Binary Models with Endogenous Explanatory Variables 1 ... - Cemfi

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B Consistent standard errors for two-step estimators Consider an estimator b θ that maximizes an objective function that depends on estimated parameters: b θ =argmax θ NX ì (θ, bγ) . i=1 The estimated parameters themselves are obtained by solving bγ =argmax γ NX ζi (γ) . i=1 Denote true values as (θ0, γ 0), and consider the following notation for score and Hessian terms: ` θ i (θ, γ) = ∂ì (θ, γ) , ζ ∂θ γ i (γ) =∂ζ i (γ) ∂γ Hθθ = − plim N→∞ 1 N NX i=1 ∂ 2 ì (θ0, γ 0) ∂θ∂θ 0 , Hθγ = − plim N→∞ Moreover, assume that the sample scores are asymptotically normal: NX Ã 1 `θ i (θ0, γ √ 0) N ζ γ i (γ ! " Ã d Υθθ → N 0, 0) Υγθ Υθγ Υγγ !# . i=1 1 N NX i=1 ∂ 2 ì (θ0, γ 0) ∂θ∂γ 0 , Hγγ = − plim N→∞ Under standard regularity conditions, an expansion of the first-order conditions for b θ gives 0= 1 √ N NX i=1 ` θ i ³ ´ bθ, bγ = 1 √ N A similar expansion of the first-order conditions for bγ gives 0= 1 √ N or equivalently NX i=1 ζ γ 1 i (bγ) = √ N √ N (bγ − γ0) =H −1 γγ 1 √ N Substituting (4) into (3) we get and √ ³ ´ Hθθ N bθ − θ0 = 1 √ N √ ³ ´ N bθ − θ0 = H −1 θθ NX i=1 NX i=1 NX i=1 NX i=1 1 N NX i=1 ∂ 2 ζ i (γ 0) ∂γ∂γ 0 . ` θ √ ³ ´ √ i (θ0, γ0) − Hθθ N bθ − θ0 − Hθγ N (bγ − γ0)+op (1) . (3) ζ γ i (γ √ 0) − Hγγ N (bγ − γ0)+op (1) ζ γ i (γ 0)+op (1) . (4) ` θ i (θ0, γ 0) − HθγH −1 γγ ¡ I,−HθγH −1¢ 1 γγ √N i=1 1 √ N NX i=1 ζ γ i (γ 0)+op (1) NX Ã `θ i (θ0, γ0) ζ γ i (γ ! + op (1) . 0) 12
Thus, √ ³ ´ N bθ d→ − θ0 N (0,V) where where V = H −1 θθ ¡ I,−HθγH −1¢ γγ Ã Υθθ Υθγ = H −1 £ ¡ −1 θθ Υθθ + Hθγ H A consistent estimator of V is: bV = bH −1 ³ θθ I,− bHθγ bHθθ = − 1 N bΥθθ = 1 N NX i=1 ` θ i NX i=1 ∂ 2 ì ´ bH −1 γγ Ã bΥθθ Υγθ Υγγ γγ ΥγγH −1 γγ bΥθγ bΥγθ bΥγγ !Ã I −H−1 γγ Hγθ ! H −1 θθ ¢ Hγθ − HθγH −1 γγ Υγθ − ΥθγH −1 ¤ −1 γγ Hγθ Hθθ . !Ã ³ ´ bθ, bγ ∂θ∂θ 0 , bHθγ = − 1 N ³ ´ bθ, bγ ` θ ³ ´ 0 bθ, i bγ , bΥθγ = 1 N I − bH −1 γγ bHγθ NX i=1 NX i=1 ∂ 2 ì ` θ i ! bH −1 θθ ³ ´ bθ, bγ ∂θ∂γ0 , Hγγ = − 1 N ³ ´ bθ, bγ ζ γ i (bγ)0 , NX i=1 bΥγγ = 1 N ∂2ζ i (bγ) . ∂γ∂γ0 NX i=1 ζ γ i (bγ) ζγ i (bγ)0 . Note that H −1 −1 θθ ΥθθHθθ is the asymptotic variance of the infeasible estimator that maximizes PN i=1 ì (θ, γ0), andthatH−1 γγ ΥγγH −1 γγ is the asymptotic variance of √ N (bγ − γ0). H −1 θθ If the information identities hold (H −1 θθ = Υθθ and H −1 γγ = Υγγ), given consistent estimates of and H−1 γγ , all we need to construct a consistent estimate of V are consistent estimates of the cross-terms Hθγ and Υθγ. 13
Page 1 and 2: 1 Introduction Binary Models with E
Page 3 and 4: In these circumstances it is natura
Page 5 and 6: 3 The control function approach 3.1
Page 7 and 8: 3.3.1 Constructing policy parameter
Page 9 and 10: Local average treatment effects (LA
Page 11: A 2SLS as a control function estima
Page 15: The proposed method is to form the

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