Endogoneity and All That

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1 Edogoeity ad All That Roger Koeker Uiversity of Illiois, Urbaa-Champaig Uiversity of Miho Jue 2017 Roger Koeker (UIUC) Edogoeity Braga / 17

2 Is there IV for QR? Amemiya (1982) ad Powell (1983) cosider aalogues of 2SLS for media regressio models Che ad Portoy (1986) ad Kim ad Muller (2004) cosider extesios to quatile regressio Abadie, Agrist ad Imbes (2002) cosider models with biary edogoous treatmet Cherozhukov ad Hase (2003) propose iverse quatile regressio Chesher (2003) cosiders triagular models with cotiuous edogoous variables. Recet surveys by Cherozhukov, Hase ad Wüthrich ad Melly ad Wüthrich i the forthcomig Hadbook of Quatile Regressio. Roger Koeker (UIUC) Edogoeity Braga / 17

3 Abadie-Agrist-Imbes QTE Model: Q Y (τ D, X, D 1 > D 0 ) = Dα(τ) + Xβ(τ) where D i deotes (potetial) treatmet status uder the (radomized) biary itet-to-treat, Z. Let π(x) = P(Z = 1 X), ad defie κ(d, Z, X) = 1 D(1 Z) 1 π(x) (1 D)Z π(x) which is 1 if D = Z for compliers ad less tha zero otherwise. We would like to estimate the model by solvig the empirical couterpart of, mi Eκρ τ (Y Dα Xβ) but egative weights are problematic so,settig ν = E(Z Y, D, X) it is proposed to replace κ by a estimate of, κ ν = E[κ U = (Y, D, X)] = 1 D(1 ν(u)) 1 π(x) (1 D)ν(U) π(x) Roger Koeker (UIUC) Edogoeity Braga / 17

4 Cherozhukov ad Hase QRIV Motivatio: Yet aother way to view two stage least squares. Model: y = Xβ + Zα + u, W u Estimator: ˆα = argmi α ˆγ(α) 2 A=W M X W ˆγ(α) = argmi γ y Xβ Zα Wγ 2 Thm ˆα = (Z P MX WZ) 1 Z P MX Wy, the 2SLS estimator. Heuristic: ˆα is chose to make ˆγ(α) as small as possible to satisfy (approximately) the exclusio restrictio/assumptio. Geeralizatio: The quatile regressio versio simply replaces 2 i the defiitio of ˆγ by the correspodig QR orm. Roger Koeker (UIUC) Edogoeity Braga / 17

5 A Liear Locatio Shift Recursive Model Y = Sα 1 + x α 2 + ɛ + λν (1) S = zβ 1 + x β 2 + ν (2) Suppose: ɛ ν ad (ɛ, ν) (z, x). Substitutig for ν from (2) ito (1), Q Y (τ 1 S, x, z) = S(α 1 + λ) + x (α 2 λβ 2 ) + z( λβ 1 ) + F 1 ɛ (τ 1 ) Q S (τ 2 z, x) = zβ 1 + x β 2 + F 1 ν (τ 2 ) π 1 (τ 1, τ 2 ) = Si Q Yi Si =Q Si + z i Q Yi Si =Q Si zi Q Si = (α 1 + λ) + ( λβ 1 )/β 1 = α 1 Roger Koeker (UIUC) Edogoeity Braga / 17

6 A Liear Locatio-Scale Shift Model Y = Sα 1 + x α 2 + S(ɛ + λν) S = zβ 1 + x β 2 + ν π 1 (τ 1, τ 2 ) = α 1 + F 1 ɛ (τ 1 ) + λf 1 ν (τ 2 ) ˆπ 1 (τ 1, τ 2 ) = Q Y (τ 1 S, x, z) = Sθ 1 (τ 1 ) + x θ 2 + S 2 θ 3 + Szθ 4 + Sx θ 5 Q S (τ 2 z, x) = zβ 1 + x β 2 + F 1 ν (τ 2 ) w i {ˆθ 1 (τ 1 )+2 ˆQ Si ˆθ 3 (τ 1 )+z iˆθ 4 (τ 1 )+x ˆθ i 5 (τ 1 )+ ˆQ Si ˆθ 4 (τ 1 ) } ˆβ 1 (τ 2 ) a weighted average derivative estimator with ˆQ Si = ˆQ S (τ 2 z i, x i ). Roger Koeker (UIUC) Edogoeity Braga / 17

