Identifying Multiple Marginal Effects with a Single Instrument

Transcription

1 Identifying Multiple Marginal Effects with a Single Instrument Carolina Caetano 1 Juan Carlos Escanciano 2 1 Department of Economics, University of Rochester 2 Department of Economics, Indiana University April 10, 2017 Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

2 Outline 1 Introduction 2 Intuition 3 Identification 4 Estimation 5 Application 6 Conclusion Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

3 Introduction Y = g(x)+u, E[U X] 0 Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

4 Introduction Y = g(x)+u, E[U X] 0 To identify g, an IV Z must satisfy 1 (validity) E[U Z] = 0 2 (relevance) The dependence between X and Z is strong enough. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

5 Introduction Y = g(x)+u, E[U X] 0 To identify g, an IV Z must satisfy 1 (validity) E[U Z] = 0 2 (relevance) The dependence between X and Z is strong enough. Relevance condition = Z must be at least as complex as X, so If X is continuous, then Z should be continuous If X is discrete with k points of support, then Z needs to have at least k points of support If X is a vector, then Z must have at least as many components as X Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

6 Introduction Y = g(x)+u, E[U X] 0 To identify g, an IV Z must satisfy 1 (validity) E[U Z] = 0 2 (relevance) The dependence between X and Z is strong enough. Relevance condition = Z must be at least as complex as X, so If X is continuous, then Z should be continuous If X is discrete with k points of support, then Z needs to have at least k points of support If X is a vector, then Z must have at least as many components as X Good continuous IVs are hard to find Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

7 Introduction Y = g(x)+u We propose a new method to identify g (up to a constant) with a binary IV. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

8 Introduction Y = g(x)+u We propose a new method to identify g (up to a constant) with a binary IV. We will use a covariate W to classify observations into groups. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

9 Introduction Y = g(x)+u We propose a new method to identify g (up to a constant) with a binary IV. We will use a covariate W to classify observations into groups. Method Conditions (validity) E[U Z, W] = E[U W] (W may be endogenous) Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

10 Introduction Y = g(x)+u We propose a new method to identify g (up to a constant) with a binary IV. We will use a covariate W to classify observations into groups. Method Conditions (validity) E[U Z, W] = E[U W] (W may be endogenous) (relevance) F(X Z = 1, W) F(X Z = 0, W) varies sufficiently with W Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

11 Introduction Y = g(x)+u We propose a new method to identify g (up to a constant) with a binary IV. We will use a covariate W to classify observations into groups. Method Conditions (validity) E[U Z, W] = E[U W] (W may be endogenous) (relevance) F(X Z = 1, W) F(X Z = 0, W) varies sufficiently with W X can be multivalued, continuous, or even a vector Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

12 Introduction Y = g(x)+u We propose a new method to identify g (up to a constant) with a binary IV. We will use a covariate W to classify observations into groups. Method Conditions (validity) E[U Z, W] = E[U W] (W may be endogenous) (relevance) F(X Z = 1, W) F(X Z = 0, W) varies sufficiently with W X can be multivalued, continuous, or even a vector Z can be binary Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

13 Introduction Y = g(x)+u We propose a new method to identify g (up to a constant) with a binary IV. We will use a covariate W to classify observations into groups. Method Conditions (validity) E[U Z, W] = E[U W] (W may be endogenous) (relevance) F(X Z = 1, W) F(X Z = 0, W) varies sufficiently with W X can be multivalued, continuous, or even a vector Z can be binary Requirements on the complexity of a variable are made only on W Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

14 Introduction Y = g(x)+u We propose a new method to identify g (up to a constant) with a binary IV. We will use a covariate W to classify observations into groups. Method Conditions (validity) E[U Z, W] = E[U W] (W may be endogenous) (relevance) F(X Z = 1, W) F(X Z = 0, W) varies sufficiently with W X can be multivalued, continuous, or even a vector Z can be binary Requirements on the complexity of a variable are made only on W Can be naturally extended to the Regression Discontinuity Design (RDD) framework Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

15 Introduction Y = g(x)+u We propose a new method to identify g (up to a constant) with a binary IV. We will use a covariate W to classify observations into groups. Method Conditions (validity) E[U Z, W] = E[U W] (W may be endogenous) (relevance) F(X Z = 1, W) F(X Z = 0, W) varies sufficiently with W X can be multivalued, continuous, or even a vector Z can be binary Requirements on the complexity of a variable are made only on W Can be naturally extended to the Regression Discontinuity Design (RDD) framework Can be extended to some non-separable models, but not all. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

