Econometrics. 3) Statistical properties of the OLS estimator

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Rosanna Donna Payne
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1 30C0000 Ecoometrcs 3) Statstcal propertes of the OLS estmator Tmo Kuosmae Professor, Ph.D.

2 Today s topcs Whch assumptos are eeded for OLS to work? Statstcal propertes Ubasedess Effcecy Cosstecy Asymptotc ormalty

3 Importat dstcto Regresso model Theoretcal model: y = β 1 + β x + ε Emprcal model: y = b 1 + b x + e What ca we fer about the parameters of the theoretcal model based o the emprcal model?

4 Types of statstcal ferece Estmato Pot estmato Iterval estmato Hypothess testg

5 Assumptos (SLR model) Assumpto SLR.1 Lear Parameters: y = β 1 + β x + ε (populato model) Assumpto SLR. Radom Samplg: {(y, x ) : = 1,,, } draw from the populato model Assumpto SLR.3 Sample Varato x: there s sample varato outcomes of x. Assumpto SLR.4 Zero Codtoal Mea: E[ε x] = 0 Assumpto SLR.5 Homoskedastcty: Var(ε x) = σ

6 Assumptos (MLR model) Assumpto MLR.1 Lear Parameters: y = β x + ε (populato model) Assumpto MLR. Radom Samplg: {(y, x 1,, x k ) : = 1,,, } draw from the populato model Assumpto MLR.3 No perfect collearty Sample Varato x. No exact lear depedece betwee explaatory varables. Assumpto MLR.4 Zero Codtoal Mea: E[ε x] = 0 Assumpto MLR.5 Homoskedastcty: Var(ε x) = σ

7 Sample vs. populato Sample average x x / 1 Populato mea Ex ( ) Sample varace Est.Var(x) Populato varace Var(x) 1 s E[( x) ] x x ( ) 1 1 Sample covarace Est.Cov(x,y) 1 r ( y y)( x x) XY 1 1 Populato covarace Cov(x,y) E[( x )( y )] XY x y 7

8 OLS estmator of slope β b 1 ( x x)( y y) 1 ( x x) Est. Cov( x, y) Est. Var( x) Note: f Est.Var(x)=0, the OLS estmator caot be computed. (Assumpto SLR.3)

9 OLS estmator of slope β b 1 ( x x)( y y) 1 ( x x) Usg SLR.1, sert y = β 1 + β x + ε b 1 ( x x) ( x ) ( x ) ( x x)

10 OLS estmator of slope β b ( x x) ( x x) ( x x) ( x x) ( x ) ( x ) 1 1 ( x x) ( x x)( ) ( x x) x x ( x x) 0 ( x x) ( x x)

11 OLS estmator of slope β b ( x x) 1 ( x x) 1 Est. Cov( x, ) Est. Var( x) Iterpretato: OLS estmator b s equal to the true parameter β + a error term The greater the sample covarace of x ad ε, the greater the error b The greater the sample varace of x, the smaller the error b

12 Desrable propertes of a estmator Fte sample propertes Ubasedess Effcecy Asymptotc propertes Cosstecy Asymptotc ormalty

13 Ubasedess of OLS The OLS estmator b s ubased f Eb ( ) I other words, the estmator does ot systematcally uder- or over-estmate the true β

14 Ubasedess of OLS The OLS estmator b s ubased f Eb ( ) Note that E( b ) ( x x) 1 E ( x x) 1 To acheve ubasedess, we eed to make sure that Est. Cov( x, ) Cov( x, ) E Est. Var ( x ) Var ( x ) 0 Note: Est.Cov ad Est.Var are ubased estmators

15 Exogeety: Ubasedess of OLS Assumpto SLR.4 Zero Codtoal Mea: E[ε x] = 0 SLR.4 mples Cov(x,ε) =0 Gve the exogeety assumpto, E ( x x) E Est. Cov( x, ) Cov( x, ) 1 ( ) ( ) ( ) Var x Var x x x 1 0 Note: sample covarace s a ubased estmator of the populato covarace

16 Ubasedess of OLS SLR.4 s the oly statstcal assumpto we eed to esure ubasedess. Volato of ths assumpto s called Edogeety (to be examed more detal later ths course). Note: assumg E(ε) = 0 does ot mply Cov(x,ε) =0. E[ε x] = 0 mples that E(ε) = 0 ad Cov(x,ε) =0. The coverse s also true the case of SLR.

17 Effcecy of OLS The OLS estmator b s effcet f t has smaller varace tha ay other lear ubased estmator Varace of the OLS estmator s ( )( ) ( ) ( ) ( ) ( )( ) ( ) x x Var b Var Var x x x x Var x x

18 Effcecy of OLS Assumpto: o autocorrelato For ay par of observatos,j, dsturbaces ε ad ε j are ucorrelated: Thus, Cov(ε,ε j ) =0 for all j=1,,; j ( )( x x) ( x x) ( x x) Var b Var Var Cov 1 ( ) ( ) (, ) j 1 j 1 1: ( x x) ( x x) j ( x x) ( x x) ( x x) Var( )

19 Effcecy of OLS Assumpto SLR.5 Homoskedastcty: Varace of dsturbace ε s costat across all observatos: Var(ε x) = σ Thus, 1 ( ) x x ( x x) 1 Var 1 4 ( x x) ( x x) 1 1 Var( b ) ( ) ( x x) ( 1) Est. Var( x)

20 Effcecy of OLS Gauss-Markov theorem: OLS estmator b 1 has smaller varace tha ay other lear ubased estmator of β 1. I other words, OLS s statstcally effcet. Some texts state that OLS s the Best Lear Ubased Estmator (BLUE) Note: we eed three assumptos Exogeety (SLR.3), Homoscedastcty (SLR.5), ad o autocorrelato.

21 Cosstecy of OLS The OLS estmator b s statstcally cosstet f t coverges probablty to β : plm b lm Pr b 0 for ay ε >0 Proof. Recall that b Est. Cov( x, ) Est. Var( x) If the assumptos Exogeety ad o autocorrelato hold, the plmb plm( Est. Cov( x, )) Cov( x, ) plm( Est. Var( x)) Var( x)

22 Asymptotc ormalty By the cetral lmt theorem, f the assumptos Exogeety, Homoscedastcty, ad o autocorelato hold, the b 1 coverges dstrbuto to the ormal dstrbuto b ~ N, ( 1) Var ( x ) a Note 1: Covergece dstrbuto s the weakest form of covergece. Covergece probablty (cosstecy) mples covergece dstrbuto, but the coverse s ot true. Note : Asymptotc ormalty of b does ot requre that dsturbaces ε are ormally dstrbuted.

23 SLR assumptos recosdered Model specfcato: y = β 1 + β x + ε Radom samplg: observed data radomly draw from the populato model. Data requremet: Est.Var(x)>0 Assumptos regardg the radom dsturbace term ε: 1) Exogeety: Dsturbaces ε ad regressor x are ucorrelated. ) No autocorrelato: Dsturbaces ε are ucorrelated wth each other. 3) Homoskedastcty: Dsturbaces ε have the same varace across all observatos. [Note: Normalty of dsturbace ε ot eeded.]

24 Assumptos <-> propertes Fte sample propertes Requred assumptos Ubasedess Exogeety Effcecy Exogeety, No autocorrelato, Homoscedastcty Asymptotc propertes Cosstecy Exogeety, No autocorrelato Asymptotc ormalty Exogeety, No autocorrelato, Homoscedastcty

25 Topc: Next tme Wed 1 Mar Statstcal ferece 5

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch