北京大学国家发展研究院2010年中级计量经济学期中考试试题

Intermediate EconometricsMidterm Exam,Spring2010The following terminology and assumptions are used throughout the exam.SST:total sum of squares;SSE:explained sum of squares;SSR:Residual sum of squares.se(β):standard error ofβ.sd(β):standard deviation ofβ.Simple linear regression model:y i=β0+β1x i+u,i=1,2,...,n.Multiple linear regression model:y i=β0+β1x i1+...+βk x ik+u,i=1,2,...,n.MLR.1:The model in population is linear.MLR.2:Random Sampling.MLR.3:Zero conditional mean.MLR.4:No perfect collinearity.The error term is homoskedasitic(MLR.5)and normally distributed(MLR.6).1True or FalseJudge whether the following statements are true(T)or false(F)and briefly explain.1.If MLR.3holds,we must have Ni=1(x2i−¯x2)u i=0for all i=1,...,N,where¯x arethe mean of x i.2.In the simple linear regression model,y is also called regressor and x the regressand.3.Significance level is the probability of accepting H0when it is true.4.Unbiasedness is the minimum requirement for an estimator in linear regression mod-els.5.If cov(x i,u)=0holds,we know that the zero conditional mean assumption holds.6.Under the classical linear assumptions,the t statistics have t distributions under thenull.7.The inconsistency inˆβ1,OLS is defined as p limˆβ1,OLS−β1=cov(x1,u)/V ar(x1)forthe simple regression model.8.The R-squred of the regression is defined as R2=SSE/SST=1−SSR/SST,andis interpreted as the fraction of the population variation in y that is explained by x.9.Consider the savings functionsav=β0+β1inc+u,u=e·(inc)1/2,where e is a random variable with E(e)=0and V(e)=σ2e.Assume that e is independent of inc,then we have E(u|inc)=0.10.p-values for t tests are the smallest significance level for which the null hypothesis isrejected.Therefore the hypothesis can be tested at any significance level.12Multiple Choice QuestionsOnly one of the choices is correct,4points for each question.1.Which of the following can cause the usual OLS t statistic to be invalid(that is,notto have t distributions under H0)?(a)A sample correlation coefficient of0.9between two independent variables thatare in the model.(b)The population error u is normally distributed with mean0and varianceσ2.(c)Omitting an important explanatory variable.(d)The random sample has only100observations.2.Consider an equation to explain salaries of CEOs in terms of annualfirm sales,returnon equity(roe,in percentage form),and return on thefirm’s stock(ros,in percentage formlog(salary)i=β0+β1log(sales)i+β2roe i+β3ros i+u i,i=1,2,...,N.(1) The OLS regression of model(1)giveslog(salary)i=ˆβ0+ˆβ1log(sales)i+ˆβ2roe i+ˆβ3ros i+ˆu i, whereˆu i=log(salary)i−log(salary)i.How many of the following statements is (are)wrong(variables with”-”on the top denote the corresponding averages)?(1)log(salary)i ˆu i=0(2)roe iˆu=0(3)log(salary)iˆu i=0(4)ˆu i(log(salary)i−log(salary)i)=0.(a)one(b)two(c)three(d)fouring the data in CEOSAL1.RAW,the following equation was obtained by OLS,where salary is measured in thousand US dollars:log(salary)i =4.32(0.32)+0.28(0.035)log(sales)i+0.0174(0.0041)roe i+0.00024(0.00054)ros i,N=209,R2=0.283,(2)where log(salary)and log(sales)indicate the natural log of CEO salary and sales in US dollar,respectively,roe is return on equity,and ros is return on stocks.How many of the following statements is(are)true?2(a)(1)The regression shows return on thefirm’s stock is a significant factor ex-plaining the CEO’s salaries,so it should be included in thefinal model.(2)It is possible that for the population model,when sales increase by10%,then the CEO’s salary increases by3%.(3)The regression shows when roe increases by1percentage point,CEO’s salarywill increase by about17.4US dollars.(4)Since the95%confidence interval for roe is about[0.009,0.025],it meansthere is95%probability that the true parameter ofβ2=0.01.(b)one(b)two(c)three(d)four4.Now you wish to construct a confidence interval for the log of sales.Since the confi-dence interval is only as good as the underlying assumption used to construct it,what are the minimum required assumptions for the construction of confidence interval for β1?(a)Zero conditional mean,(b),MLR.1-MLR.4,(c)MLR.1-MLR.3(d)MLR.1-MLR.6.5.The R square of this regression is0.28.Which of the following statement is true?(a)The higher the R2,the lower the omitted-variable bias.