator (dist ): \ log(price) = 9.40 + 0.312 log(dist) n = 135, R2 = 0.162 (a) Interpret the coefficient on log(dist). Is the sign of this estimate what you expect it to be? (b) Do you think simple regression provides an unbiased estimator of the ceteris paribus elasticity of price with respect to dist ? (Think about the city’s decision on where to put the incinerator.) (c) What other factors about a house affect its price? Might be these correlated with distance from the incinerator?

1

Econometrics

Exercise 1.3 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

Consider the standard simple regression model y=β0 +β1 x +u under the Gauss-Markov Assumptions SLR.1 to SLR.5. The usual OLS estimators βb0 and βb1 are unbiased for their respective population parameters. Let βe1 be the estimator of β1 obtained by assuming the intercept is zero. (a) Find E(β˜1 ) in terms of the xi , β0 , and β1 . Verify that βe1 is unbiased for β1 when the population intercept (β0 ) is zero. Are there other cases where βe1 is unbiased? (b) Find the variance of βe1 .[Hint: The variance does not depend on β0 .] Pn Pn 2 (c) Show that Var(βe1 ) ≤ Var( βb1 ). [Hint: For any sample of data, i=1 xi ≥ i=1 (xi − x¯)2 , with strict inequality unless x¯=0.]

(d) Comment on the tradeoff between bias and variance when choosing between βb1 and βe1 .

2

Econometrics

2. Multiple Regression Analysis: Estimation Exercise 2.1 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

In a study relating college grade point average to time spent in various activities, you distribute a survey to several students. The students are asked how many hours they spend each week in four activities: studying, sleeping, working, and leisure. Any activity is put into one of the four categories, so that for each student, the sum of hours in the four activities must be 168. (a) In the model GP A = β0 + β1 study + β2 sleep + β3 work + β4 leisure + u does it make sense to hold sleep, work and leisure fixed, while changing study ? (b) Explain why this model violates Assumption MLR.3. (c) How could you reformulate the model so that its parameters have a useful interpretation and it satisfies Assumption MLR.3?

Exercise 2.2 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

(a) Consider the SLR model y= β0 +β1 x +u under the first four Gauss-Markov assumptions. For some function g(x), for example g(x) = x2 or g(x)= log(1 + x2 ), define zi =g(xi ). Define a slope estimator as Pn (zi − z¯)yi βe1 = Pni=1 ¯)xi i=1 (zi − z

Show that βe1 is linear and unbiased. Remember, because E(u|x )=0, you can treat both xi and zi as nonrandom in your derivation. (b) Add the homoskedasticity assumption MLR.5. Show that Pn σ 2 i=1 (zi − z¯)2 ˜ V ar(β1 ) =  Pn 2 ¯)xi i=1 (zi − z

3

Econometrics (c) Show directly that, under the Gauss-Markov assumptions, Var(βb1 ) ≤ Var(βe1 ), where βb1 is the OLS estimator. [Hint: The Cauchy-Schwartz inequality in Appendix B implies that 2     n n n X X X 2 −1 −1 2 −1 (z − z ¯ )(x − x ¯ ) (z − z ¯ ) n ≤ n (x − x ¯ ) ; n i i i i i=1

i=1

i=1

notice that we can drop x¯ from the sample covariance.]

Exercise 2.3 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

The following equation represents the effects of tax revenue mix on subsequent employment growth for the population of counties in the USA: growth = β0 + β1 shareP + β2 shareI + β3 shareS + otherf actors, where growth is the percentage change in employment from 1980 to 1990, shareP is the share of property taxes in total tax revenue, shareI is the share of income tax revenues, and shareS is the share of sale tax revenues. All of these variables are measured in 1980. The omitted share, shareF , includes fees and miscellaneous taxes. By definition, the four shares add up to one. Other factors would include expenditures on education, infrastructure, and so on (all measured in 1980). (a) Why must we omit one of the tax share variables from the equation? (b) Give a careful interpretation of β1 .

