Quantitative Methods Final Exam (2017/1)

Transcription

1 Quantitative Methods Final Exam (2017/1) 1. Please write down your name and student ID number. 2. Calculator is allowed during the exam, but DO NOT use a smartphone. 3. List your answers (together with the detailed calculations). 4. Round up to two decimal places. 1. (Nonlinear Regression, 25%) Gabriel wants to investigate the effects of prenatal visits on birth weight. He considers the following regression: lbwght = β 0 + β 1 npvis + β 2 mage + β 3 cigs +u i where lbwght is the natural log of birth weight in grams, npvis is the total number of prenatal visits, mage is mother s age, and cigs is average cigarettes per day. Use STATA outputs to answer the following questions. (Notice that the order of the STATA results might not be in the same order as the questions.) (a) According to OLS results, how does the total number of prenatal visits affect the birth weight of the infant? (5%) (b) Gabriel is concerned about omitted variable bias. Therefore, he includes additional variables: drink (average drinks per week), mwhite (=1 if mother is white), and fage (the father s age). Based on the estimated results, comment if omitted variable bias was a serious concern in the estimation. (5%) (c) Gabriel is concerned that there might be nonlinear effect in total number of prenatal visits. He includes the square of npvis. According to the results, do you think there is a nonlinear effect in npvis? (5%) Gabriel thinks that the effect of prenatal visits on birth weight might differ with respect to the mother s age. Therefore, he considers the following regression: lbwght = β 0 + β 1 npvis + β 2 mage + β 3 cigs + β 4 npvis 2 + β 5 npvis mage +u i (d) According to the estimated results, is there any evidence showing the mother s age generates a differential effect of prenatal visits on birth weight (use 5% significance level)? (5%) (e) Continuing (d), does mother s age have an impact on birth weight (use 5% significance level)? (5%)

2 ANS: (a) According to the OLS results, one extra prenatal visit corresponds to an increase of 0.55% in birth weight. (b) Adding the additional variables, the coefficient on npvis changes from to , which is not a substantial change. Therefore, the original model doesn t suffer from severe omitted variable bias. (c) H 0 :β npvissq = 0 H 1 :β npvissq 0 t = 2.07,p_value = < 0.05 (d) The coefficient on the variable npvis 2 is significant at 5% significance level, therefore, there is nonlinear effect in npvis. H 0 :β npviage = 0 H 1 :β npviage 0 t = 1.80,p_value = > 0.05 Do not reject null hypothesis. We do not have enough evidence to say that mother s age influences the number of prenatal visits at 5% significance level. (e) H 0 : b mage = b npviage = 0 H 1 : one of the above doesn t hold F- statistic = 1.80, p_value = Do not reject the null hypothesis. We do not have enough evidence to reject the null hypothesis that mother s age don t influence birth weight. 2. (Panel Data, 45%) Using a panel data on school districts for the years 1992 through 1998 in Michigan, you are asked to investigate the effect of real expenditures per pupil, rexpp, on the percentage of fourth graders in a district receiving a passing score on a standardized math test, math4. Real expenditures are measured in 1997 dollars. (a) What is the average percentage of fourth graders receiving a passing score on math test in 1994? (5%) Consider the ordinary regression model: math4 it =θ t + β 1 lrexpp it + β 2 lenrol it + β 3 lunch it +u it

3 where q t denotes different year dummies, lrexpp is the natural log of rexpp, lenrol is the natural log of school enrollment, and lunch is the percentage of pupil eligible for free lunch. (b) Please given an example to justify the use of time variable q t.(5%) (c) What are the estimate effects of the spending variables? (5%) (d) Please compare the estimated results with and without time fixed effect. What are the estimate effects of the spending variables? (5%) Consider the fixed effects model: math4 it =θ t + β 1 lrexpp it + β 2 lenrol it + β 3 lunch it + a it +u it where a i is unobserved effect. (e) Please give an example to justify the use of unobserved effect a i.(5%) (f) Now, using data of year 1995 and 1996, estimate the equation using the first differencing method assuming heteroskedasticiy robust standard error. What is the sample size now? Is there evidence that spending affects math scores? Explain Carefully. (5%) (g) Using the data from 1992 to 1998, re-estimate part (f). Can you reject the hypothesis that b 1 + b 2 + b 3 = 0? (5%) (h) Estimate the fixed effect model assuming no serial correlation. Is the spending variable significantly different from zero at 5% significance level? (5%) (i) Re-estimate the fixed effect model but allow for serial correlation. Comparing with estimates in (h), is there any change in the size of standard error? Explain carefully. (5%) (j) Re-estimate (h) but add the time fixed effect. Comparing with estimates in (h), explain how the coefficient of spending changes with respect to the addition of time fixed effect? (5%) (k) Compare results in (i) and (g) with that in (b), how are the estimated coefficients of the spending variable in the first difference model and fixed effect model different from that in the ordinary model? Which factor (time or area fixed effect) is more important in affecting the spending coefficient? Explain carefully. (5%) ANS: (a) In 1994, the average percentage of fourth graders receiving satisfactory math score is 49.34%. (b) To reflect the fact that the population different time effects, we allow the intercept to differ across periods. This is why we have year dummy variables.

