centeris paribus. w w partial effect E(y w, c)á w. abil. c =(exper, abil) exper

Transcription

1 c centeris paribus y c y E(y w, c) w y 1 c c c w w w partial effect E(y w, c)á w w E(y w, c) c w c c c E(wage educ, exp er, abil) educ exper abil c =(exper, abil) exper abil ( ) c y w w y ( ) 2.2 (population model) (independent, identically distributed = i.i.d) ( ) ( ) ( )

2 c random sampling ( ) ( ) pooled cross section independent, not identically distributed = i.n.i.d OLS TSLS (sample selection bias) ( ) (panel data) wage 0 log(wage 0 i )=β 0 + β 1 educ i + β 2 exp er i + β 3 married i + u i (1) educ exper married u u u u i j 3 Y = AL β 1 K β 2 (2) log Y =loga + β 1 log L + β 2 log K (3) 1 2 Wooldridge (2002),pp.8-9

3 c TFP = loga A = e (δ t+β 3 spillover+quality i ) log(y )=δ t + β 1 log(l it )+β 2 log(k it )+β 3 spillover it + quality i + u it t =1, 2, 3 (4) spillover quality δ t quality i quality i Y x (x 1,x 2,...x k ) 1 k µ(x) E(Y x 1,x 2,...x k )=µ(x 1,x 2,...x k ) E(Y x) =µ(x) (5) µ(x) Y x Y x IQ E(wage educ, exp er, IQ) educ, exp er, IQ E(Y x) x Y E(Y x) µ(x) E(Y x) y = f(x) µ(x) x j µ(x)á x j E(Y x) x j (x 1...x j 1,x j+1...x k E(Y x) µ(x) x j x j (6) E(Y x) x j x j E(Y x) (partial effect) x j x j x i6=j ) elasticityx j E(Y x) x 1...x j 1,x j+1...x k E(Y x) x j x j E(Y x) = µ(x) x j x j µ(x) (7) 3 Wooldridge (2002, Chapter2)

4 c E(Y x) > 0 x j > 0 log [E(Y x)] log(x j ) (8) x j E(Y x) x j E(Y x) 100 E(Y x) 1 log [E(Y x)] =1.00 (9) x j E(Y x) x j E(Y x) > 0 E(Y x) > x j semielasticity Y Y = E(Y x)+u (10) E(u x) = 0 (11) (10) Y E(Y x) u E(u x) =0 (1)E(u x) =0, u (2)u x 1,x 2,...x k x 1,x 2,...x k ( x 2 1,x 2 2,x 1 x 2, exp(x 1 ) w Y x x = f(w) law of iterated expectations = LIE E(Y x) =E [E(Y w) x] (12) µ 1 (w) E(Y w), µ 2 (x) E(Y x) µ 2 (x) x µ 2 (w) µ 2 (x) =E [(µ 1 (w) x] (12) E(Y x) =E [E(Y x) w] (13) x w w x µ 2 (x) =E(Y x) x w µ 2 (x) µ 2 (x) (12) (13) x w w x E(Y x 1,x 2,z)=β 0 + β 1 x 1 + β 2 x 2 + β 3 z (14) z E(Y x 1,x 2 )=E(β 0 + β 1 x 1 + β 2 x 2 + β 3 z x 1,x 2 ) = β 0 + β 1 x 1 + β 2 x 2 + β 3 E(z x 1,x 2 ) (15)

5 c E(z x 1,x 2 ) (16) (15) E(z x 1,x 2 )=δ 0 + δ 1 x 1 + δ 2 x 2 (16) E(Y x 1,x 2 )=β 0 + β 1 x 1 + β 2 x 2 + β 3 (δ 0 + δ 1 x 1 + δ 2 x 2 ) =(β 0 + β 3 δ 0 )+(β 1 + β 3 δ 1 )x 1 +(β 2 + β 3 δ 2 )x 2 (17) z x 1 x 2 z z x 1 x 1 z E(Y x 1,x 2,z)=β 0 + β 1 x 1 + β 2 x 2 + β 3 z + β 4 x 1 z (18) LIE) E(Y x 1,x 2 )=β 0 + β 1 x 1 + β 2 x 2 + β 3 E(z x 1,x 2 )+β 4 x 1 E(z x 1,x 2 ) (19) E(z x 1,x 2 ) (16) (19) E(Y x 1,x 2 )=β 0 + β 1 x 1 + β 2 x 2 + β 3 (δ 0 + δ 1 x 1 + δ 2 x 2 )+β 4 x 1 (δ 0 + δ 1 x 1 + δ 2 x 2 ) =(β 0 + β 3 δ 0 )+(β 1 + β 3 δ 1 + β 4 δ 0 )x 1 +(β 2 + β 3 δ 2 )x 2 + β 4 δ 1 x β 4 δ 2 x 1 x 2 (20) z x 1 x 2 q unobserved heterogeneity E(Y x,q)=µ 1 (x,q) θ j (x,q) E(Y x,q)á x j = µ 1 (x,q)á x j (21) θ j (x,q) q q θ j (x,q) θ j (x,q) q q average partial effect δ j (x 0 ) E q [θ j (x,q)] (22) E q [ ] q q x w D(q x, w) =D(q w) (23) D( ) (23) w q w E(Y ) E(Y x,q,w) =E(Y x,q) (24)

