Statistical Properties of the OLS Coefficient Estimators. 1. Introduction
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1 ECOOMICS 35* -- OTE 4 ECO 35* -- OTE 4 Stattcal Properte of the OLS Coeffcent Etmator Introducton We derved n ote the OLS (Ordnary Leat Square etmator ˆβ j (j, of the regreon coeffcent βj (j, n the mple lnear regreon model gven by the populaton regreon equaton, or PRE Y β + β + u (,, ( where u an d random error term The OLS ample regreon equaton (SRE correpondng to PRE ( Y + + û (,, ( where ˆβ and ˆβ are the OLS coeffcent etmator gven by the formula ˆ y β (3 β ˆ Y ˆ (4 β, y Y Y,, and Y Y Why Ue the OLS Coeffcent Etmator? The reaon we ue thee OLS coeffcent etmator that, under aumpton A- A8 of the clacal lnear regreon model, they have everal derable tattcal properte Th note eamne thee derable tattcal properte of the OLS coeffcent etmator prmarly n term of the OLS lope coeffcent etmator ˆβ ; the ame properte apply to the ntercept coeffcent etmator ˆβ ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page of page
2 ECOOMICS 35* -- OTE 4 Stattcal Properte of the OLS Slope Coeffcent Etmator PROPERTY : Lnearty of ˆβ The OLS coeffcent etmator ˆβ can be wrtten a a lnear functon of the ample value of Y, the Y (,, Proof: Start wth formula (3 for ˆβ : y (Y Y Y Y Y becaue Defnng the obervaton weght for,,, we can rewrte the lat epreon above for ˆβ a: β ˆ Y where (,, (P ote that the formula (3 and the defnton of the weght mply that ˆβ alo a lnear functon of the y uch that β ˆ y Reult: The OLS lope coeffcent etmator ˆβ a lnear functon of the ample value Y or y (,,, where the coeffcent of Y or y ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page of page
3 ECOOMICS 35* -- OTE 4 Properte of the Weght In order to etablh the remanng properte of arthmetc properte of the weght ˆβ, t neceary to now the [K], e, the weght um to zero, becaue [K] ( ( ( [K3] ( nce by [K] above [K4] Implcaton: ( ( ( ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page 3 of page
4 ECOOMICS 35* -- OTE 4 PROPERTY : Unbaedne of ˆβ and ˆβ The OLS coeffcent etmator ˆβ unbaed, meanng that E(ˆ β β The OLS coeffcent etmator ˆβ unbaed, meanng that E(ˆ β β Defnton of unbaedne: The coeffcent etmator ˆβ unbaed f and only f E(ˆ β β ; e, t mean or epectaton equal to the true coeffcent β Proof of unbaedne of ˆβ : Start wth the formula Y Snce aumpton A tate that the PRE Y β + β + u, Y ( β β β + β + u, + u + β + u nce Y nce β + β and + u by A ow tae epectaton of the above epreon for ˆβ, condtonal on the ample value {:,, } of the regreor Condtonng on the ample value of the regreor mean that the are treated a nonrandom, nce the are functon only of the E(ˆ β E( β β β β + E[ + E(u + u ] nce β nce E(u a contant and the by aumpton A are nonrandom Reult: The OLS lope coeffcent etmator the lope coeffcent β: that, E(ˆ β β ˆβ an unbaed etmator of (P ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page 4 of page
5 ECOOMICS 35* -- OTE 4 Proof of unbaedne of ˆβ : Start wth the formula β ˆ Y ˆ β Average the PRE Y β + β + u acro : Y β + β + u (um the PRE over the obervaton Y u β + + (dvde by β β + + u where Y,, and u Y β Y u Subttute the above epreon for Y nto the formula β ˆ Y ˆ : β Y β β + β + u + ( β + u nce Y β + β + u 3 ow tae the epectaton of ˆβ condtonal on the ample value {:,, } of the regreor Condtonng on the mean that treated a nonrandom n tang epectaton, nce a functon only of the E(ˆ β E( β β β β β β + E ( + E( β + E( β + + ( β [ β ] [ E( β E(ˆ β ] + E(u β + E(u nce β nce E(u by aumpton A and A5 nce E( β a contant β and E(ˆ β β Reult: The OLS ntercept coeffcent etmator of the ntercept coeffcent β: that, E(ˆ β β ˆβ an unbaed etmator (P ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page 5 of page
6 ECOOMICS 35* -- OTE 4 PROPERTY 3: Varance of ˆβ Defnton: The varance of the OLS lope coeffcent etmator ˆβ defned a { } ( ˆ E [ E(ˆ β ] Var β Dervaton of Epreon for Var( ˆβ : Snce ˆβ an unbaed etmator of β, E( ˆβ β The varance of ˆβ can therefore be wrtten a { } ( ˆ E [ β ] Var β From part ( of the unbaedne proof above, the term [ ˆβ β], whch called the amplng error of ˆβ, gven by [ ˆ β] u β 3 The quare of the amplng error therefore [ β ] ( u 4 Snce the quare of a um equal to the um of the quare plu twce the um of the cro product, [ β ] ( u u + < u u ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page 6 of page