16 Covariance Completeness The main contribution of this paper is the use of covariance restrictions such as C(Y, Z W) = C(g(X), Z W) (1) for nonparametric identification under endogeneity. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

17 Covariance Completeness The main contribution of this paper is the use of covariance restrictions such as C(Y, Z W) = C(g(X), Z W) (1) for nonparametric identification under endogeneity. This paper also introduces an identifying assumption, called covariance completeness, which provides conditions under which the right hand side of (1) can be uniquely solved (up to a constant) in g. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

18 Covariance Completeness The main contribution of this paper is the use of covariance restrictions such as C(Y, Z W) = C(g(X), Z W) (1) for nonparametric identification under endogeneity. This paper also introduces an identifying assumption, called covariance completeness, which provides conditions under which the right hand side of (1) can be uniquely solved (up to a constant) in g. This method explores the variation in the relationship between X and Z for different values of W. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

19 Related Literature Identification and inference results in nonparametric models: Newey and Powell (2003), Hall and Horowitz (2005), Blundell, Chen and Kristensen (2007), Darolles, Fan, Florens and Renault (2011), Horowitz (2011) and Chen and Pouzo (2012) to mention just a few. Alternative approaches: Chesher (2003), D Haulfoeuille and Fevrier (2012), Torgovitsky (2012), D Haultfoeuille, Hoderlein and Sasaki (2013), Masten and Torgovitsky (2014) and Huang, Khalil and Yildiz (2015). For continuous instruments, Altonji and Matzkin (2005), Chernozukov and Hansen (2005) and Florens, Heckman, Meghir and Vytlacil (2008). None of these papers exploit the heterogeneity of the first stage conditional on covariates. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

20 Outline 1 Introduction 2 Intuition 3 Identification 4 Estimation 5 Application 6 Conclusion Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

21 Effect of maternal smoking on birth weight Y = g(x)+u Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

22 Effect of maternal smoking on birth weight birth weight in grams Y = g(x)+u cigarettes per day income, education, race, marital status, alcohol consumption, prenatal visits, etc. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

23 Effect of maternal smoking on birth weight E[U X] 0 Y = g(x)+u Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

24 Effect of maternal smoking on birth weight Y = g(x)+u E[U X] 0 Information from a binary IV, Z such that E[U Z]=0: Table : Z = 0: control group, Z = 1: treatment group X Z=0 X Z=1 ( X) Ȳ Z=0 Ȳ Z=1 (Ȳ) ,000 3, Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

25 Effect of maternal smoking on birth weight Y = g(x)+u E[U X] 0 Information from a binary IV, Z such that E[U Z]=0: Table : Z = 0: control group, Z = 1: treatment group X Z=0 X Z=1 ( X) Ȳ Z=0 Ȳ Z=1 (Ȳ) ,000 3, Can learn the average effect of the intervention: E[g(X) Z = 1] E[g(X) Z = 0]=200g Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

26 Effect of maternal smoking on birth weight Y = g(x)+u E[U X] 0 Information from a binary IV, Z such that E[U Z]=0: Table : Z = 0: control group, Z = 1: treatment group X Z=0 X Z=1 ( X) Ȳ Z=0 Ȳ Z=1 (Ȳ) ,000 3, Can learn the average effect of the intervention: E[g(X) Z = 1] E[g(X) Z = 0]=200g We would like to know the effect of an arbitrary reduction in smoking: g(x i ) g(x j ), for any pair x i x j Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

27 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 X {0,1,3} W {6,10,17} Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

28 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 X {0,1,3} W {6,10,17} F(X Z = 1,W) F(X Z = 0,W) varies with W Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

29 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 Y (W)=E[Y Z = 1,W] E[Y Z = 0,W] Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

30 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 Y (W)=E[Y Z = 1,W] E[Y Z = 0,W] Assumption 1 (validity) E[U Z,W]=E[U W] Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

31 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 Y (W)=E[Y Z = 1,W] E[Y Z = 0,W]=E[g(X) Z= 1,W] E[g(X) Z= 0,W] Assumption 1 (validity) E[U Z,W]=E[U W] Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

