OLS BLUE57KOþ²LÆÁ¯þ°Æ7KÆ May2,2010Á¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE51OLS BLUE5Á¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE5¯K JÑ ¦{£OLS¤´Ä´¦) .ëê k {ºÁ¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE5ïþIO Oþ ïþIOµÃ 5 55Á¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE5BLUE5`£Best¤5£Linear¤Ã £Unbias¤Á¯þ°Æ7KÆ 7KOþ²LÆ
XJ·Pµxi=Xi−X¯§yi=Yi−Y¯§Kµ∑xβiy1=∑ix2iβ0=Y¯−β1X¯OLS BLUE5é
ÎÒ `²dc í §·∑ µβ1=(∑Xi−X¯)(Yi−Y¯)(Xi−X¯)2§β0=Y¯−β1X¯Á¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE5é
ÎÒ `²dc í §·∑ µβ1=(∑Xi−X¯)(Yi−Y¯)(Xi−X¯)2§β0=Y¯−β1X¯XJ·Pµxi=Xi−X¯§yi=Yi−Y¯§Kµ∑xβiy1=∑ix2iβ0=Y¯−β1X¯Á¯þ°Æ7KÆ 7KOþ²LÆ
ki=∑xix2§Kβ1=∑kiYiiOLS BLUE555éuβ1
óµ∑β1=∑xiyix2∑i=x∑i(Yi−Y¯)x2∑i∑=∑xiYix2+∑xiY¯∑x2ii=∑xiYix2iÁ¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE555éuβ1
óµ∑β1=∑xiyix2∑i=x∑i(Yi−Y¯)x2∑i∑=∑xiYix2+∑xiY¯∑x2ii=∑xiYix2i ki=∑xix2§Kβ1=∑kiYiiÁ¯þ°Æ7KÆ 7KOþ²LÆ
w1∑i=n−X¯kix2§Kβ0=∑wiYiiOLS BLUE555éuβ0
óµβ0=Y¯∑−β1X¯=Yi−X¯∑∑kiYinx2∑i=1(−∑X¯ki)Ynx2iiÁ¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE555éuβ0
óµβ0=Y¯∑−β1X¯=Yi−X¯∑∑kiYinx2∑i=1(−∑X¯kinx2)Yii wi=1∑n−X¯kix2§Kβ0=∑wiYiiÁ¯þ°Æ7KÆ 7KOþ²LÆ
â∑ki=0§E( i)=0 µ∑E(βˆ1)=E(β1kiXi)∑=Exi(xi+X¯)(β1∑)∑x2i=β1E(∑x2X¯∑xii(∑x2)+Ex2)ii=β1OLS BLUE5à 5éuβ1
óµ∑E(βˆ1)=E(∑ki(β0+β1∑Xi+ i))∑=E(β0ki+β1kiXi+ki i)Á¯þ°Æ7KÆ 7KOþ²LÆ
µ∑E(βˆ1)=E(β1kiXi)∑=Exi(xi+X¯)(β1∑)∑x2i=β1E(∑x2X¯∑xii(∑x2)+Ex2)ii=β1OLS BLUE5à 5éuβ1
óµ∑E(βˆ1)=E(∑ki(β0+β1∑Xi+ i))∑=E(β0ki+β1kiXi+ki i)â∑ki=0§E( i)=0Á¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE5Ã 5éuβ1
óµ∑E(βˆ1)=E(∑ki(β0+β1∑Xi+ i))∑=E(β0ki+β1kiXi+ki i)â∑ki=0§E( i)=0 µ∑E(βˆ1)=E(β1kiXi)∑=Exi(xi+X¯)(β1∑)∑x2i=β1E(∑x2X¯∑xii(∑x2)+Ex2)ii=β1Á¯þ°Æ7KÆ 7KOþ²LÆ
∑wiXi=0∑:∑wiXi=1−xi(∑xi+X¯)x2∑n∑i=1−∑x2inx2−X¯∑∑xix2ii=1−1=0OLS BLUE5à 5∑wi=1µ∑∑wi=1∑+kniX¯=1n=1Á¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE5à 5∑wi=1µ∑∑wi=1∑+kniX¯=1n∑=1wiXi=0∑:∑wiXi=1−xi(∑xi+X¯)x2∑n∑i=1−∑x2inx2−X¯∑∑xix2ii=1−1=0Á¯þ°Æ7KÆ 7KOþ²LÆ
â∑∑wi=1§wiXi=0±9E( i)=0§ µ∑∑∑E(βˆ0)=E(β0wi)+E(β1wiXi)+wiE( i)=β0OLS BLUE5Ã 5éuβ0
óµ∑E(βˆ1)=E(∑wi(β0+β1∑Xi+ i))∑=E(β0wi+β1wiXi+wi i)Á¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE5Ã 5éuβ0
óµ∑E(βˆ1)=E(∑wi(β0+β1∑Xi+ i))∑=E(β0wi+β1wiXi+wi i)â∑wi=1§∑wiXi=0±9E( i)=0§ µ∑∑∑E(βˆ0)=E(β0wi)+E(β1wiXi)+wiE( i)=β0Á¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE5 5ÄkO β0Úβ1 ∑"éuβ1
óµD(βˆ1)=D(∑KiYi)=D[K∑i(β0+β1Xi+εi)]=D∑(β1+Kiεi)=∑K2iD(εi)=Xi2(∑X2)σ2∑i=∑X2i(X2σ2)2i=∑σ2X2iÁ¯þ°Æ7KÆ 7KOþ²LÆ
OLS BLUE5 5éuβ0
óµD(βˆ0)=D(ΣωiYi)=1(+X¯2ΣK2=Dni)σ2[Σωi(β0+β1Xi+εi)]=D(β0+Σωiεi)=1(+X¯2nΣX2)σ2=Σω2iiσ2=ΣX2+nX¯2i(=Σ(1−KniX¯2)σ2nΣX2)σ2i=Σ(1−2KiX¯+K2X¯2)σ2=ΣX2σ2in2ninΣX2iÁ¯þ°Æ7KÆ 7KOþ²LÆ
