01 Population Regression Models Transparencies

01 Population Regression Models Transparencies - Population...

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Unformatted text preview: Population Regression Models Econometrics II Douglas G. Steigerwald UC Santa Barbara Winter 2011 D. Steigerwald (UCSB) Population Models Winter 2011 1 / 12 Overview Reference: F. Hayashi Econometrics Chapter 1.1 (yt , xt 1 , . . . , xtK ) := (yt , xt0 ) data t = 1, . . . , n yt - dependent variable xt - regressors (covariates) model yt is determined by h (xt ) 4 key assumptions ! Classic regression model D. Steigerwald (UCSB) Population Models Winter 2011 2 / 12 Assumption 1: Linear Model model - linear in coe¢ cients h (xt ) = β0 + β1 xt 1 + + βK xtK model - approximate yt = β0 + β1 xt 1 + + βK xtK + ut ut - model error (latent) ut 2 R ) yt 2 R interpretation βk = ∂y t ∂xtk approximate model ! βk captures impact for small changes in xtk D. Steigerwald (UCSB) Population Models Winter 2011 3 / 12 Examples yt - wage (wt ) xt 1 - education (st ) xt 2 - experience (et ) issue 1: wt > 0 but dependent variable can take any value on R ln (wt ) = β0 + β1 st + β2 et + ut β1 = ∂ ln (w t ) ∂s t = ∂w t /w t ∂s t β1 = .02 - wages increase 2% for 1 more year of schooling (holding experience constant) issue 2: nonlinear function of experience ln (wt ) = β0 + β1 st + β2 et + β3 et2 + ut ∂ ln (w t ) ∂e t = β 2 + 2 β 3 et standard job: β2 > 0 D. Steigerwald (UCSB) β3 < 0 Population Models Winter 2011 4 / 12 Identi…cation yt = β0 + β1 xt 1 + = xt0 β + ut + βK xtK + ut goal - to learn (via estimation) β 0 = ( β0 , . . . , βK ) step 1 - identi…cation: is it possible to learn β from data of the type we observe? construct xt yt = xt xt0 β + xt ut population value E (xt yt ) = E (xt xt0 ) β + E (xt ut ) ) 0 )] 1 [E (x y ) β = [E (xt xt E (xt ut )] tt xt yt and xt xt0 observed: can learn E (xt yt ) from n 1 ∑ xt yt and E (xt xt0 ) from n 1 ∑ xt xt0 xt ut unobserved: cannot learn E (xt ut ) must assume E (xt ut ) = 0 for identi…cation must also assume E (xt xt0 ) invertible for identi…cation could use median in place of E D. Steigerwald (UCSB) Population Models Winter 2011 5 / 12 Assumption 2: Strict Exogeneity E [ut jX ] = 0 1 2 2 3 0 x1 6.7 X =4 . 5 . nK 0 xn ensures β is identi…ed model is the conditional mean of yt logic E [ut jx1 , . . . , xn ] = 0 ) E [ut jxt ] = 0 ) E [xt ut ] = 0 (LIE: detailed next) E [yt jX ] = xt0 β + E [ut jX ] = xt0 β D. Steigerwald (UCSB) Population Models Winter 2011 6 / 12 Law of Iterated Expectations E (U jZ ) = EW [E (U jZ , W )] Special case (Law of Total Expectations) E (U ) = EZ [E (U jZ )] 1 E [ut jxt ] = Ex1 ,...,xt 1 2 (E [ut jx1 , . . . , xn ]) E [ut jx1 , . . . , xn ] = 0 ) E [ut jxt ] = 0 E [xt ut ] = E (E [xt ut jxt ]) = E (xt E [ut jxt ]) 1 3 1 ,x t +1 ,...,x n E [ut jxt ] = 0 ) E [xt ut ] = 0 Cov (xt , ut ) = E [xt ut ] 1 2 E [xt ] E [ut ] E [ut jxt ] = 0 ) E [ut ] = 0 E [ut jxt ] = 0 ) Cov (xt , ut ) = 0 D. Steigerwald (UCSB) Population Models Winter 2011 7 / 12 Simple Proof of LIE let U be discrete and take M values let Z be discrete and take J values E (U ) = EZ [E (U jZ )] 1 2 3 4 5 6 E (U ) = ∑M=1 um P (U = um ) m = ∑M=1 um ∑J=1 P (U = um , Z = zj ) m j = ∑M=1 um ∑J=1 P (U = um jZ = zj ) P (Z = zj ) m j = ∑J=1 P (Z = zj ) ∑M=1 um P (U = um jZ = zj ) j m = ∑J=1 P (Z = zj ) E (U jZ = zj ) j = EZ E (U jZ = zj ) D. Steigerwald (UCSB) Population Models Winter 2011 8 / 12 Assumption 3: Linear Independence E (xt xt0 ) has rank K and X has rank K with probability 1 ensures β is identi…ed rank X < K ) linear dependence example: Cor xtj , xtk = 1 ∂y βj = ∂x t not identi…ed, (same for βk ) tj cannot vary xtj and hold xtk constant D. Steigerwald (UCSB) Population Models Winter 2011 9 / 12 Assumption 4: Conditionally Spherical Error conditionally homoskedastic 2 E ut jX = σ2 conditionally uncorrelated E (ut us jX ) = 0 notation 2 3 u1 6.7 u=4 . 5 . un Ω = E (u u 0 ) Assumption 4 implies Ω = σ2 In D. Steigerwald (UCSB) Population Models Winter 2011 10 / 12 Review yt is determined by h (xt ) what do we assume for h (xt )? h (xt ) = β0 + β1 xt 1 + + βK xtK what does βk capture? βk = ∂y t ∂xtk how does the error term arise? approximation error D. Steigerwald (UCSB) ut = yt xt0 β Population Models Winter 2011 11 / 12 what identi…es β? E (xt ut ) = 0 E (xt xt0 ) has rank K what further do we assume about the error? 2 E ut jX = σ2 E (ut us jX ) = 0 D. Steigerwald (UCSB) Population Models Winter 2011 12 / 12 ...
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