Simultaneous Causality: Part IV on Causality James J. Heckman Econ - PowerPoint PPT Presentation

References Simultaneous Causality: Part IV on Causality James J. Heckman Econ 312, Spring 2019 1 / 29

References Econometric Causality Entertains the Possibility of Simultaneous Causality Nonrecursive (Simultaneous) Models of Causality: Developed in Economics (Haavelmo, 1944) A system of linear simultaneous equations captures interdependence among outcomes Y . 2 / 29

References Linear model in terms of parameters (Γ , B ), observables ( Y , X ) and unobservables U : Γ Y + BX = U , E ( U ) = 0 , (1) Y is now a vector of internal and interdependent variables X is external and exogenous ( E ( U | X ) = 0) Γ is a full rank matrix (“completeness” totally different). Y = Γ − 1 BX + Γ − 1 U (reduced form) from a concept: � Φ θ ( X ) dF ( X ) = 0 support X ⇒ Φ θ ( X ) = 0 3 / 29

References This is a linear-in-the-parameters “all causes” model for vector Y , where the causes are X and E . The “structure” is (Γ , B ), Σ U , where Σ U is the variance-covariance matrix of U . In the Cowles Commission analysis it is assumed that Γ , B , Σ U are invariant to general changes in X and translations of U . Autonomy (Frisch, 1938) later called “SUTUA” in Holland, 1986. X , U external variables . Y internal variables . 4 / 29

References Nonlinear Systems Possible Thus we can postulate a system of equations G ( Y , X , U ) = 0 and develop conditions for unique solution of reduced forms Y = K ( X , U ) requiring that certain Jacobian terms be nonvanishing (Matzkin, “Nonparametric Identification of Simultaneous Equations,” 2007). The structural form (1) is an all causes model that relates in a deterministic way outcomes (internal variables) to other outcomes (internal variables) and external variables (the X and U ). Question: Are ceteris paribus manipulations associated with the effect of some components of Y on other components of Y possible within the model? Yes. 5 / 29

References Consider a two-agent model of social interactions. Y 1 is the outcome for agent 1; Y 2 is the outcome for agent 2. 6 / 29

References = α 1 + γ 12 Y 2 + β 11 X 1 + β 12 X 2 + U 1 , (2a) Y 1 Y 2 = α 2 + γ 21 Y 1 + β 21 X 1 + β 22 X 2 + U 2 . (2b) Social interactions model is a standard version of the simultaneous equations problem. This model is sufficiently flexible to capture the notion that the consumption of 1 ( Y 1 ) depends on the consumption of 2 if γ 12 � = 0, as well as 1’s value of X if β 11 � = 0, X 1 (assumed to be observed), 2’s value of X , X 2 if β 12 � = 0 and unobservable factors that affect 1 ( U 1 ). The determinants of 2’s consumption are defined symmetrically. Allow U 1 and U 2 to be freely correlated. Captures essence of “reflection problem.” 7 / 29

References Assume E ( U 1 | X 1 , X 2 ) = 0 (3a) and E ( U 2 | X 1 , X 2 ) = 0 . (3b) Completeness guarantees that (2a) and (2b) have a determinate solution for ( Y 1 , Y 2 ). Applying Haavelmo’s (1943) analysis to (2a) and (2b), the causal effect of Y 2 on Y 1 is γ 12 . This is the effect on Y 1 of fixing Y 2 at different values, holding constant the other variables in the equation. 8 / 29

References Symmetrically, the causal effect of Y 1 on Y 2 is γ 21 . Conditioning, i.e., using least squares, in general, fails to identify these causal effects because U 1 and U 2 are correlated with Y 1 and Y 2 . This is a traditional argument. It is based on the correlation between Y 2 and U 1 (Haavelmo, 1943). But even if U 1 = 0 and U 2 = 0, so that there are no unobservables, least squares breaks down because Y 2 is perfectly predictable by X 1 and X 2 . Question: Prove this. We cannot simultaneously vary Y 2 , X 1 and X 2 . The error term is not the fundamental source of non-identifiability in these models . 9 / 29