32 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 Y (W)=E[Y Z = 1,W] E[Y Z = 0,W]=E[g(X) Z= 1,W] E[g(X) Z= 0,W] 10=0 g(0)+0.5 g(1)+0.5 g(3) [0 g(0)+0 g(1)+1 g(3)] Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

33 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 Y (W)=E[Y Z = 1,W] E[Y Z = 0,W]=E[g(X) Z= 1,W] E[g(X) Z= 0,W] 10=0.5g(1) 0.5g(3) Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

34 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 Y (W)=E[Y Z = 1,W] E[Y Z = 0,W]=E[g(X) Z= 1,W] E[g(X) Z= 0,W] 10=0.5g(1) 0.5g(3) 22=0.2g(0)+0.2g(1) 0.4g(3) 30=0.33g(0) 0.33g(3) Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

35 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 Y (W)=E[Y Z = 1,W] E[Y Z = 0,W]=E[g(X) Z= 1,W] E[g(X) Z= 0,W] 10=0.5g(1) 0.5g(3) 22=0.2g(0)+0.2g(1) 0.4g(3) 30=0.33g(0) 0.33g(3) Only 2 equations are linearly independent. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

36 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 Y (W)=E[Y Z = 1,W] E[Y Z = 0,W]=E[g(X) Z= 1,W] E[g(X) Z= 0,W] 10=0.5g(1) 0.5g(3) 22=0.2g(0)+0.2g(1) 0.4g(3) 30=0.33g(0) 0.33g(3) Only 2 equations are linearly independent. We can identify the marginal effects: g(3) g(1)= 20, g(3) g(0)= 90 and g(1) g(0)= 70. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

37 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 If X {x 1,...,x p }, then W must assume at least p 1 values. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

38 Artificial Counterfactuals Table : Identification Idea W X 0,W X 1,W Y (W) P 0,W (0) P 0,W (1) P 0,W (3) P 1,W (0) P 1,W (1) P 1,W (3) /2 1/ /5 1/5 3/ /3 0 2/3 If X {x 1,...,x p }, then W must assume at least p 1 values. There must exist a set {w 1,...,w p 1 } such that the (P 1,wl (x 1 ) P 0,wl (x 1 ),...,P 1,wl (x p ) P 0,wl (x p )), l=1,...,p 1 are linearly independent. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

39 Outline 1 Introduction 2 Intuition 3 Identification 4 Estimation 5 Application 6 Conclusion Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

40 Identification Y = g(x)+u Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

41 Identification Y = g(x)+u Assumption 1 (validity) E[U Z,W]=E[U W] Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

42 Identification Y = g(x)+u Assumption 1 (validity) E[U Z,W]=E[U W] = E[Y Z = 1,W]=E[g(X) Z = 1,W]+E[U W] E[Y Z = 0,W]=E[g(X) Z = 0,W]+E[U W] Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

43 Identification Y = g(x)+u Assumption 1 (validity) E[U Z,W]=E[U W] E[Y Z = 1,W]=E[g(X) Z = 1,W]+E[U W] = E[Y Z = 0,W]=E[g(X) Z = 0,W]+E[U W] E[Y Z = 1,W] E[Y Z = 0,W]= g(x)d[f(x Z = 1,W) F(X Z = 0,W)] Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

44 Identification Y = g(x)+u Assumption 1 (validity) E[U Z,W]=E[U W] E[Y Z = 1,W]=E[g(X) Z = 1,W]+E[U W] = E[Y Z = 0,W]=E[g(X) Z = 0,W]+E[U W] E[Y Z = 1,W] E[Y Z = 0,W]= g(x)d[f(x Z = 1,W) F(X Z = 0,W)] A : G L 2 (X) L 2 (W), Ag=E[g(X) Z = 1,W] E[g(X) Z = 0,W] Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

45 Identification Y = g(x)+u Assumption 1 (validity) E[U Z,W]=E[U W] = E[Y Z = 1,W]=E[g(X) Z = 1,W]+E[U W] E[Y Z = 0,W]=E[g(X) Z = 0,W]+E[U W] E[Y Z = 1,W] E[Y Z = 0,W]= Ag A : G L 2 (X) L 2 (W), Ag=E[g(X) Z = 1,W] E[g(X) Z = 0,W] Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

46 Identification Y = g(x)+u Assumption 1 (validity) E[U Z,W]=E[U W] = E[Y Z = 1,W]=E[g(X) Z = 1,W]+E[U W] E[Y Z = 0,W]=E[g(X) Z = 0,W]+E[U W] E[Y Z = 1,W] E[Y Z = 0,W]= Ag A : G L 2 (X) L 2 (W), Ag=E[g(X) Z = 1,W] E[g(X) Z = 0,W] Assumption 2 (relevance) N (A)={c : c R} Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

47 Identification Y = g(x)+u Assumption 1 (validity) E[U Z,W]=E[U W] = E[Y Z = 1,W]=E[g(X) Z = 1,W]+E[U W] E[Y Z = 0,W]=E[g(X) Z = 0,W]+E[U W] E[Y Z = 1,W] E[Y Z = 0,W]= Ag A : G L 2 (X) L 2 (W), Ag=E[g(X) Z = 1,W] E[g(X) Z = 0,W] Assumption 2 (relevance) N (A) ={c : c R} = meaning depends on context Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

48 Identification Result E[Y Z = 1,W] E[Y Z = 0,W]= Necessary conditions for Assumption 2 are that g(x)d[f(x Z = 1,W) F(X Z = 0,W)] 1 X and W have the same level complexity (i.e. both are continuous). If there are functional form restrictions on g this can be relaxed (i.e. X vector, g linear, then W may be univariate.) 2 Elements of X are not functionally dependent on W (e.g. they do not have elements in common). Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

49 Identification Result E[Y Z = 1,W] E[Y Z = 0,W]= Necessary conditions for Assumption 2 are that g(x)d[f(x Z = 1,W) F(X Z = 0,W)] 1 X and W have the same level complexity (i.e. both are continuous). If there are functional form restrictions on g this can be relaxed (i.e. X vector, g linear, then W may be univariate.) 2 Elements of X are not functionally dependent on W (e.g. they do not have elements in common). In general, what is needed for Assumption 2 to hold is that F(X Z = 1,W) F(X Z = 0,W) varies sufficiently with W. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

50 Identification Result E[Y Z = 1,W] E[Y Z = 0,W]= Necessary conditions for Assumption 2 are that g(x)d[f(x Z = 1,W) F(X Z = 0,W)] 1 X and W have the same level complexity (i.e. both are continuous). If there are functional form restrictions on g this can be relaxed (i.e. X vector, g linear, then W may be univariate.) 2 Elements of X are not functionally dependent on W (e.g. they do not have elements in common). In general, what is needed for Assumption 2 to hold is that F(X Z = 1,W) F(X Z = 0,W) varies sufficiently with W. Theorem If Assumptions 1 and 2 hold, then g is identified up to a constant. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

51 Example: Gaussian Variables Suppose that (( 0 (X,W) Z N 0 ) ( 1 ρz, ρ Z 1 )). Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

52 Example: Gaussian Variables Suppose that (( 0 (X,W) Z N 0 ) ( 1 ρz, ρ Z 1 )). It can be shown that Ag(z)=(2π) 3/4 exp ( ) z2 µ 2 j=0{ j (ρ 1 ) µ j (ρ 0 ) } E[g(X)p j (X)] zj. j! where p j are the Hermite functions, and µ(ρ)=ρ/ 1 ρ 2. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

53 Example: Gaussian Variables Suppose that (( 0 (X,W) Z N 0 ) ( 1 ρz, ρ Z 1 )). It can be shown that Ag(z)=(2π) 3/4 exp ( ) z2 µ 2 j=0{ j (ρ 1 ) µ j (ρ 0 ) } E[g(X)p j (X)] zj. j! where p j are the Hermite functions, and µ(ρ)=ρ/ 1 ρ 2. ρ 1 ρ 0 implies the relevance condition. ( 1 f X Z=z,W=w (x)= 2π(1 ρ 2 z ) exp (x ρ z z) 2 ) 2(1 ρz) 2. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

54 Example: linear model with 2 endogenous variables Y = β 0 + β 1 X 1 + β 2 X 2 + U, Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

55 Example: linear model with 2 endogenous variables Y = β 0 + β 1 X 1 + β 2 X 2 + U, Standard IV methods cannot identify β 1 and β 2 with one binary Z Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

