Notes on Twin Models

Transcription

1 Notes on Twin Models Rodrigo Pinto University of Chicago HCEO Seminar April 19, 2014 This draft, April 19, :17am Rodrigo Pinto Gene-environment Interaction and Causality, April 19, :17am 1 / 32

2 Figure 1: Standard Confounding Model E X 1 X Y E Y 1 Z 1 A X 1 V A Y 1 C Y 1 A X 2 V X 2 A Y 2 C Y 2 X 2 Y 2 E Y 2

3 Figure 2: Non-parametric IV Solution Z X Y E Y 1 Z 1 A X 1 V A Y 1 C Y 1 A X 2 V X 2 A Y 2 C Y 2 X 2 Y 2 E Y 2

4 IV Properties 1 Independence: V Z and Y Z (X, V ). 2 Or: Y (x) Z X and V X. 3 Identification: Based on Assumptions on the causal link between Z and X. Rodrigo Pinto Gene-environment Interaction and Causality, April 19, :17am 4 / 32

5 In Pinto(2013): Theorem 1 In the standard confounding model with IV, where Z is the instrument that takes values on {z 1,..., z NZ }, V are unobserved variables and X is the categorical treatment, the following statements are equivalent: (i) An utility function u representing rational preferences over X, V, Z is additive in Z, V, i.e.: u(x, Z, V ) u 1 (X, V ) + u 2 (X, Z); (ii) For any z, z supp(z) and v, v supp(v ) such that: X (Z = z, V = v) = t and X (Z = z, V = v ) = t, then X (Z = z, V = v ) = t or X (Z = z, V = v) = t. (iii) Each Stata Matrices A x ; x supp(x ) is a Lonesum Matrix. (iv) Each Stata Matrices A x ; x supp(x ) is equivalent to its maximal matrix Ā t. (v) Treatment X is separable on V and Z, that is: there exist functions, f x : supp(z) [0, 1] and g x : supp(v ) [0, 1]; x supp(x ) ( such that P E X = ) 1[f x (Z) g x (V )] x = 1. x supp(x ) Rodrigo Pinto Gene-environment Interaction and Causality, April 19, :17am 5 / 32

6 Twins - Two Useful Properties 1 Confounding Dependence: Twins share, to some degree, the confounding variables causing outcomes. 2 Independent Variation: While Monozygotic twins (MZ) share the same genetic endowment, Dizygotic twins don t (DZ); Rodrigo Pinto Gene-environment Interaction and Causality, April 19, :17am 6 / 32

7 Figure 3: Twin Identification Two Properties E X 1 X 1 Y 1 E Y 1 cxx a yx c yx Z 1 V X 1 C X 1 A Y 1 C Y 1 dz/mz mz = dz = 1 dz = 0.5mz mz = dz = 1 V X 2 C X 2 A Y 2 C Y 2 c xx a yx X 2 Y 2 E Y 2 cyx

8 Classification of Confounding Variables Properties (1) and (2) suggest a partition of all confounding variables into: 1 A.Genetic Endowment: confounding variables that are equal only if twins are MZ; 2 C.Shared Environment: confounding variables that are equal for twins regardless of type (MZ or DZ); 3 E.Non-shared Environment: remaining confounding variables. 4 Exclusion Restriction: No confounding variables is equal for DZ and different for MZ; Genetic Type Are Confounding Variables equal across twins? DZ MZ Variables 1. Genetic Endowment (A X 1, AX 2 ) 2. Shared Environment (C X 1, C X 2 ) 3. Non-shared Environment (E X 1, ) 4. Exclusion Assumption

9 Statistical Properties Classification based on Properties (1) and (2) generate the following properties: 1 Genetic Endowment: 2 Shared Environment: Pr(A X 1 = A X 2 MZ = 1) = 1. Pr(C X 1 = C X 2 MZ = 1) = Pr(C X 1 = C X 2 MZ = 0) = 1. 3 Non-shared Environment (Assumption): 4 Does not justify: E X 1 (MZ = 1), (E X 1 MZ = 0). A X i Ci X, A X i Ei X, Ci X E X i ;

10 Figure 4: Standard Univariate ACE Model E X 1 X 1 Y 1 E Y 1 cxx a yx c yx Z 1 A X 1 C X 1 A Y 1 C Y 1 dz = 0.5mz mz = dz = 1 dz = 0.5mz mz = dz = 1 A X 2 C X 2 A Y 2 C Y 2 c xx a yx X 2 Y 2 E Y 2 cyx

