计量经济学章节习题.doc

CHAPTER 1THE NATURE OF REGRESSION ANALYSIS1.1 a. please use the above data to compute the inflation rate of each country. (a) These rates (%) are as follows. They are year-over-year, starting with 1974 as there is no data prior to 1973. These rates are, respectively, for Canada, France, Germany, Italy, Japan, UK and US.10.7843113.583826.84713419.4174823.173280.1577060.110360108407111.704835.96125217.0731711.694920.2445820.0912787.5848309.5671984,36005616.666619.5599390.1641790.0576217.7922089.5634103.63881419.345248.1717450.1581200.0650268.9500869.1081592.73081912468834.2253520.0830260.0759089.32069510.608704.05063315.521063.6855040.1345830.1134979.97109813.679255A7445321.305187.7014220.1786790.13498612.4835713.278016.34371419.303804.8404840.1197450.16315510.8644911.965815.31453416.313002.9380900.0853240.0616065.7955749.4874593.29557214.937291.7329260.0461220.0321244.2828697.6693232.39282210.615082.3046090.0501000.0431734.1069725.8279372.0447918.6098651.9588640.0601150.0355114.1284402.534965-0.0954206.1106520.6724300.0342030.0185874.3171813.2395570.1910224.5914400.0000000.0417750.0364964.0540542.7250211.3346044.9851190.7633590.0492900.0413734.9512993.4565922.7281286.5910702.3674240.0772290.0481834.7950503.3411032.7472536.1170213.0527290.0953440.0540325.6088563.1578953.6541896.3909773.2315980.0587040.0420811.5373862.4052484.9871025.3003531.5521740.0369660.0301031.7894012.1352314.5045054.2505591.2831480.0159800.0299360.2028401.6027872.7429473.9153090.7601350.0248030.0256062.1592441.7832651.8306645.369128-0.1676450.0336480.0283401.5852052.0215631.4981273.8706520.1679260.0245570.0295281.6254881.1889041.6974171.7452831.6764460.0312150.022945 b. please plot the inflation rate of each country, with time as lateral axis, and inflation rate as vertical axis in the Plane Rectangular Coordinate System.c. what conclusion can you draw from the inflation history from in above 7 countries?As you can see from this figure, the inflation rate of each of the countries has generally declined over the years.(c) As you can see from this figure, the inflation rate of each of the countries has generally declined over the years.d. which country change the largest in inflation rate? Can you give an explaination?(d) As a measure of variability, we can use the standard deviation. These standard deviations are 0.036, 0.044, 0.018, 0.062, 0.051, 0.060, and 0.032, respectively, for Canada, France, Germany, Italy, Japan, UK and USA. The highest variability s thus found for Italy and the lowest for Germany.2.12.1 What is the conditional expectation function or the population regression function?Conditional expectation, also called conditional mathematical expectation. For convenience, we discuss two random variables is deduced with the occasion of the eta, assuming they have the density function p (x, y), and the p (y x) under the condition of known factor = x, density function, the conditions of eta to p1 (x) of density function is deduced. Defined under the condition of factor = x, eta conditional mathematical expectation is defined as: E eta factor = x = yf (y x) dy. UndertheconditionofagivenvariableXi,interpretedvariableexpecttrajectoryiscalledpopulationregressionlineorpopulationregressioncurve.Thecorrespondingfunction：E（Y/Xi）=f(Xi)iscalledpopulationregressionfunction,PRF.2.3 What is the role of the stochastic error termin regression analysis? What is the difference between the stochastic error term and the residual?A regression model can never be a completely accurate description of reality. Therefore, there is bound to be some difference between the actual values of the regressand and its values estimated from the chosen model. This difference is simple the stochastic error term,whose various forms are discussed in the chapter. The residual is the sample counterpart of the stochastic error term.2.5What do we mean by a linear regression model?It tells how the mean or average of the sub-population of Y varies with the fixed values of the explanatory variable(s). 2.7 Are the following model the linear model? Why they are or not? (a) Taking the natural log, we find that, which becomes a linear regression model.