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. -CHAPTER 1THENATUREOFREGRESSIONANALYSIS1.1 a. please use the above data to pute 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.10.7843113.583826.84713419.4174823.173280.1577060.110360108407111.704835.96125217.0731711.694920.2445820.0912787.5848309.5671984,36005616.666619.5599390.1641790.0576217.7922089.5634103.63881419.345248.1717450.1581200.0650268.9500869.1081592.73081912468834.2253520.0830260.0759089.32069510.608704.05063315.521063.6855040.1345830.1134979.97109813.679255A7445321.305187.7014220.1786790.13498612.4835713.278016.34371419.303804.8404840.1197450.16315510.8644911.965815.31453416.313002.9380900.0853240.0616065.7955749.4874593.29557214.937291.7329260.0461220.0321244.2828697.6693232.39282210.615082.3046090.0501000.0431734.1069725.8279372.0447918.6098651.9588640.0601150.0355114.1284402.534965-0.0954206.1106520.6724300.0342030.0185874.3171813.2395570.1910224.5914400.0000000.0417750.0364964.0540542.7250211.3346044.9851190.7633590.0492900.0413734.9512993.4565922.7281286.5910702.3674240.0772290.0481834.7950503.3411032.7472536.1170213.0527290.0953440.0540325.6088563.1578953.6541896.3909773.2315980.0587040.0420811.5373862.4052484.9871025.3003531.5521740.0369660.0301031.7894012.1352314.5045054.2505591.2831480.0159800.0299360.2028401.6027872.7429473.9153090.7601350.0248030.0256062.1592441.7832651.8306645.369128-0.1676450.0336480.0283401.5852052.0215631.4981273.8706520.1679260.0245570.0295281.6254881.1889041.6974171.7452831.6764460.0312150.022945b. please plot the inflation rate of each country, with time as lateral a*is, and inflation rate as vertical a*is in the Plane Rectangular Coordinate System.c. what conclusion can you draw from the inflation history from in above 7 countriesAs 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 e*plaination(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 e*pectation function or the population regression functionConditional e*pectation, also called conditional mathematical e*pectation. For convenience, we discuss two random variables is deduced with the occasion of the eta, assuming they have the density function p (*, y), and the p (y *) under the condition of known factor = *, density function, the conditions of eta to p1 (*) of density function is deduced. Defined under the condition of factor = *, eta conditional mathematical e*pectation is defined as: E eta factor = * = yf (y *) dy. Undertheconditionofagivenvariable*i,interpretedvariablee*pecttrajectoryiscalledpopulationregressionlineorpopulationregressioncurve.Thecorrespondingfunction：EY/*i=f(*i)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 residualA regression model can never be a pletely 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 modelIt tells how the mean or average of the sub-population of Y varies with the fi*ed values of the e*planatory 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 bees 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 theoremUnder 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 meanwhen 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 OLSgiven 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 e*plain the assumption of the first column is equivalent to the second column.，since the are constants and * 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.3Givenby assumption3.6Note that :Multiplying the two ,we obtain the e*pression for r2,the squared sample correlation coefficient.3.7 Even thoughit may still matter (for causality and theory)if Y is regressed on * or * on Y ,since it ij just the product of the two that equals 1.This does not say that.3.6 Let and are the slop of Y regress to *, and * regress to Y respectively. please e*plain , meanwhile r is the correlation coefficient between * and Y.3.7 If in the question 3.6, please e*plain what the difference between regression of Y to * and regression of * to YWhat is the 10 assumption of classic linear model1 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 model1 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 e*plain the reason.5.1(a)True.TheItest isbasedonvariableswith anormaldistribution.Sincetheestimatorsof,Oandfl2arelinenbinationsofthe erroru,whichisassumedtobenormallydistributedunderCLRM,theseestimatorsarc also normallydistributed.