Universidad Carlos III de Madrid Econometría Nonlinear Regression Functions Problem Set 8

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1 Universidad Carlos III de Madrid Econometría Nonlinear Regression Functions Problem Set 8 1. The sales of a company amount to 196 millions of dollars in 2009 and increased up to 198 millions in (a) Compute the percentage of increment using the formula 100 (Sales 2010 Sales 2009 ) Sales Compare this value with the approximation 100 [ln (Sales 2010 ) ln (Sales 2009 )]. (b) Repeat a) assuming that Sales 2010 = 205, Sales 2010 = 250 and Sales 2010 = 500. (c) How good is the approximation when the change is small? Does the quality of the approximation deteriorates when the percentage change increases? 2. Imagine that a researcher collect data on the house sold in a given area during the last year and gets the following regression results: Regressor Dependent Variable: ln(price) (1) (2) (3) (4) (5) Size ( ) ln(size) 0.69 (0.054) 0.68 (0.087) 0.57 (2.03) ln(size) (0.14) Bedrooms (0.037) Swimming P ool (0.032) V iews (0.029) (0.034) (0.028) (0.034) (0.026) (0.036) (0.029) 0.69 (0.055) (0.035) (0.030) Swimming P ool V iew (0.10) State 0.13 (0.045) Intercept (0.069) 0.12 (0.035) 6.60 (0.39) 0.12 (0.035) 6.63 (0.53) 0.12 (0.036) 7.02 (0.50) 0.12 (0.035) 6.60 (0.40) Summary Statistics SER R (a) Using the results of column (1), which is the expected change in the price of a house if an attachment of 500 square feet is built to it? Construct a 95% confidence interval for the percentage change in price. (b) Comparing columns (1) and (2), which is better to use, the variable size or ln (size), to explain the house price? (c) Using column (2), which is the estimated effect of having swimming pool on the price? (Make sure of expressing the result in the correct units of measure). Build a 95% confidence interval for this effect. (d) The regression in column (3) adds the variable number of bedrooms to the regression. Which is the magnitude of the estimated effect of having an additional bedroom? Is this effect statistically significative? Why do you believe that the estimated effect is so small? (Hint: which other variables are being held constant?) (e) Is the term ln (size) 2 important? (f) Use the regression in column 85) to calculate the expected variation in price when a swimming pool is added to a house without views. Repeat for the effect of a house with good views. Is there a big difference? Is this difference statistically significant? 1

2 3. This problem is inspired by a study on the "gender gap" in the wage earnings for the highest level of corporation executives (Bertrand and Hallock 2011). The study compares the total compensation of the top executives for a large number of US corporations in the 1990 decade. (Each year these companies have to inform on the total earnings of their main five executives). (a) Let F emale be an indicator variable which is equal to 1 for women and 0 for men. A regression of the log of the earnings on the variable F emale led to the following result: ln(earnings) = F emale, SER = (0.01) (0.05) 1. The estimated coeffi cient for F emale is Explain which is the meaning of this value. 2. The SER is Explain which is the meaning of this value. 3. Is this regression suggesting that the women that have top executive positions earn less that the top executives which are male? (b) Two new variables are added to the regression, the market value of the company (a measure of the size of the company in millions of dollars) and the stock return (a measure of the performance of the company, in percentage points). ln(income) = F emale (0.03) (0.04) n = , R2 = (0.0034) ln(marketvalue) (0.003) Return, 1. The coeffi cient of ln(marketvalue) is Explain which is the meaning of this value. 2. The coeffi cient of the variable F emale is now Explain why this has changed with respect to the regression in (a). (c) Are the big companies more likely to have top female executives than the small companies? Explain. 4. X is a continuous variable that takes values between 5 and 100. Z is a binary variable. Represent the following regression functions (with values of X between 5 and 100 in the horizontal axis and the values of Ŷ in the vertical axis): (a) (b) (c). (d). Ŷ = 2, 0 + 3, 0 ln (X). Ŷ = 2, 0 + 3, 0 ln (X). 1. Ŷ = 2, 0 + 3, 0 ln (X) + 4, 0 Z, with Z = Ŷ = 2, 0 + 3, 0 ln (X) + 4, 0 Z, with Z = Ŷ = 2, 0 + 3, 0 ln (X) + 4, 0 Z 1, 0 Z ln (X), with Z = Ŷ = 2, 0 + 3, 0 ln (X) + 4, 0 Z 1, 0 Z ln (X), with Z = 0 (e) Ŷ = 1, 0 + 1, 25 X 0, 01 X2. 5. Consider the regression model Y i = β 1 X 1i + β 2 X 2i + β 3 (X 1i X 2i ) + U i. Show that (a) Y / X 1 = β 1 + β 3 X 2 (effect of a change in X 1 holding X 2 constant). (b) Y / X 2 = β 1 + β 3 X 1 (effect of a change in X 2 holding X 1 constant). (c) If X 1 changes X 1 and X 2 changes X 2, then Y = (β 1 + β 3 X 2 ) X 1 + (β 2 + β 3 X 1 ) X 2 + β 3 X 1 X We are studying the factors behind the Body Mass Index (BMI), defined as the weight (in kilograms) divided by the squared of height (in meters). For a random sample we have estimated the following model (Model 1, Output 1), BMI = β 0 + β 1 drinks + β 2 drinks 2 + β 3 female (Model 1) 2