4 (c) The spending variable, having a positive coefficient of 8.42, is statistically significant at 1% significance level. A 1% increase in spending increases the math4 pass rate by percentage points. (d) Without the time fixed effects, a 1% increase in spending increase the math4 pass rate by 0.35 percentage point. This is a large difference compared to the effect when we control for the time fixed effects, in which a 1% increase in spending increases the math4 pass rate by percentage points. (e) In this case, a i are factors that are constant within a school district over the period 1992 to For example, the distribution of income of the families in each district is relatively fixed within the periods from 1992 to (f) H 0 : b lrexpp = 0 H 1 : b lrexpp ¹ 0 t = -1.28, p-value = > 0.05 The sample size is 550. We cannot reject the null hypothesis therefore the coefficient is not significant under 5% significance level. (g) H 0 : b 1 + b 2 + b 3 = 0 H 1 : b 1 + b 2 + b 3 ¹ 0 F-statistic 7.11, p-value = < 0.05 Reject null hypothesis. Under 5% significance level, we have evidence that b 1 + b 2 + b 3 is different from zero. (h) H 0 : b lrexpp = 0 H 1 : b lrexpp ¹ 0 t = 41.96, p-value = 0.00 Reject null hypothesis. Under 5% significance level, we reject the null hypothesis that b lrexpp is not different from zero. (i) Allowing for serial correlation, the standard errors on the variables lrexpp, lenrol, and lunch are large than when estimated not allowing for serial correlation. The estimates on the coefficients of the variables are identical. (j) Adding time-fixed effect into the model, the coefficient on the spending variable decreases from to 0.31, which is a substantial drop. In addition, after adding the time-fixed effect, the coefficient is no longer significant at 5% significance level. (k) In (b), the estimation is based on the ordinary regression model without time-fixed effects, the coefficient on lrexpp is and significant at 1% significance level.

5 When we add the time-fix effect, while still being significant, the coefficient drops by a substantial amount. In the fixed-effect model, the coefficient is significant and large when we leave out the time-fixed effects. However, when we add the time-fixed effects, the coefficient drops by a substantial amount and turns to be insignificant. In short, the time effect is more important in affecting the spending coefficient than does area fixed effect. 3. (IV Regression, 40%) Charlie wants to understand how education affects earned wage. He considers the following regression log(wage) i = B 0 + β 1 educ i + β 2 exper i +u i In the STATA outputs, please find estimated results that include the dependent variable: wage (monthly wage) and independent variables: educ (years of education) and exper (year of experience). Use STATA outputs to explain the following questions (Notice that the order of STATA results might not be in the same order of these questions) (a) Before discussing the regression results, Charlie would like to know some sample statistics. Please indicate the mean of education years and standard deviation of years of experience. (5%) (b) According to OLS results, does the years of education significantly influences the earned wage? How large is the effect? (5%) (c) Charlie thinks the coefficient of educ might be biased and suggests 2SLS estimation. Please provide an explanation on why OLS results could be biased. (5%) (d) Charlie likes to use mother s years of education, meduc, as an instrument for educ. Do you think meduc is a good instrument for educ? Please explain carefully. (5%) (e) Is meduc a weak instrument of educ? (Notice Use F statistics) (5%) (f) According to 2SLS results, how does the year of education influence the earned wage? (5%) (g) Charlie wants to know whether one could really conduct the analysis using two stages regression. He first regresses educ on meduc and exper. After obtaining the predicted value (p_educ) based on estimates of the first regression, he regresses log(wage) on p_educ and exper. Compare the results of the second regression with results obtained from IVREG2, what are their differences Which estimation method is correct (5%)