56 Example: linear model with 2 endogenous variables Y = β 0 + β 1 X 1 + β 2 X 2 + U, Standard IV methods cannot identify β 1 and β 2 with one binary Z Our approach: E[U Z, W] = E[U W] = E[Y Z = 1,W] E[Y Z = 0,W]=β 1 (E[X 1 Z = 1,W] E[X 1 Z = 0,W]) + β 2 (E[X 2 Z = 1,W] E[X 2 Z = 0,W]) Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

57 Example: linear model with 2 endogenous variables Y = β 0 + β 1 X 1 + β 2 X 2 + U, Standard IV methods cannot identify β 1 and β 2 with one binary Z Our approach: E[U Z, W] = E[U W] = E[Y Z = 1,W] E[Y Z = 0,W]=β 1 (E[X 1 Z = 1,W] E[X 1 Z = 0,W]) + β 2 (E[X 2 Z = 1,W] E[X 2 Z = 0,W]) (β 1,β 2 ) is identified as long as the first stage vectors vary with W in a linearly independent manner. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

58 Covariance Completeness Y = g(x)+u, E[U Z,W]=E[U W] In general our identification relies on solving C(Y,Z W)=C(g(X),Z W) Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

59 Covariance Completeness Y = g(x)+u, E[U Z,W]=E[U W] In general our identification relies on solving Definition C(Y,Z W)=C(g(X),Z W) We say (X,Z) given W is G -covariance complete if for each g G C(g(X),Z W)=0 a.s.= g=0 a.s. When G is unrestricted (except for the location normalization) we simply say (X,Z) given W is L 2 -covariance complete or simply covariance complete. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

60 Covariance Completeness Results Theorem (identification) If The variable Y has bounded second moment. The function p(w)=e[z W = w] satisfies 0<p<1 a.s. Then, g is point-identified in G if and only if (X,Z) W is G -covariance complete. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

61 Covariance Completeness Results Theorem (identification) If The variable Y has bounded second moment. The function p(w)=e[z W = w] satisfies 0<p<1 a.s. Then, g is point-identified in G if and only if (X,Z) W is G -covariance complete. Theorem (relationship with completeness) Define the class of measurable functions F ={f(x,w)=g(x)k(x,w) : g G} then (X,Z) W is G -covariance complete if and only if it is F complete. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

62 Nonseparable Model Let the model be Y = m(x,u), where m is strictly increasing in the scalar U, and Z U W. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

63 Nonseparable Model Let the model be Y = m(x,u), where m is strictly increasing in the scalar U, and Z U W. Then C ( m 1 (Y,X),Z W ) = 0. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

64 Nonseparable Model Let the model be Y = m(x,u), where m is strictly increasing in the scalar U, and Z U W. Then C ( m 1 (Y,X),Z W ) = 0. Let then g(y,x) := Y m 1 (Y,X), C(Y,Z W)=C(g(Y,X),Z W). Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

65 Outline 1 Introduction 2 Intuition 3 Identification 4 Estimation 5 Application 6 Conclusion Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

66 Estimation (linear model) Suppose that Y = α+ X β + U then E[Y Z = 1,W] E[Y Z = 0,W]=(E[X Z = 1,W] E[X Z = 0,W]) β Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

67 Estimation (linear model) Suppose that Y = α+ X β + U then E[Y Z = 1,W] E[Y Z = 0,W]=(E[X Z = 1,W] E[X Z = 0,W]) β A natural approach would be to specify E[Y Z,W] and E[X Z,W] as linear functions of Z, W and ZW, estimate them, plug them into the identification equation and then estimate β by OLS. Call this estimator ˆβ 3Step. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

68 Estimation (linear model) Suppose that Y = α+ X β + U then E[Y Z = 1,W] E[Y Z = 0,W]=(E[X Z = 1,W] E[X Z = 0,W]) β A natural approach would be to specify E[Y Z,W] and E[X Z,W] as linear functions of Z, W and ZW, estimate them, plug them into the identification equation and then estimate β by OLS. Call this estimator ˆβ 3Step. We show that this approach is equivalent to a TSLS regression of Y onto X and W which treats W as exogenous, and uses Z and ZW as instruments of X. The estimated coefficient of X is ˆβ (so ˆβ = ˆβ 3Step ). Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

