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首页 Obtaining_Univariate_OLS_Estimates_Through_Eleme…

Obtaining_Univariate_OLS_Estimates_Through_Elementary_Algebra.pdf

Obtaining_Univariate_OLS_Estima…

xu1892
2012-07-22 0人阅读 举报 0 0 暂无简介

简介:本文档为《Obtaining_Univariate_OLS_Estimates_Through_Elementary_Algebrapdf》,可适用于人文社科领域

ObtainingunivariateOLSparameterestimatesthroughelementaryalgebraConsiderthelinearregressionmodelyt=αβxtut,t=,,TGivenparameterestimatesαˆandβˆtheresidualsaredefinedasuˆt=yt−αˆ−βˆxt,t=,,TTheOLSestimatesoftheparametersαandβaredeterminedbyminimizingtheresidualsumofsquaresRSS=T∑t=uˆtWewillshowthattheOLSestimatescanbeobtainedbyelementaryalgebra,withoutusingcalculusFirst,expresstheresidualsusingthedemeanedversionsofytandxt(thatis,yt−y¯andxt−x¯)asuˆt=(yt−y¯)−βˆ(xt−x¯)−αˆ−(y¯−βˆx¯)andnotethatsince∑Tt=(yt−y¯)=∑Tt=(xt−x¯)=wehaveT∑t=uˆt=T∑t=(yt−y¯)−βˆ(xt−x¯)Tαˆ−(y¯−βˆx¯),asthesumofcrossproductsvanishesThesecondtermisminimized,andbecomesequaltozero,whenαˆ=y¯−βˆx¯Hence,itsufficestominimizetheterm∑Tt=(yt−y¯)−βˆ(xt−x¯)withrespecttoβˆNotethatT∑t=(yt−y¯)−βˆ(xt−x¯)=SyySxxβˆ−SyxβˆwhereSyy=T∑t=(yt−y¯),Sxx=T∑t=(xt−x¯),Syx=T∑t=(yt−y¯)(xt−x¯)SinceSyydoesnotdependonβˆ,itsufficestominimizethequantitySxxβˆ−SyxβˆwithrespecttoβˆWedoso,by“completingthesquare”:Sxxβˆ−Syxβˆ=Sxx(βˆ−SyxSxxβˆ)=Sxx(βˆ−SyxSxx)−SyxSxxItisclearthattheexpressionaboveisminimizedwhenthesquare(βˆ−SyxSxx)becomeszero,ie,whenβˆ=SyxSxxItfollowsthattheOLSestimatesofαandβaregivenbyβˆ=∑Tt=(yt−y¯)(xt−x¯)∑Tt=(xt−x¯)=∑Tt=ytxt−Tx¯y¯∑Tt=xt−Tx¯,αˆ=y¯−βˆx¯

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