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1、第一章 简单回归模型y=b0+b1x+u要求：1、普通最小二乘估计方法(OLS)2、OLS的统计特性1ContentsWhatisthesimpleregressionmodel?Howtoderivetheordinaryleastsquares(OLS)estimates?PropertiesofOLSstatisticsandR2UnbiasednessofOLSandVariancesoftheOLSestimators2Whatisthesimpleregressionmodel?y=b0+b1x+u3SomeTerminologyInthesimp
2、lelinearregressionmodel,wherey=b0+b1x+u,wetypicallyrefertoyastheDependentVariable,orLeft-HandSideVariable,orExplainedVariable,orResponseVariable,orRegressand4SomeTerminology,cont.y=b0+b1x+uInthesimplelinearregressionofyonx,wetypicallyrefertoxastheIndependentVariable,orRight-HandSide
3、Variable,orExplanatoryVariable,orRegressor,orCovariate,orControlVariables5ASimpleAssumptiony=b0+b1x+uTheaveragevalueofu,theerrorterm,inthepopulationis0.Thatis,E(u)=0Thisisnotarestrictiveassumption,sincewecanalwaysuseb0tonormalizeE(u)to0wage=b0+b1educ+u6ZeroConditionalMeany=b0+b1x+uW
4、eneedtomakeacrucialassumptionabouthowuandxarerelatedWewantittobethecasethatknowingsomethingaboutxdoesnotgiveusanyinformationaboutu,sothattheyarecompletelyunrelated.Thatis,thatE(u
5、x)=E(u)=0,whichimpliesE(y
6、x)=b0+b1x,whichisoftencalledPopulationRegressionFunction(PRF)7..x1x2E(y
7、x)asal
8、inearfunctionofx,whereforanyxthedistributionofyiscenteredaboutE(y
9、x)E(y
10、x)=b0+b1xyf(y)PopulationRegressionFunctionHowtoestimatetheparametersb0andb1?8Howtoderivetheordinaryleastsquares(OLS)estimates?9OrdinaryLeastSquaresBasicideaofregressionistoestimatethepopulationparametersfromasam
11、pleLet{(xi,yi):i=1,…,n}denotearandomsampleofsizenfromthepopulationForeachobservationinthissample,itwillbethecasethatyi=b0+b1xi+ui10....y4y1y2y3x1x2x3x4}}{{u1u2u3u4xyPopulationregressionline,sampledatapointsandtheassociatederrortermsE(y
12、x)=b0+b1x11DerivingOLSEstimatesToderivetheOLSes
13、timatesweneedtorealizethatourmainassumptionofE(u
14、x)=E(u)=0alsoimpliesthatCov(x,u)=E(xu)=0Why?RememberfrombasicprobabilitythatCov(X,Y)=E(XY)–E(X)E(Y)12DerivingOLScontinuedWecanwriteour2restrictionsjustintermsofx,y,b0andb1,sincey=b0+b1x+u,u=y–b0–b1xE(y–b0–b1x)=0E[x(y–b0–b1x)]=0Thesear
15、ecalledmomentrestri