11 Properties of the Standard ACE Model 1 Variance Decomposition: heritable ( axx 2 ) axx 2 +cxx 2 +a, shared xx 2 environment ( cxx 2 ) ( or non-shared environment exx 2 ) axx 2 +cxx 2 +a. xx 2 a 2 xx +c 2 xx +a 2 xx 2 Concept: Decompose the confounding variables V 1, V 2 into: Variables that are independent between twins (E X 1, E X 1 ) : (E2, E1) (A1, A2, C1, C2, Y 2) and E1 E2 Variables that are equal for both twins types (C X 1, C X 1 ) : Pr(C X 1 = C X 2 MZ) = Pr(C X 1 = C X 2 DZ) = 1 Variables that are not equal for both twins types (A X 1, AX 1 ) : Pr(A X 1 = A X 2 MZ) = 1, Pr(A X 1 = A X 2 DZ) 1 3 Critics: No Gene Environment Interaction (A X 1, AX 2 ) (C X 1, C X 2 ). Rodrigo Pinto Gene-environment Interaction and Causality, April 19, :17am 11 / 32

12 Figure 5: What about the Bivariate Case? E X 1 X 1 Y 1 E Y 1 Z 1 A X 1 V X 1 A Y 1 C Y 1 mz = dz A X 2 V X 2 A Y 2 C Y 2 X 2 Y 2 E Y 2

13 Figure 6: Bivariate ACE Diagram E X 1 X 1 Y 1 E Y 1 cxx a yx c yx Z 1 A X 1 C X 1 A Y 1 C Y 1 dz = 0.5mz mz = dz dz = 0.5mz mz = dz A X 2 C X 2 A Y 2 C Y 2 c xx a yx X 2 Y 2 E Y 2 cyx

14 Basic Goal of Economics Main Goal: Evaluating the impact of X on Y instead of Variance Decomposition. Question: How Twin Data can help on identifying the causal impact of X on Y? Analysis based on the Bivariate ACE Diagram Approach: Add a causal link X Y in the Bivariate ACE Model. Problem: Model not identified. Solution: Assume that all confounding effects are equally shared by MZ Twins (ACE-β Model). Estimation: Linear Effect of X on Y can be estimated by Twin Fixed Effects using MZ Data. Rodrigo Pinto Gene-environment Interaction and Causality, April 19, :17am 14 / 32

15 Figure 7: Bivariate ACE Diagram E X 1 X 1 Y 1 E Y 1 cxx a yx c yx Z 1 A X 1 C X 1 A Y 1 C Y 1 dz = 0.5mz mz = dz dz = 0.5mz mz = dz A X 2 C X 2 A Y 2 C Y 2 c xx a yx X 2 Y 2 E Y 2 cyx

16 Figure 8: Bivariate ACE Diagram With Causal Link X Y E X 1 X 1 β Y 1 E Y 1 cxx a yx c yx Z 1 A X 1 C X 1 A Y 1 C Y 1 dz = 0.5mz mz = dz dz = 0.5mz mz = dz A X 2 C X 2 A Y 2 C Y 2 c xx a yx X 2 Y 2 E2 Y β cyx

17 Figure 9: ACE-β Diagram E X 1 X 1 β Y 1 E Y 1 cxx a yx c yx Z 1 A X 1 C X 1 A Y 1 C Y 1 dz = 0.5mz mz = dz dz = 0.5mz mz = dz A X 2 C X 2 A Y 2 C Y 2 c xx a yx X 2 Y 2 E2 Y β cyx

18 Interpreting the ACE-β Model Identification: Strong assumption of no confounding effects under outcome difference for linear models. Neglects the potential use of Property (2) of Twins data as an identification tool, only needs MZ to estimate β. Generate an IV: ( X ) (A, C) but X X. Estimation: Standard Twin Fixed Effects. Pending Critics: No Gene Environment Interaction; Rodrigo Pinto Gene-environment Interaction and Causality, April 19, :17am 18 / 32

19 Figure 10: Non-parametric Monozygotic and Dizygotic ACE E 1 A X 1 X 1 G 1 Y 1 A Y 1 MZ C A X 2 X 2 G 2 Y 2 A Y 2 E 2

20 Characteristics of the Nonparametric ACE Model Setting the Problem No IV: MZ des not have IV properties. Conditional Independence Relations: Model generates 341 total conditional independence relations and 141 unique ones. Additional Restrictions: solely as a DAG. Model cannot be characterized Four Additional Independence Relations: A X i {T \ {A X i, A X i }} (A X j, MZ = 1); i, j {1, 2} Equality in Distributions: X 1 = d X 2, Y 1 = d Y 2, A X 1 = d A X 2, A Y 1 = d A Y 2, E 1 = d E 2. Rodrigo Pinto Gene-environment Interaction and Causality, April 19, :17am 20 / 32