(b). The following transformation, known as the logit transformation, makes this model a linear regression model:(c) a linear regression mdel(d) a nonlinear regression model(e) a nonlinear regression model, as is raised to the third power.BLUE（名词解释及证明）Terms definition and verifications.What is the Gauss-Markov theorem?Under the assumption of Classic Linear Model (CLM), the Ordinary Least Square (OLS) can make the estimators is the smallest deviation among the unbiased linear estimator, which are called best linear unbiased estimator(BLUE). Under Assumptions MLR.1 through MLR.5, are the best linear unbiased estimators(BLUEs) of .What does the BLUE mean?when the standard set of assumptions holds, The best linear unbiased estimator means that the estimators have the characteristics as follows: linear parameter, unbiased estimators of parameters, and effective parameters, namely the unbiased parameters have the smallest deviation. minimum variance unbiased estimators,How to deduce the estimators of OLS?given the partial difference of to the sum of residual square, we get the following two equations:Please give the verification of the BLUE?(1) linear parameterswhere, So is the linear function of Yi; since it is the weighted function of Yi with the wight of ki. , it is a linear estimator. in the same reason, is a linear estimator.(2) unbiased estimatorssowhat is the variance of ? due to we know , thenassume , and (),(due to the definition of). (3) efficiency: minimum variance unbiased estimators (a)the OLS estimator of is unbiased estimator.thenwhereso is an unbiased estimator of true variance .(b) minimum varianceassume another linear estimator of true parameter here may be not equal to ki.let , thenwhen ki (the weight of OLS), the variance ofis the variance of OLS .so do .3.1 Please explain the assumption of the first column is equivalent to the second column.，since the are constants and X is nonstochastic.is zero by assumption.(2)Giveall i，j(),then,because the error terms are not correlated by assumption =0，since each has zero mean by assumption.（3）Givenby assumption3.6Note that :Multiplying the two ,we obtain the expression for r2,the squared sample correlation coefficient.3.7 Even thoughit may still matter (for causality and theory)if Y is regressed on X or X on Y ,since it ij just the product of the two that equals 1.This does not say that.3.6-3.73.6 Let and are the slop of Y regress to X, and X regress to Y respectively. please explain , meanwhile r is the correlation coefficient between X and Y.3.7 If in the question 3.6, please explain what the difference between regression of Y to X and regression of X to Y?What is the 10 assumption of classic linear model?1 Parameter is linear(Linear in Parameters), 2 random regressor (Random Sampling) 3.the mean of error terms is zero (Zero Conditional Mean 4. Homoskedasticity 5. error term is no-correlation 6. the covariance is zero between error term and regressor 7 the number of observation is less than the number of regressors 8. regressor is various 9. the model is miss specification 10. non multicollinearity(No Perfect Collinearity)What is the 11 standard set of assumptions of classic normal linear model?1 Parameter is linear(Linear in Parameters), 2 random regressor (Random Sampling) 3.the mean of error terms is zero (Zero Conditional Mean 4. Homoskedasticity 5. error term is no-correlation 6. the covariance is zero between error term and regressor 7 the number of observation is less than the number of regressors 8. regressor is various 9. the model is miss specification 10. non multicollinearity(No Perfect Collinearity) 11 Normal Sampling Distributions5.1 Please judge what is true, false, or not sure? and explain the reason.5.1 (a) True. The I test is based on variables with a normal distribution.Since the estimators of ,O and fl2 are linen combinations of the error u, which is assumed to be normally distributed under CLRM,these estimators arc also normally distributed.