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1、MultipleRegressionAnalysisy=b0+b1x1+b2x2+...bkxk+u1.Estimation1ParallelswithSimpleRegressionb0isstilltheinterceptb1tobkallcalledslopeparametersuisstilltheerrorterm(ordisturbance)Stillneedtomakeazeroconditionalmeanassumption,sonowassumethatE(u
2、x1,x2,…,xk)=0Stillminimi
3、zingthesumofsquaredresiduals,sohavek+1firstorderconditions2theOLSregressionlineorthesampleregressionfunction(SRF).istheOLSinterceptestimateandaretheOLSslopeestimates.Stilluseordinaryleastsquarestogettheestimates:3OLSFirstOrderConditionsThisminimizationproblemcanbesol
4、vedusingmultivariablecalculas.Thisleadstok+1linearequationink+1unknown:……4AFittedorPredictedValueForobservationi,thefittedvalueisTheresidualforobservationiisdefinedasinthesimpleregressioncase,Theproperties123Thepoint()isalwaysontheOLSregressionline:5InterpretingMulti
5、pleRegression6A“PartiallingOut”Interpretation7“PartiallingOut”continuedPreviousequationimpliesthatregressingyonx1andx2givessameeffectofx1asregressingyonresidualsfromaregressionofx1onx2Thismeansonlythepartofxi1thatisuncorrelatedwithxi2arebeingrelatedtoyisowe’reestimat
6、ingtheeffectofx1onyafterx2hasbeen“partialledout”8SimplevsMultipleRegEstimate9Goodness-of-Fit10Goodness-of-Fit(continued)Howdowethinkabouthowwelloursampleregressionlinefitsoursampledata?Cancomputethefractionofthetotalsumofsquares(SST)thatisexplainedbythemodel,callthis
7、theR-squaredofregressionR2=SSE/SST=1–SSR/SST11Goodness-of-Fit(continued)12MoreaboutR-squaredR2canneverdecreasewhenanotherindependentvariableisaddedtoaregression,andusuallywillincreaseBecauseR2willusuallyincreasewiththenumberofindependentvariables,itisnotagoodwaytocom
8、paremodels13AssumptionsforUnbiasednessPopulationmodelislinearinparameters:y=b0+b1x1+b2x2+…+bkxk+uWecanusearandomsampleofsizen,{(xi1,xi2,…,xik,yi):i=1,2,…,n},fromthepopulationmodel,sothatthesamplemodelisyi=b0+b1xi1+b2xi2+…+bkxik+uiE(u
9、x1,x2,…xk)=0,implyingthatallofthe
10、explanatoryvariablesareexogenousNoneofthex’sisconstant,andtherearenoexactlinearrelationshipsamongthem14TooManyorTooFewVariablesWhat