(b) True.SolongasE(u1)=0,theOLSestimatorsareunbiased.No probabilisticassumptionsarerequiredtoestablishunbiasedness.(c)True.6.1.(d)True.Thepvalueisthesmallestlevelofsignificanceatwhich thenullhypothesiscanberejected.Thetermslevelofsignificance andsizeofthe testaresynonymous.(e)True.ThisfollowsfromEq.(1)ofApp.3A,Sec.3A.1.(f)False.Allwecansayisthat thedataathanddoesnotpermit ustorejectthenullhypothesis.(g)False.Alargera2maybecounterbalancedbyalarger.It isonlyifthelatterisheldconstant, thestatementcanbetrue.(h)False.Theconditionalmeanofarandomvariabledependson thevaJuestakenbyanother(conditioning)variable.Onlyifthe twovariablesareindependent,that theconditionaland unconditionalmeanscanbethe same.(I)True.ThisisobviousfromEq.(j)True.If*hasnoinfluenceon1,6will bezero,inwhichcase se = (0.8355) ( ) t = ( ) (9.6536) r2=0.8944 n=13a. fill the figure in the blank ( ).b. e*plain the coefficient 0.6416c. do you refuse the assumption that true slop coefficient is zero which test you use and why what is the statistic figure of p valued. assumed there is no r2 in the regression, can you get it from the other figure5.3(a)seofthe slopecoefficient is:=0.0664theivalueunderH0:fl1=0,is:=0.8797(b)Onaverage,meanhourly wagegoesupbyabout64centsforan additionalyearofschooling.(c) Here n=13,sodf= 11.lithenullhypothesisweretrue,theestimated(valueis9.6536. Theprobabilityofobtainingsuch a(valueise*tremelysmall;thepvalueispracticallyzero. Therefore,onecanrejectthenullhypothesisthateducation hasnoeffectonhourlyearnings.(d)TheESS =74.9389;RSS=8.8454;numeratordf=1, denominatordf =11,，F =93.1929. The pvalueofsuchanF underthe nullhypothesisthat thereisnorelationshipbetween the twovariablesis0.000001,whichise*tremelysmall.We canthusrejectthenullhypothesiswithgreatconfidence.NotethattheF valueisappro*imatelythe squareofthetvalue underthesamenullhypothesis.(e)Inthebivariatecase,givenHo:2=0,thereisthefollowing relationshipbetweenthervalueandr2: r2 =。Since the1valueis givenas 9.6536,=2We obtain: r2(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 putation.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 Te*aco (%) * = market rate of return (%)a. what is the difference of two regression modelsb. given the above results, will you keep the inception of the first model and whyc. How can you e*plain the slop coefficients two modelsd. What is the theory of two modelse. Can you pare the two models r2 and whyf. in the first model, the Jarque-Bera statistic value is 1.1167, and value of the second model is 1.1170. What conclusion you can drawg.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 e*planation 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 modelsd.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 Te*aco mon stock over the sample period.(d).Whatisthetheoryunderlyingthetwomodels Answer :Asdiscussedinthechapter,thismodelrepresentsthecharacteristiclineofinvestmenttheory.inthepresentcasethemodelrelatesthemonthlyreturnontheTe*acostocktothemonthlyreturnonthemarket,asrepresentedbyabroadmarketinde*.(e).Canyouparether2termsofthetwomodelsWhyorwhynotAnswer :No,thetwor2sarenotparable.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(g).Thetvalueoftheslopecoefficientinthezerointerceptmodelisabout2.95,whereasthatwiththeinterceptpresentisabout2.81.Canyoura-tionalizethisresultAnswer :AsperTheilsremarkdiscussedinthechapter,iftheintercepttermisabsentfromthemodel,thenrunningtheregressionthroughtheoriginwillgivemoreefficientestimateoftheslopecoefficient,whichitdoesinthepresentcase.6.3 Considering this regression model: Y0，*0 (a)Is it a linear regression modelKey:Since the model is linear in the parameters,it is a linear regression model.bHowcanyouestimatethismodelKey:DefinedY*=(1/Y)and*=(1/*)anddo anOLSregressionofY*on*.(C)With the * tending to infinity,what will Y doKey:As * tends to infinity,Y tends to 1/1 .(d)Canyougiveane*ampleofthismodelmaybeapplicableKey:Perhaps this model may be appropriate to e*plain low consumption of a modity when ine is large,such as an inferior good.7.1 Think about the following data and make some estimation like Y*1*2112321833Yi =1+2*2i+1iYi =1+3 *3i+2iYi =1+2 *2i+3 *3i+iE*planatory note : estimating the parameter and no need to estimate the standard error .