Exercise 2.4 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

The following model is a simplified version of the multiple regression model used by Biddle and Hamermesh (1990) to study the trade-off between time spent sleeping and working and to look at other factors affecting sleep: sleep = β0 + β1 totwrk + β2 educ + β3 age + u where sleep and totwrk are measured in minutes per week and educ and age are measured in years. (a) If adults trade-off sleep for work, what is the sign of β1 ? (b) What signs do you think β2 and β3 will have?

4

Econometrics (c) Consider the following estimated equation: [ = 3, 638.25 − 0.148totwrk − 11.13educ + 2.2age n = 706, R2 = 0.113 sleep If someone works five more hours per week, by how many minutes is sleep predicted to fall? Is this a large trade-off? (d) Discuss the sign and magnitude of the estimated coefficient on educ. (e) Would you say totwrk, educ, and age explain much of the variation in sleep? What other factors might affect the time spent sleeping? Are these likely to be correlated with totwrk ?

5

Econometrics

3. Multiple Regression Analysis: Inference Exercise 3.1 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

In exercise 2.4 we estimated the following equation: [ = 3, 638.25 − 0.148totwrk − 11.13 educ + 2.20 sleep age n = 706, R2 = 0.113 (112.28)

(0.017)

(5.88)

(1.45)

where we now report standard errors along with the estimates. (a) Is either educ or age individually significant at the 5% level against a two-sided alternative? Show your work. (b) Dropping educ and age from the equation gives [ = 3, 586.38 − 0.151 totwrk n = 706, R2 = 0.103 sleep (38.91)

(0.017)

Are educ and age jointly significant in the original equation at 5% level? Justify your answer? (c) Does including educ and age in the model greatly affect the estimated trade-off between sleeping and working? (d) Suppose that the sleep equation contains heteroskedasticity. What does this mean about the tests computed in parts (a) and (b)?

Exercise C1 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

Use the data in WAGE2.RAW for this problem. As usual, be sure all of the following regressions contain an intercept. (a) Run a simple regression of IQ on educ to obtain the slope coefficient, say δe1 .

(b) Run the simple regression of log(wage) on educ, and obtain the slope coefficients, βe1 . (c) Run the multiple regression of log(wage) on educ and IQ, and obtain the slope coefficients, βb1 and βb2 , respectively.

(d) Verify that βe1 = βb1 + βb2δe1 .

6

Econometrics

Exercise 3.2 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

Consider the following model: log (scrap) = β0 + β1 hrsemp + β2 log (sales) + β3 log (employ ) + u

where hrsemp is annual hours of training per employee, sales is annual firm sales (in dollars), and employ is the number of firm employees. Using 29 observations the estimated equation is: \ log (scrap) = 12.46 − 0.029 hrsemp − 0.962 log (sales) + 0.761log (employ ) (5.69)

(0.453)

(0.023)

(0.407)

n = 29, R2 = 0.262

Using 43 observations the estimated equation is: \ log (scrap) = 11.74 − 0.042 hrsemp − 0.951 log (sales) + 0.992log (employ ) (4.57)

(0.370)

(0.019)

(0.360)

2

n = 43, R = 0.310

(a) Compare the two estimated equations. (b) Show that the population model can also be written as log (scrap) = β0 + β1 hrsemp + β2 log (sales/employ ) + θ3 log (employ ) + u

where θ3 = β2 + β3 . Interpret the null hypothesis H0 : θ3 = 0. (c) When the equation from part (b) is estimated, we obtain log\ (scrap) = 11.74− 0.042 hrsemp − 0.951 log (sales/employ ) + 0.041log (employ ) (4.57)

(0.019)

(0.370)

(0.205)

n = 43, R2 = 0.310

Controlling for worker training and for the sales-to-employee ratio, do bigger firms have larger statistically significant scrap rates? (d) Test the hypothesis that a 1% increase in sales/employ is associated with a 1% drop in the scrap rate.