(b) True. So long as E(u1) =0, the OLS estimators are unbiased.No probabilistic assumptions are required to establish unbiasedness.(c) True. In this case the Eq. (1) in App. 3%, Sec. 3A.l, will be absent. This topic is discussed more fully in Chap. 6, Sec. 6.1.(d) True. The p value is the smallest level of significance at which the null hypothesis can be rejected. The terms level of significance and size of the test are synonymous.(e) True. This follows from Eq. (1) of App. 3A, Sec. 3A.1.(f) False. All we can say is that the data at hand does not permit us to reject the null hypothesis.(g) False. A larger a2 may be counterbalanced by a larger. It is only if the latter is held constant, the statement can be true.(h) False. The conditional mean of a random variable depends on the vaJues taken by another (conditioning) variable. Only if the two variables are independent, that the conditional and unconditional means can be the same.(I) True. This is obvious from Eq. (3.1.7).(j) True. Refer of Eq. (3.5.2). IfX has no influence on 1, 6 will be zero, in which case 5.3 from the data of 2.6 about the revenue and education level. we get the equation (3.7.3) se = (0.8355) ( ) t = ( ) (9.6536) r2=0.8944 n=13a. fill the figure in the blank ( ).b. explain the coefficient 0.6416?c. do you refuse the assumption that true slop coefficient is zero? which test you use and why? what is the statistic figure of p value?d. assumed there is no r2 in the regression, can you get it from the other figure?5.3 (a) se of the slope coefficient is:=0.0664the ivalue under H0: fl1=0,is: =0.8797(b) On average, mean hourly wage goes up by about 64 centsfor an additional year of schooling.(c) Here n=13, so df= 11. lithe null hypothesis were true,the estimated (value is 9.6536. The probability of obtaining such a (value is extremely small; the p value is practically zero. Therefore, one can reject the null hypothesis that education has no effect on hourly earnings.(d) The ESS =74.9389; RSS = 8.8454; numerator df= 1, denominator df =11,，F =93.1929. The p value of such an F under the null hypothesis that there is no relationship between the two variables is 0.000001, which is extremely small. We can thus reject the null hypothesis with great confidence.Note that the F value is approximately the square of the t value under the same null hypothesis.(e) In the bivariate case, given Ho: 2=0, there is the following relationship between the r value and r2: r2 =。Since the 1value is given as 9.6536,=2We obtain: r 2 (9.6536)2 =0.8944(9.6536) 116.1considering the regression model: meanwhile , . the regression line must pass the original point. Is the conclusion right or wrong? give your computation.6.2according to the data from Jan of 1978 to Dec.of 1987, we get the regression result as =0.00681+0.75815 se = (0.02596) (0.27009) t = (0.26229) (2.80700) p 值 = (0.7984) (0.0186) r2=0.4406 =0.76214 se = (0265799) t = (2.95408) p 值 = (0.0131) r2=0.43684meanwhile Y=the monthly return of stock of Texaco (%) X = market rate of return (%)a. what is the difference of two regression models?b. given the above results, will you keep the inception of the first model and why?c. How can you explain the slop coefficients two models?d. What is the theory of two models?e. Can you compare the two models r2? and why?f. in the first model, the Jarque-Bera statistic value is 1.1167, and JB value of the second model is 1.1170. What conclusion you can draw?g.the slop coefficient of t value of the model without intercept is about 2.95, and that of model with intercept is 2.81, can you make a reasonable explanation of the result? (a).What is the difference between the two regression models?(b).Given the preceding results, would you retain the intercept term in the first model? Why or why not? Answer :In the first equation an intercept term is included,since the intercept in the first model is not statistically significiant,say at the 5% level ,it maybe dropped from the model.