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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 Fall '08
 Staff
 Econometrics, Regression Analysis, Xt, D. Steigerwald, (UCSB)

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