69 Estimation (linear model) Suppose that Y = α+ X β + U then E[Y Z = 1,W] E[Y Z = 0,W]=(E[X Z = 1,W] E[X Z = 0,W]) β A natural approach would be to specify E[Y Z,W] and E[X Z,W] as linear functions of Z, W and ZW, estimate them, plug them into the identification equation and then estimate β by OLS. Call this estimator ˆβ 3Step. We show that this approach is equivalent to a TSLS regression of Y onto X and W which treats W as exogenous, and uses Z and ZW as instruments of X. The estimated coefficient of X is ˆβ (so ˆβ = ˆβ 3Step ). ˆβ is consistent even if E[Y Z,W],E[X Z,W] or E[U W] are not linear. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

70 Estimation (linear model) Suppose that Y = α+ X β + U then E[Y Z = 1,W] E[Y Z = 0,W]=(E[X Z = 1,W] E[X Z = 0,W]) β A natural approach would be to specify E[Y Z,W] and E[X Z,W] as linear functions of Z, W and ZW, estimate them, plug them into the identification equation and then estimate β by OLS. Call this estimator ˆβ 3Step. We show that this approach is equivalent to a TSLS regression of Y onto X and W which treats W as exogenous, and uses Z and ZW as instruments of X. The estimated coefficient of X is ˆβ (so ˆβ = ˆβ 3Step ). ˆβ is consistent even if E[Y Z,W],E[X Z,W] or E[U W] are not linear. The variance of the TSLS is also consistent. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

71 Estimation (Nonparametric case) E[Y Z = 1,W] E[Y Z = 0,W]=E[g(X) Z = 1,W] E[g(X) Z = 0,W] Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

72 Estimation (Nonparametric case) E[Y Z = 1,W] E[Y Z = 0,W]=E[g(X) Z = 1,W] E[g(X) Z = 0,W] Suppose that g(x)= j=1 β jρ j (X). Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

73 Estimation (Nonparametric case) E[Y Z = 1,W] E[Y Z = 0,W]=E[g(X) Z = 1,W] E[g(X) Z = 0,W] Suppose that g(x)= j=1 β jρ j (X). Then E[Y Z = 1,W] E[Y Z=0,W]= j=1 β j [E[ρ j (X) Z = 1,W] E[ρ j (X) Z = 0,W]] Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

74 Estimation (Nonparametric case) E[Y Z = 1,W] E[Y Z = 0,W]=E[g(X) Z = 1,W] E[g(X) Z = 0,W] Suppose that g(x)= j=1 β jρ j (X). Then E[Y Z = 1,W] E[Y Z = 0,W]= j=1 β j [E[ρ j (X) Z = 1,W] E[ρ j (X) Z = 0,W]] A natural approach is to suppose that E[V Z,W]= l=1 α V,Z,lρ l (W), for V = Y,ρ 1 (X),...,ρ J (X). Estimate each of these components using the ρ l (W) up to l=l, substitute them into the identification equation and estimate β 1,...,β J by OLS. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

75 Estimation (Nonparametric case) E[Y Z = 1,W] E[Y Z = 0,W]=E[g(X) Z = 1,W] E[g(X) Z = 0,W] Suppose that g(x)= j=1 β jρ j (X). Then E[Y Z = 1,W] E[Y Z = 0,W]= j=1 β j [E[ρ j (X) Z = 1,W] E[ρ j (X) Z = 0,W]] A natural approach is to suppose that E[V Z,W]= l=1 α V,Z,lρ l (W), for V = Y,ρ 1 (X),...,ρ J (X). Estimate each of these components using the ρ l (W) up to l=l, substitute them into the identification equation and estimate β 1,...,β J by OLS. Analogously to the linear case, this is equivalent to running a TSLS regression of Y onto ρ 1 (X),...,ρ J (X) and ρ 1 (W),...,ρ L (W) treating the ρ l (W) as exogenous and using Z and Z ρ 1 (W),...,Z ρ L (W) as instruments of the ρ j (X). Then ĝ(x)= J j=1 ˆβ j ρ j (X). Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

76 Asymptotic results (nonparametric case) Theorem (nonparametric case) Given our identification assumptions and the standard assumptions for estimation in the nonparametric IV literature (Blundell, Chen and Kristensen (2007)), then ) J ĝ n g =O P (J r + τ n, n where J is the number of elements in the basis. r is g s level of smoothness τ n := sup g Gn g / (A A) 1/2 g is a measure of ill-posedness. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

