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Abstract: This review presents empirical researchers with recent advances in causal inference, and stresses the paradigmatic shifts that must be undertaken in moving from traditional statistical analysis to causal analysis of multivariate data. Special emphasis is placed on the assumptions that underly all causal inferences, the languages used in formulating those assumptions, the conditional nature of all causal and counterfactual claims, and the methods that have been developed for the assessment of such claims. These advances are illustrated using a general theory of causation based on the Structural Causal Model (SCM) described in Pearl (2000a), which subsumes and unifies other approaches to causation, and provides a coherent mathematical foundation for the analysis of causes and counterfactuals. In particular, the paper surveys the development of mathematical tools for inferring (from a combination of data and assumptions) answers to three types of causal queries: (1) queries about the effects of potential interventions, (also called “causal effects ” or “policy evaluation”) (2) queries about probabilities of counterfactuals, (including assessment of “regret, ” “attribution” or “causes of effects”) and (3) queries about direct and indirect effects (also known as “mediation”). Finally, the paper defines the formal and conceptual relationships between the structural and potential-outcome frameworks and presents tools for a symbiosis analysis that uses the strong features of both.

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This paper unifies three complementary approaches to defining, identifying, and estimating causal effects: the classical structural equations approach of the Cowles Commision; the treatment effects framework of Rubin (1974) and Rosenbaum and Rubin (1983); and the Directed Acyclic Graph (DAG) approach of Pearl. The settable system framework nests these prior approaches, while affording significant improvements to each. For example, the settable system approach permits identification of causal effects without requiring exogenous instruments; instead, a weaker conditional exogeneity condition suffices. It removes the stable unit treatment value assumption of the treatment effect approach and provides significant insight into the selection of covariates. It generalizes the DAG ap-proach by accommodating mutual causality and attributes. We provide a variety of results ensuring structural identification of general covariate-conditioned average causal effects, laying the founda-tion for parametric and nonparametric estimation of effects of interest and new tests for structural identification.

We study the interrelations between (conditional) independence and causal relations in settable systems. We provide definitions in terms of functional dependence for direct, indirect, and total causality as well as for (indirect) causality via and exclusive of a set of variables. We then provide necessary and sufficient causal and stochastic conditions for (conditional) dependence among random vectors of interest in settable systems. Immediate corollaries ensure the validity of Reichenbach’s principle of common cause and its informative extension, the conditional Reichenbach principle of common cause. We relate our results to notions of d-separation and D-separation in the artificial intelligence literature.

This paper unifies three complementary approaches to defining, identifying, and estimating causal effects: the classical structural equations approach of the Cowles Commision; the treatment effects framework of Rubin (1974) and Rosenbaum and Rubin (1983); and the Directed Acyclic Graph (DAG) approach of Pearl. The settable system framework nests these prior approaches, while affording significant improvements to each. For example, the settable system approach permits identification and estimation of causal effects without requiring exogenous instruments, generalizing the classical structural equations approach; it relaxes the stable unit treatment value assumption of the treatment effect approach and provides significant insight into the selection of covariates; and it accommodates mutual causality, generalizing the DAG approach. We provide necessary and sufficient conditions for identification of covariate-conditioned average causal effects, parametric and nonparametric estimation results, and new tests for unconfoundedness.

This paper reviews recent advances in the foundations of causal inference and introduces a systematic methodology for defining, estimating and testing causal claims in experimental and observational studies. It is based on non-parametric structural equation models (SEM) – a natural generalization of those used by econometricians and social scientists in the 1950-60s, and provides a coherent mathematical foundation for the analysis of causes and counterfactuals. In particular, the paper surveys the development of mathematical tools for inferring the effects of potential interventions (also called “causal effects” or “policy evaluation”), as well as direct and indirect effects (also known as “mediation”), in both linear and non-linear systems. Finally, the paper clarifies the role of propensity score matching in causal analysis, defines the relationships between the structural and

Judea Pearl’s Causal Model is a rich framework that provides deep insight into the nature of causal relations. As yet, however, the Pearl Causal Model (PCM) has not had much impact on economics or econometrics. This may be due in part to the fact that the PCM is not as well suited to analyzing economic structures as might be desired. We o¤er the settable systems framework as an extension of the PCM that embodies features of central interest to economists and econometricians: optimization, equilibrium, and learning. Because these are common features of physical, natural, or social systems, our framework may prove generally useful. In particular, settable systems o¤er a number of advantages relative to the PCM for machine learning. Important distinguishing features of the settable systems framework are its countable dimensionality, its treatment of attributes, the absence of a …xed-point requirement, and the use of partitioning and partition-speci…c response functions to accommodate the behavior of optimizing and interacting agents. A series of closely related machine learning examples and examples from game theory and machine learning with feedback demonstrates limitations of the PCM and motivates the distinguishing features of settable systems.

We analyze the relations between Granger (G) non-causality and a notion of structural causality arising naturally from a general nonseparable recursive dynamic structural system. Building on classical notions of G non-causality, we introduce interesting and natural extensions, namely weak G non-causality and retrospective weak G non-causality. We show that structural non-causality and certain (retrospective) conditional exogeneity conditions imply (retrospective) (weak) G non-causality. We strengthen structural causality to notions of (retrospective) strong causality and show that (retrospective) strong causality implies (retrospective) weak G causality. We provide practical conditions and straightforward new methods for testing (retrospective) weak G non-causality, (retrospective) conditional exogeneity, and structural non-causality. Finally, we apply our methods to explore structural causality in industrial pricing, macroeconomics, and …nance.

This paper aims to acquaint researchers in the quantitative social and behavior sciences with recent advances in causal inference which provide a systematic methodology for defining, estimating, testing, and defending causal claims in experimental and observational studies. These advances are illustrated using a general theory of causation based on nonparametric structural equation models (SEM) – a natural generalization of those used by econometricians and social scientists in the 1950-60s, which provides a coherent mathematical foundation for the analysis of causes and counterfactuals. In particular, the paper surveys the development of mathematical tools for inferring (from a combination of data and assumptions) answers to three types of causal queries: (1) queries about the effects of potential interventions, (also called “causal effects” or “policy evaluation”) (2) queries about probabilities of counterfactuals, (including assessment of “regret,” “attribution” or “causes of effects”) and (3) queries about direct and indirect effects (also known as “mediation”). Finally, the paper clarifies the role of propensity score matching in causal analysis, defines the relationships between the structural and potential-outcome frameworks, and develops symbiotic tools that use the strong features of both.