References Under completeness, the reduced form outcomes of the model after social interactions are solved out can be written as 9 Y 1 = π 10 + π 11 X 1 + π 12 X 2 + E 1 , (4a) Y 2 = π 20 + π 21 X 1 + π 22 X 2 + E 2 . (4b) E ( E 1 | X ) = 0 E ( E 2 | X ) = 0 10 / 29

References Least squares can identify the ceteris paribus effects of X 1 and X 2 on Y 1 and Y 2 because E ( E 1 | X 1 , X 2 ) = 0 and E ( E 2 | X 1 , X 2 ) = 0. Simple algebra: π 11 = β 11 + γ 12 β 21 π 12 = β 12 + γ 12 β 22 , π 21 = γ 21 β 11 + β 21 , , 1 − γ 12 γ 21 1 − γ 12 γ 21 1 − γ 12 γ 21 and π 22 = γ 21 β 12 + β 22 1 − γ 12 γ 21 U 1 + γ 12 U 2 E 1 = , 1 − γ 12 γ 21 γ 21 U 1 + U 2 E 2 = . 1 − γ 12 γ 21 11 / 29

References Without any further information on the variances of ( U 1 , U 2 ) and their relationship to the causal parameters, we cannot identify the causal effects γ 12 and γ 21 from the reduced form regression coefficients. This is so because holding X 1 , X 2 , U 1 and U 2 fixed in (2a) or (2b), it is not possible to vary Y 2 or Y 1 , respectively, because they are exact functions of X 1 , X 2 , U 1 and U 2 . This exact dependence holds true even if U 1 = 0 and U 2 = 0 so that there are no unobservables. 12 / 29

References There is no mechanism yet specified within the model to independently vary the right hand sides of Equations (2a) and (2b). The mere fact that we can write (2a) and (2b) means that we “can imagine” independent variation. Causality is in the mind. Question: Can we still define the causal effect of Y 2 on Y 1 and Y 1 on Y 2 , even if we cannot identify them? 13 / 29

References We “can imagine” a model Y = ϕ 0 + ϕ 1 X 1 + ϕ 2 X 2 , but if part of the model is ( ∗ ) X 1 = X 2 , no causal effect of X 1 holding X 2 constant is possible in principle within the rules of the model. If we break restriction ( ∗ ) and permit independent variation in X 1 and X 2 , we can define the causal effect of X 1 holding X 2 constant. But we can imagine such variation. 14 / 29

References In some conceptualizations, no causality is possible; in others it is. Distinguish identification from causation. The X effects on Y 1 and Y 2 , identified through the reduced forms, combine the direct effects (through β ij ) and the indirect effects (as they operate through Y 1 and Y 2 , respectively). If we assume exclusions ( β 12 = 0) or ( β 21 = 0) or both, we can identify the ceteris paribus causal effects of Y 2 on Y 1 and of Y 1 on Y 2 , respectively, if β 22 � = 0 or β 11 � = 0, respectively. 15 / 29

References Consider Standard Identification Analyses Suppose β 12 = 0 and β 21 = 0 β 11 γ 12 β 22 π 11 = π 12 = 1 − γ 12 γ 21 1 − γ 12 γ 21 γ 21 β 11 β 22 π 21 = π 22 = 1 − γ 12 γ 21 1 − γ 12 γ 21 16 / 29

References π 12 = γ 12 π 22 π 21 = γ 21 π 11 ∴ we identify β 11 and β 22 . 17 / 29

References Suppose instead only β 12 = 0 β 22 π 22 = 1 − γ 12 γ 21 γ 12 β 22 π 12 = 1 − γ 12 γ 21 π 12 = γ 12 π 22 Then can form left-hand side of y 1 − γ 12 y 2 = β 11 X 1 + β 12 X 2 + U 1 . ∴ can identify β 11 = 0 from OLS. 18 / 29

