We examine how structural systems can yield observed variables instrumental in identifying and estimating causal effects. We provide an exhaustive characterization of potentially identifying conditional exogeneity relationships and demonstrate how structural relations determine exogeneity and exclusion restrictions that yield moment conditions supporting identification. This provides a comprehensive framework for constructing instruments and covariates. We introduce notions of conditioning and conditional extended instrumental variables (XIVs). These permit identification but need not be traditional instruments, as they may be endogenous. We distinguish between observed XIVs and proxies for unobserved XIVs. A main message is the importance of sufficiently specifying causal relations governing the unobservables. JEL classification: C10, C13, C21, C31, C51Une classeétendue de variables instrumentales pour l'estimation des effets de causalité. Les auteurs examinent comment des systèmes structurels peuvent engendrer des variables observées instrumentales dans l'identification et l'estimation d'effets de causalité. On fournit une caractérisation exhaustive des relations d'exogénéité conditionnelle potentiellement porteuses d'identification, et on démontre comment des relations structurelles déterminent les restrictions d'exogénéité et d'exclusion qui engendrent les conditions de moment qui supportent l'identification. Cela fournit un cadre compréhensif pour construire instruments et covariables. On introduit des notions de variables instrumentaleś etendues conditionnantes et conditionnelles (XIVs). Celles-ci permettent l'identification, mais n'ont pasàêtre des instruments traditionnels, parce qu'elles peuventêtre endogènes. On distingue entre les XIVs observées et les variables concomitantes avec les XIVsThe authors thank
We study the scope of local indirect least squares (LILS) methods for nonparametrically estimating average marginal e¤ects of an endogenous cause X on a response Y in triangular structural systems that need not exhibit linearity, separability, or monotonicity in scalar unobservables. One main …nding is negative: in the fully nonseparable case, LILS methods cannot recover the average marginal e¤ect. LILS methods can nevertheless test the hypothesis of no e¤ect in the general nonseparable case. We provide new nonparametric asymptotic theory, treating both the traditional case of observed exogenous instruments Z and the case where one observes only error-laden proxies for Z. Acknowledgement 0.1 We thank Stefan Hoderlein, Xun Lu, Andres Santos, and Suyong Song for helpful comments and suggestions. Any errors are the authors' responsibility. S. M. Schennach acknowledges support from the National Science Foundation via grants SES-0452089 and SES-0752699. This is a revised version of a paper titled "Estimating Average Marginal E¤ ects in Nonseparable Structural Systems." JEL Classi…cation Numbers: C13, C14, C31.
We provide necessary and sufficient conditions for effect identification, thereby characterizing the limits to identification. Our results link the nonstructural potential outcome framework for identifying and estimating treatment effects to structural approaches in economics. This permits economic theory to be built into treatment effect methods. We elucidate the sources and consequences of identification failure by examining the biases arising when the necessary conditions fail, and we clarify the relations between unconfoundedness, conditional exogeneity, and the necessary and sufficient identification conditions. A new quantity, the exogeneity score, plays a central role in this analysis, permitting an omitted variable representation for effect biases. This analysis also provides practical guidance for selecting covariates and insight into the price paid for making various identifying assumptions and the benefits gained.
We study the consequences of substituting an error-laden proxy W for an instrument Z on the interpretation of Wald, local instrumental variable (LIV), and instrumental variable (IV) estimands in an ordered discrete choice structural system with heterogeneity. A proxy W need only satisfy an exclusion restriction and that the treatment and outcome are mean independent from W given Z. Unlike Z, W need not satisfy monotonicity and may, under particular specifications, fail exogeneity. For example, W could code Z with error, with missing observations, or coarsely. We show that Wald, LIV, and IV estimands using W identify weighted averages of local or marginal treatment effects (LATEs or MTEs). We study a necessary and sufficient condition for nonnegative weights. Further, we study a condition under which the Wald or LIV estimand using W identifies the same LATE or MTE that would have been recovered had Z been observed. For example, this holds for binary Z and therefore the Wald estimand using W identifies the same “average causal response,” or LATE for binary treatment, that would have been recovered using Z. Also, under this condition, LIV using W can be used to identify MTE and average treatment effects for e.g., the population, treated, and untreated.
We study the connections between causal relations and conditional independence within the settable systems extension of the Pearl causal model (PCM). Our analysis clearly distinguishes between causal notions and probabilistic notions, and it does not formally rely on graphical representations. As a foundation, we provide definitions in terms of suitable functional dependence for direct causality and for indirect and total causality via and exclusive of a set of variables. Based on these foundations, we provide causal and stochastic conditions formally characterizing conditional dependence among random vectors of interest in structural systems by stating and proving the conditional Reichenbach principle of common cause, obtaining the classical Reichenbach principle as a corollary. We apply the conditional Reichenbach principle to show that the useful tools of d-separation and D-separation can be employed to establish conditional independence within suitably restricted settable systems analogous to Markovian PCMs.
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