Single-Equation GMM: Estimation

Transcription

1 Sigle-Equatio GMM: Estimatio Lecture for Ecoomics 241B Douglas G. Steigerwald UC Sata Barbara Jauary 2012

2 Iitial Questio Iitial Questio How valuable is ivestmet i college educatio? ecoomics - measure value i terms of wage Stochastic Model W = β 0 + β 1 S + C 0 γ + U C 0 = A, A 2, M, R - vector of cotrols Key: S is edogeous

3 Stadard IV Momet Assumptio W t = β 0 + β 1 S t + C 0 t γ + U t S t edogeous C t exogeous z t istrumet for S t Classic OLS Momet Assumptio E Xt 0 U t = 0 with X 0 t = S t, Ct 0 issue - S t edogeous implies E (Xt 0U t ) 6= 0 Replacemet : Istrumet Momet Assumptio E Zt 0 U t = 0 with Z 0 t = z t, Ct 0 We do ot model how S t is related to z t

4 Stadard IV MoM Estimator Populatio Momet Assumptio E Zt 0 U t = 0 with Z 0 t = z t, Ct 0 Istrumetal Variable (IV) estimator solves Zt 0 (W t X t B IV ) = 0 ) B IV = Z 0 1 t X t Zt 0 W t t=1 Should we replace S t with z t i the stochastic model? No, it would lead to omitted variable bias W t = β 0 + β 1 z t + C 0 t γ + [U t + β 1 (S t z t )]

5 Stadard IV Two-Stage Least Squares Estimator B 2SLS ofte used i place of B IV First Stage S t = α 0 + α 1 z t + C 0 t δ + V t OLS yields Ŝ t = ˆα 0 + ˆα 1 z t + C 0 t ˆδ provides a atural test for the assumptio Cov (S t, z t ) 6= 0 key - must coditio o C t, because β 1 captures chages i schoolig give C t Secod Stage W t = β 0 + β 1 Ŝ t + C 0 t γ + U t + β 1 ˆV t OLS yields B 2SLS

6 Stadard IV Aalysis of Secod Stage Secod Stage W t = β 0 + β 1 Ŝ t + C 0 t γ + U t + β 1 ˆV t 1 edogeous regressor ad 1 istrumet ) B 2SLS idetical to B IV Why ca we replace S t with Ŝ t? By costructio ˆV t ucorrelated with Ŝ t, C 0 t (vital to iclude C t i rst stage) Is B IV cosistet ad ubiased? Cosistet but biased (ˆα 0, ˆα 1, ˆδ all deped o S t ) Exact ID - o ite mea! (Kial 1980)

7 Multiple Istrumets Over Ideti catio Istrumets z t1 z t2 Edogeous Covariate S t Should we use z t1 or z t2 as a istrumet for S t? Both! Method is 2SLS

8 Multiple Istrumets Two Stage Least Squares First Stage S t = α 0 + α 1 z t1 + α 2 z t2 + α 3 A t + α 4 A 2 t + α 5 M t + α 6 R t + V t yields liear combiatio of z 1 ad z 2 that maximizes correlatio with S Secod Stage (uchaged) W t = β 0 + β 1 Ŝ t + C 0 t γ + U t + β 1 ˆV t

9 Multiple Istrumets 2SLS as GMM Classic Momet Coditios Wage Model W t = X 0 t θ + U t X 0 t = (1, S t, C 0 t ) θ 0 = (β 0, β 1, γ 0 ) 6 momet coditios (C t has 4 elemets) E X t W t X 0 t θ = 0 E [W t Xt 0 θ] = 0 (1) E [S t (W t Xt 0 θ)] = 0 (2) E [C t (W t Xt 0 θ)] = 0 (3 6) 6 momet coditios, 6 coe ciets, exact ideti catio

10 Multiple Istrumets 2SLS as GMM Istrumet Momet Coditios Istrumet vector Z 0 t = (1, z t1, z t2, C 0 t ) 7 momet coditios E Z t W t X 0 t θ = 0 replace E [S t (W t Xt 0 θ)] = 0 with E [z t1 (W t Xt 0θ)] = 0 ad E [z t2 (W t Xt 0θ)] = 0 7 momet coditios, 6 coe ciets, over ideti catio

11 Multiple Istrumets 2SLS as GMM Geeralized Method of Momets (GMM) Populatio momet Sample aalog 1 E Z t W t X 0 t θ = 0 Z t W t X 0 t GMM = 0 (1) t=1 # momet coditios > # coe ciets ) caot select ˆθ GMM to satisfy (1) select ˆθ GMM to miimize sum of squared momet coditios

12 Multiple Istrumets 2SLS as GMM GMM : Optimizatio Problem ˆθ GMM satis es " arg mi ˆθ 1 Z t W t X 0 ˆθ! 0 t t=1 Ω 1 Z t W t X 0!# t ˆθ t=1 Ω - weight matrix for the momets e ciet GMM : Ω is the covariace matrix of the momet vector Miimum Distace Estimators satisfy arg mi g ˆθ 0 Ωg ˆθ ˆθ with g ˆθ P! 0 ˆθ GMM is a miimum distace estimator, where g sample mea : g ˆθ = 1 t=1 Z t W t Xt 0 ˆθ ˆθ is a

13 Multiple Istrumets 2SLS as GMM GMM : Solutio ˆθ GMM satis es the FOC ˆθ 1! 0 Z t Xt 0 Ω 1 t=1 1 Z t W t t=1! Z t Xt 0 ˆθ GMM = 0 t=1 solutio ˆθ GMM = X 0 Z ΩZ 0 X 1 X 0 Z ΩZ 0 W Z = (L) Z 0 1. Z X (K ) = X 0 1. X

14 Multiple Istrumets 2SLS as GMM Two Stage Least Squares Estimator X 0 = (S, C 0 ) Z 0 = (z, C 0 ) ˆX 0 = Ŝ, C 0 First Stage ˆX = Z Z 0 Z 1 Z 0 X P Z X PZ 0 = P Z (symmetric) P Z P Z = P Z (idempotet) Secod Stage B 2SLS = ˆX 0 ˆX 1 ˆX 0 W = X 0 P Z X 1 X 0 P Z W

15 Multiple Istrumets 2SLS as GMM 2SLS ad GMM Secod Stage B 2SLS = X 0 P Z X 1 X 0 P Z W = X 0 Z Z 0 Z 1 1 Z 0 X X 0 Z Z 0 Z 1 Z 0 W = X 0 Z σ 2 Z 0 Z 1 1 Z 0 X X 0 Z σ 2 Z 0 Z 1 Z 0 W Coditioal homoskedasticity assumptio Var (Z 0 UjZ ) = σ 2 Z 0 Z (scalar σ 2 t=1 z2 t ) B 2SLS is a GMM estimator with optimal weight Coditioal heteroskedasticity Var (Z 0 UjZ ) 6= σ 2 Z 0 Z (scalar t=1 σ 2 t z 2 t ) B 2SLS is a GMM estimator with o-optimal weight