3 where drinks = number of days during the last year in which the individual has drunk 5 or more glasses of alcohol. drinks 2 = squared of drinks. female = is a dummy variable that takes a value one for women and zero otherwise. a) Using Model 1, What is the marginal impact of drinks on expected BMI? Is this effect constant? For which values of drinks is this effect positive (negative)? Explain. b) Using Model 1, what is the predicted difference in BMI betwen a man with drinks = 2 and a woman with drinks = 6? Alternatively, the following model has been estimated (Model 2, Output 2) BMI = β 0 + β 1 drinks + β 2 drinks 2 + β 3 female + β 4 drinks female + β 5 drinks 2 female (Model 2) c) Using Model 2, What is the predicted difference in expected BMI between a man with drinks = 2 and a woman with drinks = 6? d) Using Model 2, what is the impact of a marginal change in drinks on BMI for a man? What is the impact of a marginal change in drinks on BMI for a woman? e) Using Model 2 as benchmark (unrestricted model), explain and test that the effect of a marginal change in drinks on BMI is linear for men. f) Using Model 2 as benchmark (unrestricted model), explain and test that the effect of a marginal change in drinks on BMI is linear for women. g) Using Model 2 as benchmark (unrestricted model), explain and test that the effect of a marginal change in drinks on BMI is linear for an individual whatever the gender. OUTPUT 1: OLS, using observations Coeff S.E t p-value const drinks drinks e female Sum of the Squared Residuals R OUTPUT 2: OLS, using observations Coeff S.E t p-value const drink drink e female drinks*female drinks2*female

4 Sum of the Squared Residuals R OUTPUT 3: OLS, using observations Coeff S.E t p-value const drinks drinks e 005 Sum of the Squared Residuals R OUTPUT 4: OLS, using observations Coeff S.E. t p-value const drinks drinks*female female Sum of the Squared Residuals R OUTPUT 5: OLS, using observations Coeff S.E. t p-value const drink female drinks*female drinks2*male e 005 Sum of the Squared Residuals R NOTE: male is a dummy variable that takes a value one when the person is a man and zero otherwise. 7. We are interested in explaining a worker s wage in terms of the number of years of education (educ) and years of experience (exper) using the following model: log (wage) = β 0 + β 1 educ + β 2 exper + u, where we assume that u satisfies the classical assumptions of OLS and is homoskedastic. The estimated parameters by OLS for a sample of n = 935 observations are displayed in the first column of Table 2. 4

5 Table 2. OLS estimates. Dependent variable: log(wage) Variables Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 educ ( ) ( ) ( ) ( ) ( ) ( ) exper ( ) ( ) ( ) ( ) ( ) educ married ( ) exper married ( ) married ( ) ( ) ( ) ( ) ( ) black ( ) ( ) ( ) married black ( ) ( ) ( ) exper e-05 ( ) const ( ) ( ) ( ) ( ) ( ) ( ) Observations R Several extensions of this model were considered to address the effects of being married (with the binary variable married) and/or being black (with binary black) or possible nonlinearity on the effect of years of experience. Using the appropriate output from Table 2 answer the following questions: (a) Test whether the wage regressions for married workers and unmarried workers are the same. Write down the estimated regression for each group and interpret the coeffi cient of educ. (b) Based on a statistical test, do the effects of education and experience depend on the civil status? (c) Given a number of years of education and experience, which is the group defined by the variables married and black with the lowest average log(wage)? Does this group have a significantly lower wage than the group of single non-black workers? [Use Model 4 for this question] (d) What we conclude about the possible nonlinearity of the relationship of log (wage) with respect to the years of experience? Can you conclude that years of experience has no significant effect on log (wage) in Model 5? Make two statistical tests to answer these questions. 8. Using the database TeachingRatings of Stock and Watson carry out the following exercises: (a) Estimate a regression of the variable Course_Eval on the variables Beauty, Intro, OneCredit, F emale, Minority and NNEnglish (b) Add the variables Aget and Age 2 to the regression. Is there evidence that the variable Age has a nonlinear effect on the variable Course_Eval? Is there evidence that the variable Age has any effect on the variable Course_Eval? (c) Change the regression in a) so that the effect of the variable Beauty on the variable Course_Eval is different for men and women. Is the difference between men and women effects on the variable Beauty statistically significative? (d) Professor Smith is a man. He gets a plastic surgery to increase his beauty which increment his beauty index from one standard deviation below the average up to one standard deviation above the average. Which is the value of the variable Beauty which would correspond to him before the surgery? And after the surgery? Using the regression in c), build a 95% confidence interval for the increment in course evaluation. (e) Repeat d) for Professor Jones, who is a women. 5