77 Outline 1 Introduction 2 Intuition 3 Identification 4 Estimation 5 Application 6 Conclusion Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

78 Application: Chay and Greenstone JPE (2005) Valuation of air quality as reflected by house prices. Model: log(p 1980 ) log(p 1970 )=g(tsp 1980 TSP 1970 )+U P= average house price ($40,000 in 1970) TSP= average level of pollution (total suspended particulates) (TSP 1980 TSP 1970 normal with mean -7 and standard deviation 22) Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

79 Application: Chay and Greenstone JPE (2005) Valuation of air quality as reflected by house prices. Model: log(p 1980 ) log(p 1970 )=g(tsp 1980 TSP 1970 )+U P= average house price ($40,000 in 1970) TSP= average level of pollution (total suspended particulates) (TSP 1980 TSP 1970 normal with mean -7 and standard deviation 22) IV Z = 1 if county was regulated by the Environmental Protection Agency (EPA) in 1975, Z = 0 otherwise (Z = 1{TSP 1975 TSP}). Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

80 Application: Chay and Greenstone JPE (2005) Valuation of air quality as reflected by house prices. Model: log(p 1980 ) log(p 1970 )=g(tsp 1980 TSP 1970 )+U P= average house price ($40,000 in 1970) TSP= average level of pollution (total suspended particulates) (TSP 1980 TSP 1970 normal with mean -7 and standard deviation 22) IV Z = 1 if county was regulated by the Environmental Protection Agency (EPA) in 1975, Z = 0 otherwise (Z = 1{TSP 1975 TSP}) counties over the U.S., information on house prices, TSP and several other variables. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

81 Application: Chay and Greenstone JPE (2005) Valuation of air quality as reflected by house prices. Model: log(p 1980 ) log(p 1970 )=g(tsp 1980 TSP 1970 )+U P= average house price ($40,000 in 1970) TSP= average level of pollution (total suspended particulates) (TSP 1980 TSP 1970 normal with mean -7 and standard deviation 22) IV Z = 1 if county was regulated by the Environmental Protection Agency (EPA) in 1975, Z = 0 otherwise (Z = 1{TSP 1975 TSP}) counties over the U.S., information on house prices, TSP and several other variables. W: % spending on highways, health, and education. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

82 Application: Chay and Greenstone JPE (2005) Valuation of air quality as reflected by house prices. Model: log(p 1980 ) log(p 1970 )=g(tsp 1980 TSP 1970 )+U P= average house price ($40,000 in 1970) TSP= average level of pollution (total suspended particulates) (TSP 1980 TSP 1970 normal with mean -7 and standard deviation 22) IV Z = 1 if county was regulated by the Environmental Protection Agency (EPA) in 1975, Z = 0 otherwise (Z = 1{TSP 1975 TSP}) counties over the U.S., information on house prices, TSP and several other variables. W: % spending on highways, health, and education. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

83 Estimates: Linear Case Table : Binary IV, exogenous covariates IV Highway Health Education All 3 no exogenous covariates (.140) (.138) (.136) (.134) (.135) exogenous covariates (.093) (.094) (.093) (.093) (.093) Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

84 Nonlinear Case: Binary IV with exogenous covariates Change in the Geometric Mean TSP between 1970 and Change in the Geometric Mean TSP between 1970 and Change in the Geometric Mean TSP between 1970 and Change in the Geometric Mean TSP between 1970 and 1980 Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

85 Outline 1 Introduction 2 Intuition 3 Identification 4 Estimation 5 Application 6 Conclusion Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20

86 Conclusion Proposed a strategy to identify marginal effects of complex variables using a binary IV. Identification requires that the change in the distribution of the endogenous variable between treatment and control groups conditional on a covariate varies sufficiently with the covariate. We provide estimators and asymptotics. All estimators are TSLS regressions which can be implemented directly with packaged software. Method can be extended to some nonseparable models. Method may lead to overidentification even in situations where the classic IV and RDD provide just-identification. Method extends naturally to the RDD with multi-valued X. Caetano and Escanciano (Rochester and Indiana) Marginal Effects with Binary IV 04/10/ / 20