计量经济分析（第六版）答案 Midterm01.docx
NAMEEconometrics I, Fall, 2001, Professor W. Greene Midterm Examination.1. Consider random sampling from the normal distribution with mean 日 and variance a2. Xi N|i,o2, i =For a random variable X which is normally distributed, the variable Y = ex has a lognormal distribution with meanEy = +<j2/2= y and VarY=#+措(炉 _We are interested in estimating y = EY. The obvious candidate would be thesample mean, Y = £匕=一£/' But, there is another estimator. Using the z=l Z=1original data, let X and s2 be the sample mean and variance of the X's. Then,it,人X+q2 / 2the alternative estimator is / = e .a. Are either of these estimators unbiased? Explain.b. Are either of these estimators consistent? Explain2. Referring to the variables in question 1, we also have the following exact results: Var X = cy2 / n, Vars2 = 2cy4/n, Cov X ,s2 = 0.VarK = VarY / n was given above in question 1.a. How would you find the asymptotic variance of / = ex+s .b. How would you determine which of the two proposed estimators of EYJ is more efficient asymptotically?3. In computing the least squares estimator in the regression of y on l,xi,X2,d,t where "1" is a constant term, xi and X2 are two variables, d is a dummy variable and t is a time trend (1,2,T), let XI = xi,X2 and let X2 =l,d,t. Which of the following would produce an unbiased estimator of the parameters in the modelY = Xlpi + X2p2 + £Explain in each case. Hint: None of these require long matrix algebra proofs. Just think about each one. Answers should only be one or two lines of explanation in each case.(a) Regress y on XI and X2(b) Regress X2 on XI and compute fitted values for X2, then regress Y onXI and these fitted values for X2.(c) Regress y on XI and compute residuals, y* Regress y* on XI and X2.(d) Regress X2 on XI and compute residuals. Regress y on these residuals.4. Suppose you have used least squares to estimate a demand equation for a monopolist. You also know that marginal cost is constant, at, say, MC Your estimated demand equation isPrice = a + bQuantityThe optimal (profit maximizing) quantity equates marginal revenue, a + 2bQ, to MC, producing a solution Q* = (MC - a)/2b. (b is negative). You have the least squares coefficent estimates and the 2x2 estimated covariance matrix in hand.a = 60100 -10-102b = -3Estimated covariance matrix =MC =24How would you form a confidence interval for Q*5. In the regression results below, the variables are:N = 150, sample of the work force behavior of married women, taken in 1975, used in the Mroz paper referenced in your text,Y = hourly wage, the dependent variable. This is in $ per hour.KIDS 二 a dummy variable = 1 if there are children under 18 living in the home.AGE 二 age in yearsAGESQ = square of ageEDUC = educationa. Is the effect of having children in the home statistically significant? Explainb What is effect of an additional year in “age" on the expected wage fbr a 30 year old woman?c. How would you form a confidence interval for the estimated effect that you found in part b?d. How would you test the hypothesis that age has no effect on WAGE in this model? Explain in detail.6. Referring to the model in question 4, one might hypothesize that the presence of young children completely alters the wage/education profile. To investigate that possibility, the following three regressions give results for the full sample, families with children and families with no children.a. How would you test the hypothesis that the same regression applies to both subgroups?b. Can you see any evidence in the results below whether women with children have higher or lower wages than women without children?All Families, KIDS appears in the reeression. Use for Question 5.+I Ordinary least squares regression Weighting variable = none |I Dep. var. = WAGE Mean= 3.792050000, SD,=2.342414629|I Model size: Observations =150, Parameters =5, Deg.Fr.=145 |I Residuals: Sum of squares= 674.2375198, Std.Dev.=2.15637 |I Fit:R-squared= .175294, Adjusted R-squared =.15254 |I Model test: F 4 f145 =7.71, Prob value =.00001 |+I Variable|Coefficient|Standard Error|t-ratio|P|T|>t|Mean ofX|+Constant-6.7875247695.0447230-1.345.1806KIDS-1.126731368.45458453-2.479.0143.66000000AGE.3119934568.233749811.335.184142.786667AGESQ-.003957994169.0027314125-1.449.14951901.0267EDUC.4349928716.0836213975.202.000012.640000This is theestimated covariance matrixfor the coefficientestimates.123451125.4492.3275-1.1410.0131-.130221.3275.2066-.0259.0004-.004331-1.1410-.0259.0546-.0006.002241.0131.0004-.0006.00000460614 -.0000253330351-.1302-.0043.0022-.00002533303.0070All families+I Ordinary least squares regression Weighting variable = none |I Dep. var. = WAGE Mean= 3.792050000, SD=2.342414629|I Model size: Observations =150, Parameters =4, Deg.Fr.=146 |I Residuals: Sum of squares= 702.8039766, Std.Dev.=2.19402 |I Fit:R-squared= .140353, Adjusted R-squared =.12269 |I Model test: F3,146 =7.95, Prob value =.00006 |+I Variable|Coefficient|Standard Error|t-ratio|P|T|>t|Mean ofX|+4-+Constant-5.0016342585.0801906-.985. 3265AGE.1707522643.23065611.740.460342.786667AGESQ-.001952378674.0026543577-.736.46321901.0267EDUC.4113360731.0845255754.866.000012.640000Families with Children+I Ordinary least squares regression Weighting variable = none |I Dep. var. = WAGE Mean= 3.618494949, S.D.=2.168827626|I Model size: Observations =99, Parameters =4, Deg.Fr.=95 |I Residuals: Sum of squares= 411.8075395, Std.Dev.=2.08202 |I Fit:R-squared= .106657z Adjusted R-squared =.07845 |I Model test: F 3Z95 =3.78z Prob value =. 01304 |+4-+I Variable|Coefficient|Standard Error|t-ratioIP|T|>t|Mean ofX|+Constant-16.369942018.3220409-1.967.0521AGE.7832912943.401493691.951.054039.707071AGESQ-.009508912588.0049393377-1.925.05721622.6970EDUC.3372655578.113692172.966.003812.797980Families with no chldren+I Ordinary least squares regression Weighting variable = none |I Dep. var. = WAGE Mean= 4.128950980, S.D.=2.637440831|I Model size: Observations =51z Parameters =4, Deg.Fr.=47 |I Residuals: Sum of squares= 233.2955027, Std.Dev.=2.22794 |I Fit:R-squared= .329234, Adjusted R-squared =.28642 |I Model test: F3,47 =7.69, Prob value =. 00028 |+4-IVariable|Coefficient|Standard Error|t-ratio|P|T|>tIMean ofX|+Constant6.4550570909.4372178.684.4973AGE-.3032343154.39863070-.761.450648.764706AGESQ.002569488111.0043602988.589.55852441.3137EDUC.5017377974.128003413.920.000312.333333