(a)2=2why or why notb3=3why 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 R2 Answer: Where NR=number of new regressors. Divided the numerator and denominator by TSS and recall that R2=ESS/TSS and (1-R2)=RSS/TSS.Answer: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. is a perfect linear bination of theremaining * variables. How would you show that in this case it is impossibleto estimate the k regression coefficientsAnswer：If is a perfect linear bination of the remaining e*planatory variables,then there are (k-1) equations with k unknowns.With more unknowns than equations,unique solutions are not possible.Dependent Variable: CM VariableCoefficient Std. Errort-StatisticProb. FLR -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 75.97807S.E. of regression 39.13127 Akaike info criterion 10.23218Sum squared resid 91875.38 Schwarz criterion10.36711Log likelihood -323.4298 F-statistic 59.16767Durbin-Watson stat 2.170318 Prob(F-statistic) 0.000000What changes do you see And how do you account for themb. Is it worth adding the variable TFR to the model Whyc. Since all the individual t coefficients are statistically significant, canwe say that we do not have a collinearity problem in the present caseAnswer:(a)Although the numerical values of the intercept and the slope coefficients of PGNP and FLR have changed, their signs have not. Also, these variables are still statiscally significant. These changes are due to the addition of the TFR variable, suggesting that there may be some collineariy among the regressors. (b)Since the t value of the TFR coefficient is very significant (the p value is only 0032), it seems TFR belongs in the model. The positive sign of this coefficient also makes sense in that the larger the number of children born to a woman, the greater the chances of increased child mortality. (c)This is one of those happy occurrences where despite possible cllinearity, the individual coefficients are still statistically significant.10.5. Consider the following model:= + + + + + where Y = consumption, * = ine, and t = time. The precedingmodelpostulates that consumption e*penditure at time t is a function not onlyof ine at time t but also of ine through previous periods. Thus,consumption e*penditure in the first quarter of 2000 is a function of inein that quarter and the four quarters of 1999. Such models arecalled distributed lag models, and we shall discuss them in a laterchapter.a. Would you e*pect multicollinearity in such models and whyb. If collinearity is e*pected, how would you resolve the problemAnswer:(a) Yes. Economic time series data tend to move in the same direction. Here, the lagged variables of ine will generally move in the same direction.(b) As discussed briefly in Chapter 10 and further discussed in Chapter 17, the first difference transformation may alleviate the problem.10.7. In data involving economic time series such as GNP, money supply,prices, ine, unemployment, etc., multicollinearity is usuallysuspected.WhyAnswer：As discussed in Question 10.5,economic variables are often influenced by similiar factors such as business cycles and trend.Therefore,in regression analysis,using variables such as GNP and money supply,one should e*pect m11.1 briefly e*plain what statements are true, false or not sure.a.the heteroscedasticity appears, OLS estimator are biased and non efficient.b. heteroscedasticity appears, t-test and F-test are not effective any more.c. heteroscedasticity appears, OLS will over estimate the standard error of estimators.d. OLS regression residential are system pattern, it means heteroscedasticity.f. If the regression model is miss specification, ( for e*ample, miss an important variable), the OLS residential show apparent pattern.g. if a model miss a non constant variance of regressor, the OLS residential will be heteroscedasticity.(a)False.The estimators are inefficient but are unbiased.(b)True.Typically,var(2*)var(2，which means the confidence interval will be overestimated.The is to say t-test and F-test will provide inaccurate results for us.(c)False.Typically,but not always,will the variance be over estimated.Sometimes the estimators are underestimated.(d)False.Besides heteroscedasticity ,such a pattern may result from autocorrelation,model specification errors.(e)True.Since the true i2 are not directly observable,some assumption about th
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