6 9. Use the data base CollegeDistance to answer the following questions: (a) Run a regression of the variable ED on the variables Dist, F emale, Bytest, T uition, Black, Hispanic, Incomehi, Ownhome, DadColl, MomColl, Cue80 and Stwmfg80. If the variable Dist increases from 2 to 3 (i.e. from 20 to 30 miles), how much do you expect the years of education to change? (b) Run a regression of the variable ln(ed) on the variables Dist, F emale, Bytest, T uition, Black, Hispanic, Incomehi, Ownhome, DadColl, MomColl, Cue80 and Stwmfg80. If the variable Dist increases from 2 to 3 (i.e. from 20 to 30 miles), how much do you expect the years of education to change? (c) Run a regression of the variable ED on the variables Dist, Dist 2, F emale, Bytest, T uition, Black, Hispanic, Incomehi, Ownhome, DadColl, MomColl, Cue80 and Stwmfg80. If the variable Dist increases from 2 to 3 (i.e. from 20 to 30 miles), how much do you expect the years of education to change? If the variable Dist increases from 6 to 7 (i.e., from 60 to 70), how much do you expect the years of education to change? (d) Do you prefer regression c) to regression a)? Explain. SOLUTIONS: 1. (a) The exact answer is % and the approximation is %. (b) If Sales 2010 =205, % and %, resp. Sales 2010 =250, % and %. Sales 2010 = 500, 155.1% and %. 2. (a) 21%, [17.276%, %]. (b) ln(size). (c) 7.1% [0.436%, %]. (d) 0.36%, No significant t = (e) No significant, t = (f) Without views: 7.1%. With views: 7.32%. 3. (a) 3. Yes. (b) 2. The regression in (a) has an omitted variable problem. (c) No. 6. a) The impact is not constant. The value for which the marginal impact of drinks changes sign is drinks = An increase in the consumption of alcohol is associated with a reduction in the BMI. Nevertheless, an increase in the consumption of alcohol for values of drinks over drinks is associated with an increase in BMI. b) E[BMI female = 0, drinks = 2] E[BMI female = 1, drinks = 6] = c) E[BMI female = 0, drinks = 2] E[BMI female = 1, drinks = 6] = d) Marginal Effect = BMI ˆ drinks = (ˆβ 1 + β ˆ 4 female) + 2 drinks (ˆβ 2 + β ˆ 5 female) M. Effect for a man = BMI ˆ drinks = ˆβ ˆβ 2 drinks = drinks = drinks M. Effect for a woman = BMI ˆ drinks = (ˆβ 1 + β ˆ 4 ) + 2 drinks (ˆβ 2 + β ˆ 5 ) = drinks 6

7 e) We can reject that the marginal effect is constant and we confirm is linear for men. f) We can reject that the impact of drinks is constant and we confirm is linear for women. g) We can reject that the impact of drinks is constant in general so that is linear at least for some type of individuals. 7. (a) For this test we have to consider the non restricted regression (Model 2) and test H 0 : β 3 = β 4 = β 5 = 0 H 1 : H 0 false and we can reject the null hypothesis that both regressions are the same (F = ). (b) For this test we have to consider again the non restricted regression (Model 2) and test H 0 : β 3 = β 4 = 0 H 1 : H 0 falsa so that we can not reject the null hypothesis and we do not find evidence supporting that the effects of education and experience depend on the civil status (F = ). (c) For the first question we check the coeffi cients of married, black and married black in Model 4, and the different subgrups, E [log (wage) \educ, exper, married = 0, black = 0] = β 0 + β 1 educ + β 2 exper E [log (wage) \educ, exper, married = 1, black = 0] = β 0 + β 1 educ + β 2 exper + β m E [log (wage) \educ, exper, married = 0, black = 1] = β 0 + β 1 educ + β 2 exper + β b E [log (wage) \educ, exper, married = 1, black = 1] = β 0 + β 1 educ + β 2 exper + β m + β b + β mb for a fixed level of educ and exper and the combination which provides a lowest predicted value of log (wage) is black = 1 and married = 0. For the comparison the one-sided test required is H 0 : β b = 0 H 1 : β b < 0 which is performed with a t test (t = ) which is significant at the 1% and therefore the null hypothesis of wages equality is rejected in favour of black having a lower wage within the group of nonmarried. (d) For the first question we have to test in Model 5 if H 0 : β exper 2 = 0 H 1 : β exper 2 0 through a t test (t = 0.084) which is not significant at the 5% level, so that the quadratic (non linear) term is not significant once we have already included the linear term. For the second question we have to test in Model 5 if H 0 : β exper = β exper 2 = 0 H 1 : H 0 false compared to the restricted regression (Model 6) (F = ), so that we reject the null hypothesis that the years of experience have no effect on log (wage), despite the two t statistics are not individually significative. 7

8 8. (b) No and No. (c) Yes. (d) Increment: Confidence interval: [0.22, 0.51]. (e) Increment: Confidence interval: [0.02, 0.27]. 9. (a) Reduction of years. (b) 0.26% in both cases. (c) and years in each case. (d) (c) better than (a). 8