6 c (23)(24) (21) δ j (x 0 )=E w E(Y x 0, w)á x j (25) E w [ ] w q (25) E(Y x, w) (Y,x, w) x 0 ˆµ 2 (x 0, w) x j µ 2 (x, w) E(Y x, w) E w µ2 (x 0, w)á x j = E E θj (x 0,q) w ª = δ j (x 0 ) (26) OLS 4 Y = β 0 + β 1 x 1 + β 2 x β k x k + u (27) Y,x 1,x 2,...x k u β 0, β 1, β 2,...β k u (omitted variables) (measurement error) β j E(u) =0, Cov(x j,u)=0 j =1, 2,...,k (28) x j u 1 omitted variables E(Y x,q) q q x j x j (self-selection) 2 measurement error 4 Wooldridge (2002, Chapter 4)

7 c simultaneity x j Y Y x j x j Y 1 Y = β 0 + β 1 x 1 + β 2 x β k x k + γq + u (29) E(v x 1,x 2,...x k,q) = 0 (30) q v structural error 1-1 q (29) Y = β 0 + β 1 x 1 + β 2 x β k x k + u (31) u γq + v (32) q x i u x i β i OLS plims q x 1,x 2,...x k q = δ 0 + δ 1 x δ k x k + r (33) E(r) =0,cov(x j,r)=0,j=1, 2,...,k (33) (31)(32) Y =(β 0 + γδ 0 )+(β 1 + γδ 1 )x 1 +(β 2 + γδ 2 )x (β k + γδ k )x k + v + γr (34) u + γr plim ˆβ j = β j + γδ j q x k δ j (j 6= k) plim ˆβ j = β j j 6= k plim ˆβ k = β k + γ [cov(x k,q)ávar(x k )] (35) δ k = cov(x k,q) (sign γ > 0 x k q x k q x k x k

8 c q (proxy variable) (redundant ignorable) z q z E(Y x,q,z)=e(y x,q) (36) Y z q x j z (partial out) L(q 1,x 1,...,x k,z)=l(q 1,z) (37) q r q = θ 0 + θ 1 z + r (38) E(r) =0,cov(z,r) =0 z q θ 1 6=0 (37) cov(x j,r)=0 j =1, 2,...,k (39) z q z x j q (29) (38) Y =(β 0 + γθ 0 )+β 1 x 1 + β 2 x β k x k + γθ 1 z +(γr + v) (40) u γr + v x j OLS Y = β 0 + β 1 x β k x k + v (41) Y Y = y e 0 (42) (42) (41) y = β 0 + β 1 x β k x k + v + e 0 (43) x 1,x 2,...x k Y OLS e 0 x j OLS e 0 v var(v + e 0 )=σ 2 v + σ 2 0 > σ 2 v OLS

9 c Y = β 0 + β 1 x 1 + β 2 x β k x k + v (44) x 1,x 2,...x k 1 x k v x k e k = x k x k, E(e k = 0) (45) v x k x k e k 1) Cov(x k,e k ) = 0 (46) (45) e k x k x k = x k x k (44) Y = β 0 + β 1 x 1 + β 2 x β k x k +(v β k e k ) (47) v e k x j v β k e k x j x k x k OLS β j 2 Cov(x k,e k ) = 0 (48) classical errors-in-variables=cev x k x k = x k + e k x k e k (48) x k e k Cov(x k,e k )=E(x k e k )+E(e 2 k)=σ 2 ek (49) x k e k x k e k (47) OLS v x k x k (v β k e k ) Cov(x k,v β k e k )= β k Cov(x k,e k )= β k σ 2 ek OLS References [1] Wooldridge, J.M. (2002) Econometric Analysis of Cross Section and Panel Data, Cambridge, MA: The MIT Press.