7 The Geeral Recursive Model Strotz & Wold (1960) Y = ϕ 1 (S, x, ɛ, ν; α) S = ϕ 2 (z, x, ν; β) Suppose: ɛ ν ad (ɛ, ν) (z, x). Solvig for ν ad substitutig we have the coditioal quatile fuctios, Q Y (τ 1 S, x, z) = h 1 (S, x, z, θ(τ 1 )) Q S (τ 2 z, x) = h 2 (z, x, β(τ 2 )) Extesios to more tha two edogoous variables are straightforward. Roger Koeker (UIUC) Edogoeity Braga / 17

8 The (Chesher) Weighted Average Derivative Estimator ˆθ(τ 1 ) = argmi θ ρ τ1 (Y i h 1 (S, x, z, θ(τ 1 ))) ˆβ(τ 2 ) = argmi β ρ τ2 (S i h 2 (z, x, β(τ 2 ))) where ρ τ (u) = u(τ I(u < 0)), givig structural estimators: ˆπ 1 (τ 1, τ 2 ) = ˆπ 2 (τ 1, τ 2 ) = w i { S ĥ 1i Si =ĥ 2i + zĥ1i } Si =ĥ 2i, z ĥ 2i w i { x ĥ 1i Si =ĥ 2i zĥ1i Si =ĥ 2i z ĥ 2i x ĥ 2i }, Roger Koeker (UIUC) Edogoeity Braga / 17

9 2SLS as a Cotrol Variate Estimator Y = Sα 1 + X 1 α 2 + u Zα + u S = Xβ + V, where X = [X 1.X 2 ] Set ˆV = S Ŝ M X Y 1, ad cosider the least squares estimator of the model, Y = Zα + ˆVγ + w Claim: ˆα CV (Z M ˆV Z) 1 Z M ˆV Y = (Z P X Z) 1 Z P X Y ˆα 2SLS. Roger Koeker (UIUC) Edogoeity Braga / 17

10 Proof of Cotrol Variate Equivalece M ˆV = M M X S = I M X S(S M X S) 1 S M X S M ˆV = S S M X = S P X X 1 M ˆV = X 1 X 1 M X = X 1 = X 1 P X Reward for iformatio leadig to a referece prior to Dhrymes (1970). Recet work o the cotrol variate approach by Bludell, Powell, Smith, Newey ad others. Roger Koeker (UIUC) Edogoeity Braga / 17

11 Quatile Regressio Cotrol Variate Estimatio I Locatio scale shift model: Y = S(α 1 + ɛ + λν) + x α 2 S = zβ 1 + x β 2 + ν. Usig ˆν(τ 2 ) = S ˆQ S (τ 2 z, x) as a cotrol variate, Y = w α(τ 1, τ 2 ) + λs( ˆQ S Q S ) + S(ɛ F 1 ɛ (τ 1 )), where w = (S, x, Sˆν(τ 2 )) α(τ 1, τ 2 ) = (α 1 (τ 1, τ 2 ), α 2, λ) α 1 (τ 1, τ 2 ) = α 1 + F 1 ɛ (τ 1 ) + λf 1 ν (τ 2 ). ˆα(τ 1, τ 2 ) = argmi a ρ τ1 (Y i w i a). Roger Koeker (UIUC) Edogoeity Braga / 17

12 Quatile Regressio Cotrol Variate Estimatio II Y = ϕ 1 (S, x, ɛ, ν; α) S = ϕ 2 (z, x, ν; β) Regardig ν(τ 2 ) = ν F 1 ν (τ 2 ) as a cotrol variate, we have Q Y (τ 1 S, x, ν(τ 2 )) = g 1 (S, x, ν(τ 2 ), α(τ 1, τ 2 )) Q S (τ 2 z, x) = g 2 (z, x, β(τ 2 )) ˆν(τ 2 ) = ϕ 1 2 (S, z, x, ˆβ) ϕ 1 2 ( ˆQ s, z, x, ˆβ) ˆα(τ 1, τ 2 ) = argmi a ρ τ1 (Y i g 1 (S, x, ˆν(τ 2 ), a)). Roger Koeker (UIUC) Edogoeity Braga / 17