6 (h) Charlie thinks the father s year of education, feduc, is another instrument candidate. Therefore, he treats both meduc and feduc as instruments. In this case, can you check whether the IV results pass the over-identification test? Can you treat these two instruments as exogenous under the 5% level of significance (5%) ANS: (a) Mean = , standard deviation = (b) H 0 : b educ = 0 H 1 : b educ ¹ 0 t = The p-value of coefficient of educ is less than 0.001, which rejects the null hypothesis. Education has significant effects toward wages; 1 year more of education increases average wage for 7.78%. (c) It is reasonable. When it comes to this kind of regression, the bias of ability shows as well. For people have to pass entrance exam to achieve higher education, which leads to the result that only capable people could get higher education. So we can find sample selection bias. (d) There are 2 conditions to be as an instrument variable. Here, we know that first mother s year of education is related to the year of education of her child, and second the ability is not correlated to mother s year of education. Under these 2 condition, meduc could be used as an instrument for educ. (e) According to the result we know that the F-statistic in the first stage is (>10), so meduc is not a weak instrument of educ. (f) According to the results of IV, educ is doubled and it s significant. For wage is under natural logarithm, it shows that if you increase one year of education it would increase the average wage by 14.97%. (g) Compare the results of 2SLS and IV, the coefficients are similar, but the standard deviation shrinks. This is due to some bias under 2SLS. (h) Using the J-statistic, the p-value of wage is (>0.05), so the null hypothesis that all variables are exogenous is not rejected. Under 5% level of significance, they could be seen as exogenous.

7 1. //Final Exam Problem 1 2. use "/Users/samchang/Desktop/Economtrics final/bwght2.dta", clear 3. d Contains data from /Users/samchang/Desktop/Economtrics final/bwght2.dta obs: 1,832 vars: Jan :47 size: 29,312 storage display value variable name type format label variable label mage byte %10.0g mother's age, years npvis byte %10.0g total number of prenatal visits fage byte %10.0g father's age, years bwght int %10.0g birth weight, grams cigs byte %10.0g avg cigarettes per day drink byte %10.0g avg drinks per week mwhte byte %9.0g =1 if mother white mblck byte %9.0g =1 if mother black moth byte %9.0g =1 if mother is other lbwght float %9.0g log(bwght) npvissq int %9.0g npvis^2 Sorted by: 4. sum Variable Obs Mean Std. Dev. Min Max mage 1, npvis 1, fage 1, bwght 1, cigs 1, drink 1, mwhte 1, mblck 1, moth 1, lbwght 1, npvissq 1, reg lbwght npvis mage cigs, r Linear regression Number of obs = 1,656 F(3, 1652) = 7.18 Prob > F = R-squared = Root MSE = /10/17, 3:47 PM Page 1 of 13

8 lbwght Coef. Std. Err. t P> t [95% Conf. Interval] npvis mage cigs _cons reg lbwght npvis mage cigs drink mwhte fage, r Linear regression Number of obs = 1,643 F(6, 1636) = 4.91 Prob > F = R-squared = Root MSE = lbwght Coef. Std. Err. t P> t [95% Conf. Interval] npvis mage cigs drink mwhte fage _cons reg lbwght npvis npvissq mage cigs, r Linear regression Number of obs = 1,656 F(4, 1651) = 5.44 Prob > F = R-squared = Root MSE = lbwght Coef. Std. Err. t P> t [95% Conf. Interval] npvis npvissq mage cigs _cons gen npviage = npvis*mage (68 missing values generated) 1/10/17, 3:47 PM Page 2 of 13

9 9. reg lbwght npvis npvissq mage cigs npviage, r Linear regression Number of obs = 1,656 F(5, 1650) = 4.74 Prob > F = R-squared = Root MSE = lbwght Coef. Std. Err. t P> t [95% Conf. Interval] npvis npvissq mage cigs npviage _cons test mage npviage ( 1) mage = 0 ( 2) npviage = 0 F( 2, 1650) = 1.80 Prob > F = //Final Exam Problm 2 Panel Data 13. use "/Users/samchang/Desktop/Economtrics final/mathpnl.dta", clear 14. d Contains data from /Users/samchang/Desktop/Economtrics final/mathpnl.dta obs: 3,850 vars: Jan :28 size: 142,450 storage display value variable name type format label variable label distid float %9.0g district identifier lunch float %9.0g % eligible for free lunch math4 float %9.0g % satisfactory, 4th grade math year int %9.0g y92 byte %9.0g =1 if year == 1992 y93 byte %9.0g =1 if year == 1993 y94 byte %9.0g =1 if year == 1994 y95 byte %9.0g =1 if year == 1995 y96 byte %9.0g =1 if year == 1996 y97 byte %9.0g =1 if year == 1997 y98 byte %9.0g =1 if year == /10/17, 3:47 PM Page 3 of 13