(c).How would you interpret the slope coefficients in the two models?d.What is the theory underlying the two models? Answer :For each model ,a one percentage point increase in the monthly market rate of return lead on average to about 0.76 percentage point increase in the monthly rate of return on Texaco common stock over the sample period.(d).Whatisthetheoryunderlyingthetwomodels? Answer :Asdiscussedinthechapter,hepresentcasethemodelrelatesthemonthlyreturnontheTexacostocktothemonthlyreturnonthemarket,asrepresentedbyabroadmarketindex.(e).Canyoucomparether2termsofthetwomodels?Whyorwhynot?Answer :No,thetwor2sarenotcomparable.Ther2oftheinterceptlessmodelistherawr2.(f).The JarqueBera normality statistic for the first model Answer :in this problem is 1.1167 and for the second model it is 1.1170. What conclusions can you draw from these statistics?Since we have a reasonably large sample, we could use the JarqueBera test of normality. The JB statistic for the two models is about the same,namely,1.12 and the p value of obtaining such a JB value is about 0.57.Hence do not reject the hypothesis that the error terms follow a normal distribution.(g).Thetvalueoftheslopecoefficientinthezerointerceptmodelisabout2.95,whereasthatwiththeinterceptpresentisabout2.81.Canyoura-tionalizethisresult?Answer :AsperTheilsremarkdiscussedinthechapter,iftheintercepttermisabsentfromthemodel,thenrunningtheregressionthroughtheoriginwillgivemoreefficientestimateoftheslopecoefficient,whichitdoesinthepresentcase.6.3 Considering this regression model:（ Y0，X0 ）(a)Is it a linear regression model?Key:Since the model is linear in the parameters,it is a linear regression model.（b）Howcanyouestimatethismodel?Key:DefinedY*=(1/Y)andX*=(1/X)anddo anOLSregressionofY*onX*.(C)With the X tending to infinity,what will Y do?Key:As X tends to infinity,Y tends to 1/1 .(d)Canyougiveanexampleofthismodelmaybeapplicable?Key:Perhaps this model may be appropriate to explain low consumption of a commodity when income is large,such as an inferior good.7.1 Think about the following data and make some estimation like YX1X2112321833Yi =1+2X2i+1i Yi =1+3 X3i+2iYi =1+2 X2i+3 X3i+iExplanatory note : estimating the parameter and no need to estimate the standard error .(a)2=2?why or why not?（b）3=3?why or why not ?What important conclusion can you get from this question ?7.2 using following data, estimating the parameter and standard error and R2 and adjusted R27.3 Proving (7.9.1)is equal to (7.9.2)8.2 Prove that the F-ratio of (8.5.16) equals to the F-ratio of (8.5.18).(Note:ESS/TSS=R2) Answer: F=(ESSnew-ESSold)/NR / RSSnew/(n-k) (8.5.16) Where NR=number of new regressors. Divided the numerator and denominator by TSS and recall that R2=ESS/TSS and (1-R2)=RSS/TSS. Substituting these expressions into (8.5.16) ,you will obtain (8.5.18)8.3 prove that (8.5.18) and (8.7.10) of the F-test are equivalentAnswer:This is a definitional issue. As noted in the chapter,the unrestricted regression is known as the long, or new, regression,and the restricted regression is known as the short regression. These two differ in the number of regressors included in the models.10.1.In the k-variable linear regression model there are k normalequationsto estimate the k unknowns. These normal equations are given inAppendix C. Assume that is a perfect linear combination of theremaining X variables. How would you show that in this case it is impossibleto estimate the k regression coefficients?Answer：If is a perfect linear combination of the remaining explanatory variables,then there are (k-1) equations with k unknowns.With more unknowns than equations,unique solutions are not possible.10.3.Refer to the child mortality example discussed in Chapter 8. The example there involved the regression of child mortality (CM) rate on percapita GNP (PGNP) and female literacy rate (FLR). Now suppose we add the variable, total fertility rate (TFR). This gives the following regression results.Dependent Variable: CM VariableCoefficient Std. Errort-StatisticProb. C 168.3067 32.891655.1170030.0000PGNP -0.005511 0.001878 -2.9342750.0047FLR -1.768029 0.248017 -7.128663 0.0000TFR 12.86864 4.190533 3.070883 0.0032R-squared 0.747372 Mean dependent var 141.5000Adjusted R-squared 0.734740 S.D. dependent var 75.978