7

Econometrics

Exercise 3.3 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

Consider the following table: Dependent Variable: log(salary ) Independent Variables (1) (2) . 158 log(sales) .224 (.027)

(.040)

(3) .188

(.040)

.100

log(mktval )

−

.112

(.050)

(.049)

profmarg

−

−.0023

−.0022

ceoten

−

−

.0171

comten

−

−

−.0092

intercept

4.94

4.62

(0.25)

(0.25)

Observations R-squared

177 .281

177 .304

177 .353

(.0022)

(0.20)

(.0021)

(.0055)

(.0033)

4.57

The variable mktval is market value of the firm, profmarg is profit as a percentage of sales, ceoten is years as CEO with the current company, and comten is total years with the company. (a) Comment on the effect of profmarg on CEO salary. (b) Does market value have a significant effect? Explain. (c) Interpret the coefficients on ceoten and comten. Are these explanatory variables statistically significant?

Exercise 3.4 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

Consider the following estimated equation which can be used to study the effects of skipping class on college GPA: \A = 1.39 + .412hsGP A − .015ACT + .083 skipped colGP (.33)

(.094)

(.011)

(.026)

n = 141, R2 = 0.234 (a) Using the standard normal approximation, find the 95% confidence interval for βhsGP A . (b) Can you reject the hypothesis H0 : βhsGP A = 0.4 against a two-sided alternative at 5% level? (c) Can you reject the hypothesis H0 : βhsGP A = 1 against a two-sided alternative at 5% level? 8

Econometrics

Exercise 3.5 Consider the results in Table 1. (a) In the regression in column (3), the estimated value of β1 is −1.00. What does a value of −1.00 mean in this regression? (b) Using the results in column (3), construct a 99% confidence interval for β1 . (c) Construct the R2 for the regression in column (3). (d) The R¯2 in the regression in column (3) is much higher than the regression in column (1). Does this mean that it has eliminated any potential omitted variable bias? Explain. (e) Let β4 denote the coefficient on the variable ”Percent on public income assistance”. (i) Is β4 statistically significant in the regression in column (4)? Construct a 95% confidence interval for β4 using the column (4) regression. (ii) Is β4 statistically significant in the regression in column (5)? Construct a 95% confidence interval for β4 using the column (5) regression. (iii) Explain why the answers to (e.1) and (e.2) are so different? (f) An F -statistic testing the null hypothesis H0 : β2 = β4 = 0 is carried out for the regression specification in column (5). The value of the statistic is 6.88. (i) Is H0 : β2 = β4 = 0 rejected at the 1% significance level? (ii) Is the point H0 : β2 = β4 = 0 contained in the 99% confidence set for β2 and β4 ? Explain. (g) In the regression represented in column (1), is the student-teacher ratio uncorrelated with the regression error? Is the correlation positive or negative? (h) Comparing columns (1) and (2), do you think the student-teacher ratio and the percent of English learners is positively or negatively correlated? Explain. (i) Suppose that the sample size were doubled, so that n = 840. How would you expect the standard errors of the ordinary least squares estimators to change? Explain.

9

Econometrics

4. Multiple Regression Analysis: Dummy Variables Exercise 4.1 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

Consider the following estimated equation: [ = 3, 840.83 − .163totwrk − 11.71 educ − 8.7 age + 0.128 2 sleep age + 87.75male (235.11)

(.018)

(5.86)

(11.21)

(.134)

(34.33)

n = 706, R2 = .123, R¯2 = .117 The variable sleep is total minutes per week spent sleeping at night, totwrk is total weekly minutes spent working, educ and age are measured in years, and male is a gender dummy. (a) All other factors being equal, is there evidence that men sleep more than women? How strong is the evidence? (b) Is there a statistically significant trade-off between working and sleeping? What is the estimated trade-off? (c) What other regression do you need to run to test the null hypothesis that, holding other factors fixed, age has no effect on sleeping?

Exercise 4.2 (Wooldridge, J.

M., Introductory Econometrics, 4th edition)

Consider the following estimated equation: sc at = 1, 028.10+19.30hsize−2.19hsize2 −45.09f emale−169.81 black+62.31f emale·black (6.29)

(3.83)

(.53)

(4.29)

(12.71)

(18.15)

n = 4, 137, R2 = .0858 The variable sat is the combined SAT score, hsize is size of the student’s high school graduating class, in hundreds, female is a gender dummy variable, and black is a race dummy variable equal to one for blacks and zero otherwise. (a) Is there strong evidence that hsize2 should be included in the model? From this equation, what is the optimal high school size? (b) Holding hsize fixed, what is the estimated difference in SAT score between nonblack females and nonblack males? How statistically significant is this estimated difference? 10

Econometrics (c) What is the estimated difference in SAT score between nonblack males and black males? Test the null hypothesis that there is no difference between their scores, against the alternative that there is a difference. (d) What is the estimated difference in SAT score between black females and nonblack females? What would you need to do to test whether the difference is statistically significant?