(b)Since ros is an insignificant variable,dropping it from the model will not affectthe value of R2.(c)The econometrian shouldfind every way to obtain a R2of his regression asgreater as possible.(d)The R2only indicates whether the regressors are good at explaining the variationof the dependent variable in the sample,but does not tell us whether an included variable is statistically significant.6.Which of the following can cause the OLS estimator forβ1to be biased?(a)Heteroskedasticity(b)Omitting an important variable.(c)A sample correlation coefficient of0.98between two independent variables areboth included in the model.(d)Rescale sales using the levels directly,not the logs.7.Suppose you wish to test the null hypothesis that neither return on equity or thereturn on stock affects CEO’s salary,you should3(a)use t -statistics to test H 0:βk =0vs H 1:βk =0,k =2,3.(b)calculate the F -statistic.If the statistic exceeds the critical value,conclude thatyou have found evidences against the statement that CEO salary is responsive neither to roe nor to ros .(c)use t -statistics for each hypothesis and reject the null hypothesis once the statis-tic exceeds the critical value for a single hypothesis.(d)Calculate the F -statistic.If the statistic does not exceed the critical value,youhave found evidences supporting the statement that CEO salary is responsive both to roe and to ros .8.The low R square leads you to suspect that there is omitted variable bias.The omitted variable bias(a)will always be present as long as the regression R 2is a small number with respectto 1.(b)is always there but is negligible in most of the economic examples.(c)exists if the omitted variable is correlated with the included regressor but is nota determinant of the dependent variable(d)exists if the omitted variable is correlated with the included regressor and is adeterminant of the dependent variable9.Under the Gauss-Markov assumptions,let ˜βj denote estimators that solve equations of the form n i =1g j (x i ) y i −˜β0−˜β1x i 1−...−˜βk x ik =0and let ˆβj be the OLS estimator.Then which of the following is correct?(a)The OLS estimator has the smallest asymptotic variance,that is Avar ˆβj −βj <Avar ˜βj −βj .(b)The OLS estimator is asymptotically normally distributed.(c)The OLS estimator is consistent.(d)As n →∞,we must have V ar √n ˆβj ≤V ar √n ˜βj .10.Define p −value for F statistic as p =P ( >F ),where we let denote a F randomvariable with (q,n −k −1)degrees of freedom.Which of the following statement about F statistic is wrong?(a)It is the probability of observing a value of F at least as large as we did underthe null.4(b)A large p−value is evidence against the null hypothesis.(c)Once p-value is computed,the F test can be carried out at any significancelevel.(d)If the p−value for the overall significance of a model is0.314,it means the chanceof observing a value of the F statistic as large as we did under the null is31.4%. 3Analytical Questions1.(40points)A researcher is interested in studying returns to education using thefollowing modelhrearn=β0+β1edu+β2exp er+β3age+u,n=616(3)where hrearn is hourly earnings measured in US dollars,exper is years of working experience,edu is years of education received by the employee,and age is his/her age.Denote the residuals from the auxilliary regressionedu=γ0+γ1exp er+γ2age+v(4)as vedu and the residuals from regressing(3)as uedu,and the residuals fromhrearn=θ0+θ1vedu(5)asθedu.The mean of the following variables are provided but with some missinghrearn 6.23hrearn262.16hrearn*edu80.54edu12.51edu2164.07edu*age464age37.6age21578hrearn*age245.49exper18.6vedu27.133hrearn*vedu 3.11vedu vedu*exper vedu*ageθeduθedu222uedu219.85(a)(5’)Fill in the blanks and explain how you get those values.(b)(5’)Calculateˆβ1,OLS andˆθ1,OLS using the above information.(c)(5’)Interpret the meaning ofˆβ1,OLS.(d)(5’)Calculate the standard errorˆθ1,OLS.(e)(5’)Construct the95%confidence interval forθ1.(f)(5’)Based on the above estimations,what is the lowest possible rate of returnand what is the highest possible rate of return,allowing the possibility that wehave not covered the true rate of return as5%?5(g)(5’)The R squares for(3),(4)and(5)are0.1502,0.0435,and0.058,respectively.Calculate the adjusted R square for each model.(h)(5’)Calculate the F statistics for the null that all explanatory variables areinsignificant for(3),(4)and(5).6。