10 lenrol float %9.0g log(enrol) cmath4 float %9.0g math4 - math4_1 clunch float %9.0g lunch - lunch[_n-1] lrexpp float %9.0g log(rexpp) Sorted by: distid year 15. sum if year == 1994 Variable Obs Mean Std. Dev. Min Max distid lunch math year y y y y y y y lenrol cmath clunch lrexpp reg math lrexpp lenrol lunch, r Linear regression Number of obs = 3,850 F(3, 3846) = Prob > F = R-squared = Root MSE = math4 Coef. Std. Err. t P> t [95% Conf. Interval] lrexpp lenrol lunch _cons reg math4 lrexpp lenrol lunch i.year, r Linear regression Number of obs = 3,850 F(9, 3840) = Prob > F = R-squared = /10/17, 3:47 PM Page 4 of 13

11 Root MSE = math4 Coef. Std. Err. t P> t [95% Conf. Interval] lrexpp lenrol lunch year _cons bysort distid: gen clrexpp = lrexpp - lrexpp[_n-1] (550 missing values generated) 20. bysort distid: gen clenrol = lenrol -lenrol[_n-1] (550 missing values generated) reg cmath4 clrexpp clenrol clunch if year==1996 Source SS df MS Number of obs = 550 F(3, 546) = 2.73 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = cmath4 Coef. Std. Err. t P> t [95% Conf. Interval] clrexpp clenrol clunch _cons test clrexpp+clenrol+clunch=0 ( 1) clrexpp + clenrol + clunch = 0 F( 1, 546) = 5.67 Prob > F = /10/17, 3:47 PM Page 5 of 13

12 reg cmath4 clrexpp clenrol clunch if year==1996, r Linear regression Number of obs = 550 F(3, 546) = 2.44 Prob > F = R-squared = Root MSE = cmath4 Coef. Std. Err. t P> t [95% Conf. Interval] clrexpp clenrol clunch _cons test clrexpp+clenrol+clunch=0 ( 1) clrexpp + clenrol + clunch = 0 F( 1, 546) = 1.85 Prob > F = reg cmath4 clrexpp clenrol clunch if year==1996, r cluster(distid) Linear regression Number of obs = 550 F(3, 549) = 2.44 Prob > F = R-squared = Root MSE = (Std. Err. adjusted for 550 clusters in distid) cmath4 Coef. Std. Err. t P> t [95% Conf. Interval] clrexpp clenrol clunch _cons test clrexpp+clenrol+clunch=0 ( 1) clrexpp + clenrol + clunch = 0 F( 1, 549) = 1.85 Prob > F = /10/17, 3:47 PM Page 6 of 13

13 reg cmath4 clrexpp clenrol clunch, r Linear regression Number of obs = 3,300 F(3, 3296) = 5.14 Prob > F = R-squared = Root MSE = cmath4 Coef. Std. Err. t P> t [95% Conf. Interval] clrexpp clenrol clunch _cons test clrexpp + clenrol + clunch=0 ( 1) clrexpp + clenrol + clunch = 0 F( 1, 3296) = 7.11 Prob > F = reg cmath4 clrexpp clenrol clunch i.year, r Linear regression Number of obs = 3,300 F(8, 3291) = Prob > F = R-squared = Root MSE = cmath4 Coef. Std. Err. t P> t [95% Conf. Interval] clrexpp clenrol clunch year _cons /10/17, 3:47 PM Page 7 of 13

14 34. test clrexpp + clenrol + clunch=0 ( 1) clrexpp + clenrol + clunch = 0 F( 1, 3291) = 0.87 Prob > F = xtreg math4 lrexpp lenrol lunch, i(distid) fe Fixed-effects (within) regression Number of obs = 3,850 Group variable: distid Number of groups = 550 R-sq: Obs per group: within = min = 7 between = avg = 7.0 overall = max = 7 F(3,3297) = corr(u_i, Xb) = Prob > F = math4 Coef. Std. Err. t P> t [95% Conf. Interval] lrexpp lenrol lunch _cons sigma_u sigma_e rho (fraction of variance due to u_i) F test that all u_i=0: F(549, 3297) = 5.99 Prob > F = xtreg math4 lrexpp lenrol lunch, i(distid) fe vce(cluster distid) Fixed-effects (within) regression Number of obs = 3,850 Group variable: distid Number of groups = 550 R-sq: Obs per group: within = min = 7 between = avg = 7.0 overall = max = 7 F(3,549) = corr(u_i, Xb) = Prob > F = (Std. Err. adjusted for 550 clusters in distid) math4 Coef. Std. Err. t P> t [95% Conf. Interval] 1/10/17, 3:47 PM Page 8 of 13