References Symmetrically if β 21 = 0 can identify β 22 , σ 2 1 = Var ( U 1 ) Suppose Cov ( U 1 , U 2 ) = 0. Both equations identified. � U 1 + γ 12 U 2 � � γ 21 U 1 + U 2 � Cov ( E 1 , E 2 ) = Cov 1 − γ 12 γ 21 1 − γ 12 γ 21 = γ 21 σ 2 1 + γ 12 σ 2 2 (1 − γ 12 γ 21 ) 2 σ 1 2 + γ 2 12 σ 2 2 Var ( E 1 ) = (1 − γ 12 γ 21 ) 2 γ 2 21 σ 2 1 + σ 2 2 Var ( E 2 ) = (1 − γ 12 γ 21 ) 2 Suppose we add this to β 12 = 0. By previously analysis, we know γ 12 , σ 2 1 . 19 / 29

References Control Function Principle � σ 11 γ 21 + σ 12 � E ( U 1 |E 2 ) = E 2 + 0 1 − γ 21 γ 21 � σ 11 γ 12 + σ 12 � ˆ U 1 = E 2 + V 1 1 − γ 21 γ 21 � �� � control function V 1 : portion of U 1 not correlated with Y 2 . If no exclusions in first equation, perfect multicollinearity, i.e., Y 1 = γ 12 ( ˆ Y 2 ) + β 11 X 1 + β 12 X 2 + γ 12 ˆ E 2 + U 1 controls for source of bias. 20 / 29

References Then we know = γ 21 σ 2 1 + γ 12 σ 2 a = Cov ( E 1 , E 2 ) 2 σ 2 1 + γ 2 12 σ 2 Var ( E 1 ) 2 = γ 21 σ 2 1 + γ 12 σ 2 b = Cov ( E 1 , E 2 ) 2 γ 2 1 + σ 2 1 + σ 2 Var ( E 2 ) 2 2 equations in 2 unknowns 2 , γ 21 (in principle) letting “ ˆ Can solve: σ 2 ’’ denote estimate a )( σ 2 1 + γ 2 12 σ 2 2 ) = γ 21 σ 2 1 + γ 12 σ 2 (ˆ 2 (ˆ b )( γ 2 21 σ 2 1 + σ 2 2 ) = γ 21 σ 2 1 + γ 12 σ 2 2 2 ) = ˆ σ 2 γ 2 12 σ 2 b ( γ 2 σ 2 1 + σ 2 a (ˆ ˆ 1 + ˆ 21 ˆ 2 ) ˆ σ 2 γ 2 12 σ 2 c = Var ( E 1 ) 1 + ˆ 2 ˆ Var ( E 2 ) = γ 2 σ 2 1 + σ 2 21 ˆ 2 21 / 29

References In a General Nonlinear Model Y 1 = g 1 ( Y 2 , X 1 , X 2 , U 1 ) Y 2 = g 2 ( Y 1 , X 1 , X 2 , U 2 ) , exclusion is defined as ∂ g 1 ∂ X 1 = 0 for all ( Y 2 , X 1 , X 2 , U 1 ) and ∂ g 2 ∂ X 2 = 0 for all ( Y 1 , X 1 , X 2 , U 2 ). 22 / 29

References Assuming the existence of local solutions, we can solve these equations to obtain = ϕ 1 ( X 1 , X 2 , U 1 , U 2 ) Y 1 Y 2 = ϕ 2 ( X 1 , X 2 , U 1 , U 2 ) By the chain rule we can write � ∂ Y 2 � ∂ϕ 2 ∂ g 1 = ∂ Y 1 = ∂ϕ 1 . ∂ Y 2 ∂ X 1 ∂ X 1 ∂ X 1 ∂ X 1 We may define causal effects for Y 1 on Y 2 using partials with respect to X 2 in an analogous fashion. 23 / 29