7 ECOOMICS 35* -- OTE 4 For eample, f the ummaton nvolved only three term, the quare of the um would be 3 u u + u + u ( 3 3 u + u + u + uu + uu + uu ow ue aumpton A3 and A4 of the clacal lnear regreon model (CLRM: (A3 Var(u E(u σ for all,, ; > (A4 Cov(u,u, E(u u, for all 6 We tae epectaton condtonal on the ample value of the regreor : E [( β ] E(u + < σ E(u σ nce E(u u nce E(u nce E(u u, σ, by (K for by (A4 by (A3 Reult: The varance of the OLS lope coeffcent etmator ˆβ σ σ σ Var(ˆ β where TSS (P3 ( TSS The tandard error of ˆβ the quare root of the varance: e, σ σ σ e(ˆ β Var(ˆ β TSS ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page 7 of page
8 ECOOMICS 35* -- OTE 4 PROPERTY 4: Varance of ˆβ (gven wthout proof Reult: The varance of the OLS ntercept coeffcent etmator ˆβ Var(ˆ σ σ β (P4 ( The tandard error of ˆβ the quare root of the varance: e, e(ˆ σ β Var(ˆ β Interpretaton of the Coeffcent Etmator Varance Var ( β ˆ and Var ( β $ meaure the tattcal precon of the OLS coeffcent etmator ˆβ and β $ Var(ˆ σ σ β β ; Var(ˆ Determnant of Var(ˆ β and Var( β $ Var(ˆ β and Var( $ are maller: β ( the maller the error varance σ, e, the maller the varance of the unoberved and unnown random nfluence on Y ; ( the larger the ample varaton of the about ther ample mean, e, the larger the value of (,,, ; (3 the larger the ze of the ample, e, the larger ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page 8 of page
9 ECOOMICS 35* -- OTE 4 PROPERTY 5: Covarance of ˆβ and β $ Defnton: The covarance of the OLS coeffcent etmator ˆβ and defned a Cov( ˆβ, ˆβ E{[ ˆβ - E( ˆβ ][ β $ - E( β $ ]} Dervaton of Epreon for Cov( ˆβ, β $ : Snce β ˆ Y ˆ β, the epectaton of ˆβ can be wrtten a $ β E(ˆ β Y E(ˆ β Y β nce E(ˆ β β Therefore, the term E(ˆ β can be wrtten a E(ˆ β [Y ] Y + β (ˆ β β [Y β ] Y + β Snce E(ˆ β β, the term E(ˆ tae the form β E(ˆ β β 3 The product [ β ˆ E(ˆ ][ E(ˆ ] thu tae the form β β [ˆ β E(ˆ β ][ˆ β E(ˆ β ] (ˆ β (ˆ β β (ˆ β β β ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page 9 of page
10 ECOOMICS 35* -- OTE 4 4 The epectaton of the product [ β ˆ E(ˆ ][ E(ˆ ] therefore E β {[ˆ β E(ˆ β ][ˆ β E(ˆ β ]} E[ (ˆ β β ] E(ˆ β Var(ˆ β σ β β b/c a contant b/c E(ˆ β β Var(ˆ β σ b/c Var(ˆ β Reult: The covarance of ˆβ and $ β σ Cov (ˆ β, β (P5 Interpretaton of the Covarance Cov( ˆβ, β $ Snce both σ and are potve, the gn of Cov( ˆβ, ˆβ depend on the gn of ( If >, Cov( ˆβ, ˆβ < : the amplng error ( β ˆ β and ( β are of oppote gn ( If <, Cov( ˆβ, ˆβ > : the amplng error ( β ˆ β and ( β are of the ame gn ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page of page
11 ECOOMICS 35* -- OTE 4 THE GAUSS-MARKOV THEOREM Importance of the Gau-Marov Theorem: The Gau-Marov Theorem ummarze the tattcal properte of the OLS coeffcent etmator $β j (j, More pecfcally, t etablhe that the OLS coeffcent etmator ˆβ j (j, have everal derable tattcal properte Statement of the Gau-Marov Theorem: Under aumpton A-A8 of the CLRM, the OLS coeffcent etmator $β j (j, are the mnmum varance etmator of the regreon coeffcent βj (j, n the cla of all lnear unbaed etmator of β j That, under aumpton A-A8, the OLS coeffcent etmator $β j are the BLUE of βj (j, n the cla of all lnear unbaed etmator, where BLUE Bet Lnear Unbaed Etmator Bet mean mnmum varance or mallet varance So the Gau-Marov Theorem ay that the OLS coeffcent etmator $β j are the bet of all lnear unbaed etmator of βj, where bet mean mnmum varance ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page of page
12 ECOOMICS 35* -- OTE 4 Interpretaton of the G-M Theorem: Let ~ β j be any other lnear unbaed etmator of β j Let $β j be the OLS etmator of βj; t too lnear and unbaed Both etmator β ~ j and $β j are unbaed etmator of βj: E( β $ j β j and E( β ~ j β j ~ 3 But the OLS etmator $β j ha a maller varance than β j : Var( $ Var( ~ ~ β β $β j effcent relatve to β j j j Th mean that the OLS etmator any other lnear unbaed etmator of β j ~ $β j tattcally more prece than β j, ECO 35* -- ote 4: Stattcal Properte of OLS Etmator Page of page
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