Exercise 4.3 Consider the regression results given in Table 2 on the determinants of unemployment duration in the Portuguese labor market, where the dependent variable is the logarithm of elapsed unemployment duration, measured in months (logdur) and UBELEG is a dummy variable which is equal to one if the unemployed individual can collect unemployment benefits, zero otherwise. The sample comprises 9317 unemployed individuals. (a) Provide a rigorous interpretation for the regression coefficient on UB-ELEG. (b) Test the overall significance of the regression, providing the value for the F statistic. Consider now an extended version of the previous model in Table 3, where UR is the unemployment rate (measured in percentage points), SCH stands for education (measured in years), and age is measured in years. (c) Give a precise interpretation of all regression coefficients. (d) How do you explain the reduction in the estimated coefficient for UB-ELEG? The model was re-specified using seven age dummies instead the variable age (regression results given in Table 4). In this specification A2 is equal to 1 for individuals aged between 25 and 29 years; A3 equals 1 for individuals aged between 30 and 34; A4 equals 1 for individuals aged between 35 and 39; A5 equals 1 for individuals aged between 40 and 44; A6 equals 1 for individuals aged between 45 and 49; A7 equals 1 for individuals aged between 50 and 54; and A8 equals 1 for individuals aged 55 or older. (e) Which specification is preferable? Why? Is the reduction in the sum of squared residuals large enough to justify the inclusion of the seven variables? (f) How would you test that there are no differences in the mean duration of unemployment between individuals aged 30 to 34 and those aged between 45 and 49? 11

Econometrics Consider, finally, a regression where the dependent variable (EMP) is a dummy which takes the value 1 if the unemployed worker found a suitable job in following period and zero if he stayed unemployed otherwise. The UI variable is equal to 1 if the individual was collecting unemployment benefits and zero otherwise. The dur variable is the measure of elapsed unemployment duration, in months. The estimation results are presented in Table 5. (g) Interpret all the regression coefficients. (h) Discuss the validity of the standard errors, t-student, and F statistics. (i) Describe, briefly, how you would implement a feasible generalized least square estimator for this regression model?

Exercise 4.4 The players of a basketball team can play in three positions: guard, forward or center. The variable exper is the number of years as a professional player, points is the number of points scored per game, guard is a binary variable equal to one if the player is a guard and 0 otherwise, and forward is a binary variable equal to one if the player is a forward and 0 otherwise. Standard errors are reported in parentheses. Consider the following estimated equation: \ = 4.76 + 1.28 exper − 0.072 exper 2 + 2.31guard + 1.54f orward points (1.18)

(0.33)

(1.00)

(0.024)

(1.00)

2

n = 269, R2 = 0.091, R = 0.077 (a) Interpret the regression coefficients. (b) Why do you not include all three position dummy variables in the model? (c) Holding experience fixed, does a guard score more than a center? How much more? Is the difference statistically significant? (d) Considering also the following estimated equation, test if experience influences points. \ = 8.52 points + 2.40 guard + 1.68f orward (0.86)

(1.03)

(1.03)

2

n = 269, R2 = 0.020, R = 0.012 12

Econometrics (e) Consider a new equation with marital status (marr) as an additional explanatory variable: \ = 4.70 + 1.23exper − 0.07 exper 2 + 2.29 guard + 1.54f orward + 0.58marr points (1.18)

(0.33)

(1.00)

(0.02)

(1.00)

(0.74)

Holding position and experience fixed, are married players more productive (based on points per game)? (f) Consider the two estimation outputs given in Tables 6 and 7 for a new model which includes interactions of marital status with both experience variables. Is there strong evidence that marital status af...