21 Figure 11: Non-parametric Monozygotic and Dizygotic ACE E 1 X 1 G 1 Y 1 A X C A Y X 2 G 2 Y 2 E 2

22 Characteristics of the Nonparametric MZ ACE Model Setting the Problem Conditional Independence Relations: Model generates 32 unique conditional independence relations. Four Additional Independence Relations: A X i {T \ {A X i, A X i }} (A X j, MZ = 1); i, j {1, 2} Additional Distribution Equalities: X 1 = d X 2, Y 1 = d Y 2, E 1 = d E 2, Rodrigo Pinto Gene-environment Interaction and Causality, April 19, :17am 22 / 32

23 Identification of the Intergenerational Elasticity Equation (IGE) Under Gene-Envinonment Interactions IGE: Often modeled as an AR(1): X }{{} Parental Income = β Y }{{} Descendants Income + ɛ }{{} Exogenous Error Term (1) Problems: AR(1) is a simplistic approach of a complex process; β is not a causal effect of X in Y, but rather a summary of confounding factors that operate on the income of both parents and children. Question: How Twin Data can help identify the IGE process? Rodrigo Pinto Gene-environment Interaction and Causality, April 19, :17am 23 / 32

24 Figure 12: IGE AR(1) Diagram E X 1 X 1 β Y 1 E Y 1 Z 1 A X 1 C X 1 A Y 1 C Y 1 A X 2 C X 2 A Y 2 C Y 2 X 2 Y 2 E Y 2

25 Figure 13: Underlying IGE Process Diagram E X 1 X 1 Y 1 E Y 1 Z 1 V X 1 C X 1 V Y 1 C Y 1 A X 2 C X 2 A Y 2 C Y 2 X 2 Y 2 E Y 2

26 Figure 14: Underlying IGE Process Diagram (Parental Twins) E X 1 X 1 Y 1 E Y 1 Z 1 V X 1 C X 1 V Y 1 C Y 1 mz/dz V X 2 C X 2 V Y 2 C Y 2 X 2 Y 2 E Y 2

27 Figure 15: ACE/IGE Partitioning V (Twin Parents) E X 1 X 1 Y 1 E Y 1 cxx Z 1 A X 1 C X 1 A Y 1 C Y 1 1mz/0.5dz 1mz/1dz A X 2 C X 2 A Y 2 C Y 2 c xx X 2 Y 2 E Y 2

28 Figure 16: ACE/IGE Diagram (Twin Parents) E X 1 X 1 e ax e cx Y 1 E Y 1 cxx a ax c ax Z 1 A X 1 C1 X c A Y 1 C1 Y cx 1mz/0.5dz 1mz/1dz a cx a cx A X 2 C X 2 A Y 2 C Y 2 c xx a ax X 2 e Y 2 E2 Y cx e ax c cx c ax

29 Problems with the ACE/IGE Model Equal Variances by Twin Type: Var(X MZ) = Var(X DZ). Even Worse: The Model is not Identified! Solution: Generalize. Twin Siblings can affect each other income. Allow for cross fertilization due to twin s genetic endowment. Rodrigo Pinto Gene-environment Interaction and Causality, April 19, :17am 29 / 32

30 Figure 17: ACE/IGE Diagram (Twin Parents) E X 1 X 1 e ax e cx Y 1 E Y 1 cxx a ax c ax Z 1 A X 1 C1 X c A Y 1 C1 Y cx 1mz/0.5dz 1mz/1dz a cx a cx A X 2 C X 2 A Y 2 C Y 2 c xx a ax X 2 e Y 2 E2 Y cx e ax c cx c ax

31 Figure 18: Modified ACE/IGE Diagram (Twin Parents) e ax E X 1 X 1 e cx Y 1 E Y 1 cxx a ax Z 1 A X 1 C1 X c A Y 1 C1 Y cx r xx c ax 1mz/0.5dz 1mz/1dz a cx a cx r xx A X 2 C X 2 A Y 2 C Y 2 c xx a ax X 2 e Y 2 E2 Y cx e ax c cx c ax

32 Figure 19: Bivariate ACE/IGE Diagram With GxE Interaction e ax E X 1 X 1 e cx Y 1 E Y 1 cxx a ax c ax r yy Z 1 A X 1 C1 X c A Y 1 C1 Y cx r xx a cx mz/dz mz/dz mz/dz mz/dz a cx r xx A X 2 C X 2 A Y 2 C Y 2 c xx a ax X 2 e Y 2 E2 Y cx e ax c cx r yy c ax