15 lrexpp lenrol lunch _cons sigma_u sigma_e rho (fraction of variance due to u_i) xtreg math4 lrexpp lenrol lunch i.year, i(distid) fe Fixed-effects (within) regression Number of obs = 3,850 Group variable: distid Number of groups = 550 R-sq: Obs per group: within = min = 7 between = avg = 7.0 overall = max = 7 F(9,3291) = corr(u_i, Xb) = Prob > F = math4 Coef. Std. Err. t P> t [95% Conf. Interval] lrexpp lenrol lunch year _cons sigma_u sigma_e rho (fraction of variance due to u_i) F test that all u_i=0: F(549, 3291) = 6.53 Prob > F = xtreg math4 lrexpp lenrol lunch i.year, i(distid) fe vce(cluster distid) Fixed-effects (within) regression Number of obs = 3,850 Group variable: distid Number of groups = 550 R-sq: Obs per group: 1/10/17, 3:47 PM Page 9 of 13

16 within = min = 7 between = avg = 7.0 overall = max = 7 F(9,549) = corr(u_i, Xb) = Prob > F = (Std. Err. adjusted for 550 clusters in distid) math4 Coef. Std. Err. t P> t [95% Conf. Interval] lrexpp lenrol lunch year _cons sigma_u sigma_e rho (fraction of variance due to u_i) // Final Exam Question 3 IV Regressions 43. use "/Users/samchang/Desktop/Economtrics final/wage2 copy.dta", clear 44. d Contains data from /Users/samchang/Desktop/Economtrics final/wage2 copy.dta obs: 935 vars: 5 10 Jan :43 size: 7,480 storage display value variable name type format label variable label educ byte %9.0g years of education exper byte %9.0g years of work experience meduc byte %9.0g mother's education feduc byte %9.0g father's education lwage float %9.0g natural log of wage Sorted by: 1/10/17, 3:47 PM Page 10 of 13

17 45. sum Variable Obs Mean Std. Dev. Min Max educ exper meduc feduc lwage reg lwage educ exper, r Linear regression Number of obs = 935 F(2, 932) = Prob > F = R-squared = Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ exper _cons ivreg2 lwage (educ=meduc) exper, r IV (2SLS) estimation Estimates efficient for homoskedasticity only Statistics robust to heteroskedasticity Number of obs = 857 F( 2, 854) = Prob > F = Total (centered) SS = Centered R2 = Total (uncentered) SS = Uncentered R2 = Residual SS = Root MSE =.4103 lwage Coef. Std. Err. z P> z [95% Conf. Interval] educ exper _cons Underidentification test (Kleibergen-Paap rk LM statistic): Chi-sq(1) P-val = /10/17, 3:47 PM Page 11 of 13

18 Weak identification test (Kleibergen-Paap rk Wald F statistic): Stock-Yogo weak ID test critical values: 10% maximal IV size % maximal IV size % maximal IV size % maximal IV size 5.53 Source: Stock-Yogo (2005). Reproduced by permission. NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors. Hansen J statistic (overidentification test of all instruments): (equation exactly identified) Instrumented: educ Included instruments: exper Excluded instruments: meduc 48. reg educ meduc exper, r Linear regression Number of obs = 857 F(2, 854) = Prob > F = R-squared = Root MSE = educ Coef. Std. Err. t P> t [95% Conf. Interval] meduc exper _cons predict educhat (option xb assumed; fitted values) (78 missing values generated) 50. reg lwage educhat exper, r Linear regression Number of obs = 857 F(2, 854) = Prob > F = R-squared = Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educhat exper _cons /10/17, 3:47 PM Page 12 of 13

19 51. ivreg2 lwage (educ=meduc feduc) exper, r IV (2SLS) estimation Estimates efficient for homoskedasticity only Statistics robust to heteroskedasticity Number of obs = 722 F( 2, 719) = Prob > F = Total (centered) SS = Centered R2 = Total (uncentered) SS = Uncentered R2 = Residual SS = Root MSE =.4098 lwage Coef. Std. Err. z P> z [95% Conf. Interval] educ exper _cons Underidentification test (Kleibergen-Paap rk LM statistic): Chi-sq(2) P-val = Weak identification test (Kleibergen-Paap rk Wald F statistic): Stock-Yogo weak ID test critical values: 10% maximal IV size % maximal IV size % maximal IV size % maximal IV size 7.25 Source: Stock-Yogo (2005). Reproduced by permission. NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors. Hansen J statistic (overidentification test of all instruments): Chi-sq(1) P-val = Instrumented: educ Included instruments: exper Excluded instruments: meduc feduc 52. end of do-file 1/10/17, 3:47 PM Page 13 of 13