13 Asymptopia Theorem: Uder regularity coditios, the weighted average derivative ad cotrol variate estimators of the Chesher structural effect have a asymptotic liear (Bahadur) represetatio, ad after efficiet reweightig of both estimators, the cotrol variate estimator has smaller covariace matrix tha the weighted average derivative estimator. Remark: The cotrol variate estimator imposes more striget restrictios o the estimatio of the hybrid structural equatio ad should thus be expected to perform better whe the specificatio is correct. The advatages of the cotrol variate approach are magified i situatios of overidetificatio. Roger Koeker (UIUC) Edogoeity Braga / 17

14 Asymptotics for WAD Theorem The ˆπ (τ 1, τ 2 ) has the asymptotic liear (Bahadur) represetatio, (ˆπ (τ 1, τ 2 ) π(τ 1, τ 2 )) = W 1 J W 2 J σ i1 ḣ i1 ψ τ1 (Y i1 ξ i1 ) 1 σ i2 ḣ i2 ψ τ2 (Y i2 ξ i2 ) = N(0, ω 11 W 1 J 1 1 J 1 J 1 1 W 1 + ω 22 W 2 J 1 2 J 2 J 1 2 W 2 ) 1 J j = lim σ 2 ij ḣ ij ḣ 1 ij, J j = lim σij f ij (ξ ij )ḣ ij ḣ ij, W 1 = θ π(τ 1, τ 2 ), W 2 = β π(τ 1, τ 2 ), ḣ i1 = θ h i1, ḣ i2 = β h i2, ω jj = τ j (1 τ j ). Roger Koeker (UIUC) Edogoeity Braga / 17

15 Asymptotics for CV Theorem The ˆα (τ 1, τ 2 ) has the Bahadur represetatio, (ˆα (τ 1, τ 2 ) α(τ 1, τ 2 )) = D D 1 1 = N(0, ω 11 D 1 1 D 1 1 D D 12 D ω 22 D σ i1 ġ i1 ψ τ1 (Y i1 ξ i1 ) 1 σ i2 ġ i2 ψ τ2 (Y i2 ξ i2 ) D 12 D 1 2 D 2 1 D 2 D 1 12 D 1 ) D j = lim 1 σ 2 ijġijġ ij, D j = lim 1 σ ij f ij (ξ ij )ġ ij ġ ij, D 12 = lim 1 σ i1 f i1 η i ġ i1 ġ i2, ġ i1 = α g i1, ġ i2 = β g i2, η i = ( g 1i / ν i2 (τ 2 ))( νi2 ϕ i2 ) 1. Roger Koeker (UIUC) Edogoeity Braga / 17

16 ARE of WAD ad CV Efficiet weights: σ ij = f ij (ξ ij ) (ˆπ (τ 1, τ 2 ) π(τ 1, τ 2 )) N(0, ω 11 W 1 J 1 1 W 1 + ω 22 W 2 J 1 2 W 2 ) (ˆα (τ 1, τ 2 ) α(τ 1, τ 2 )) N(0, ω 11 D ω 22 D 1 1 D 12D 1 2 D 12D 1 1 ). The mappig: π = Lˆα, Lα = π. W 1 J 1 1 W 1 LD 1 1 L W 2 J 1 2 W 2 LD 1 1 D 12D 1 2 D 12D 1 1 L. Theorem Uder efficiet reweightig of both estimators, Avar( π ) Avar( ˆπ ). Roger Koeker (UIUC) Edogoeity Braga / 17

17 Coclusios Triagular structural models facilitate causal aalysis via recursive coditioig, directed acyclic graph represetatio. Recursive coditioal quatile models yield iterpretable heterogeeous structural effects. Cotrol variate methods offer computatioally ad statistically efficiet strategies for estimatig heterogeeous structural effects. Weighted average derivative methods offer a less restrictive strategy for estimatio that offers potetial for model diagostics ad testig. Roger Koeker (UIUC) Edogoeity Braga / 17

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