Simultaneous Causality: Part III James J. Heckman Econ 312, Spring - PowerPoint PPT Presentation

References Simultaneous Causality: Part III James J. Heckman Econ 312, Spring 2019 This Draft, May 20, 2019 1 / 22

References Nonrecursive (Simultaneous) Models of Causality: Developed in Economics (Haavelmo, 1944) A system of linear simultaneous equations captures interdependence among outcomes Y . 2 / 22

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. 3 / 22

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) also ”SUTUA” 1986 4 / 22

References 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. See Heckman et al. (2010). 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 ). Without some restrictions, ceteris paribus manipulations associated with the effect of some components of Y on other components of Y are not possible within the model. 5 / 22

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 / 22

References Y 1 = α 1 + γ 12 Y 2 + β 11 X 1 + β 12 X 2 + U 1 , (2a) = α 2 + γ 21 Y 1 + β 21 X 1 + β 22 X 2 + U 2 . (2b) Y 2 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. 7 / 22

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 / 22

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 . We cannot simultaneously vary Y 2 , X 1 and X 2 . 9 / 22

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

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 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 / 22

References Without any further information on the variances of ( U 1 , U 2 ) and their relationship to the causal parameters, we cannot isolate 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 in principle 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 / 22

References There is no mechanism yet specified within the model to independently vary the right hand sides of Equations (2a) and (2b). Some economists suggest that the mere fact that we can write (2a) and (2b) means that we “can imagine” independent variation. 13 / 22

References By the same token, 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. 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. 14 / 22

References Thus if β 12 = 0, from the reduced form π 12 = γ 12 . π 22 If β 21 = 0 , we obtain π 21 = γ 21 . π 11 In a general nonlinear model, = g 1 ( Y 2 , X 1 , X 2 , U 1 ) Y 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 ). 15 / 22

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. 16 / 22

References Alternatively, we could assume β 11 = β 22 = 0 and β 12 � = 0, β 21 � = 0 to identify γ 12 and γ 21 . These exclusions say that the social interactions only operate through the Y ’s. Agent 1’s consumption depends only on agent 2’s consumption and not on his value of X 2 . Agent 2 is modeled symmetrically versus agent 1. Observe that we have not ruled out correlation between U 1 and U 2 . 17 / 22

References When the procedure for identifying causal effects is applied to samples, it is called indirect least squares (Tinbergen, 1930). The analysis for social interactions in this section is of independent interest. It can be generalized to the analysis of N person interactions if the outcomes are continuous variables. 18 / 22

References The intuition for these results is that if β 12 = 0, we can vary Y 2 in Equation (2a) by varying the X 2 . Since X 2 does not appear in the equation, under exclusion, we can keep U 1 , X 1 fixed and vary Y 2 using X 2 in (4b) if β 22 � = 0. Notice that we could also use U 2 as a source of variation in (4b) to shift Y 2 . The roles of U 2 and X 2 are symmetric. However, if U 1 and U 2 are correlated, shifting U 2 shifts U 1 unless we control for it. The component of U 2 uncorrelated with U 1 plays the role of X 2 . 19 / 22

References Symmetrically, by excluding X 1 from(2b), we can vary Y 1 , holding X 2 and U 2 constant. These results are more clearly seen when U 1 = 0 and U 2 = 0. 20 / 22

References A hypothetical thought experiment justifies these exclusions. If agents do not know or act on the other agent’s X , these exclusions are plausible. An implicit assumption in using (2a) and (2b) for causal analysis is invariance of the parameters (Γ , β, Σ U ) to manipulations of the external variables. 21 / 22

References This definition of causal effects in an interdependent system generalizes the recursive definitions of causality featured in the statistical treatment effect literature (Holland, 1988, and Pearl, 2009. The key to this definition is manipulation of external inputs and exclusion, not randomization or matching. 22 / 22

References Frisch, R. (1938). Autonomy of economic relations: Statistical versus theoretical relations in economic macrodynamics. Paper given at League of Nations. Reprinted in D.F. Hendry and M.S. Morgan (1995), The Foundations of Econometric Analysis , Cambridge University Press. Haavelmo, T. (1943, January). The statistical implications of a system of simultaneous equations. Econometrica 11 (1), 1–12. Haavelmo, T. (1944). The probability approach in econometrics. Econometrica 12 (Supplement), iii–vi and 1–115. Heckman, J. J., R. L. Matzkin, and L. Nesheim (2010, September). Nonparametric identification and estimation of nonadditive hedonic models. Econometrica 78 (5), 1569–1591. Holland, P. W. (1988). Causal inference, path analysis and recursive structural equation models. In C. Clogg and G. Arminger (Eds.), Sociological Methodology , pp. 449–484. Washington, DC: American Sociological Association. 22 / 22