This paper builds on the structural equations, treatment effect, and machine learning literatures to provide a causal framework that permits the identification and estimation of causal effects from observational studies. We begin by providing a causal interpretation for standard exogenous regressors and standard “valid” and “relevant” instrumental variables. We then build on this interpretation to characterize extended instrumental variables (EIV) methods, that is methods that make use of variables that need not be valid instruments in the standard sense, but that are nevertheless instrumental in the recovery of causal effects of interest. After examining special cases of single and double EIV methods, we provide necessary and sufficient conditions for the identification of causal effects by means of EIV and provide consistent and asymptotically normal estimators for the effects of interest.

This paper aims at assisting empirical researchers benefit from recent advances in causal inference. The paper 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, and the conditional nature of causal claims inferred from nonexperimental studies. These emphases are illustrated through a brief survey of recent results, including the control of confounding, the assessment of causal effects, the interpretation of counterfactuals, and a symbiosis between counterfactual and graphical methods of analysis.

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.

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.

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.

In this paper we examine the ability to perform causal reasoning with equilibrium models. We explicate a postulate, which we term the Manipulation Postulate, that is required in order to perform causal inference, and we prove that there exists a general class of recursive equilibrium models that are guaranteed to violate the Manipulation Postulate. In addition, we show that all models in this class possess a set of variables V ′ whose manipulation will cause an instability such that no equilibrium model will exist for the system. We define the Structural Stability Principle which provides a graphical criterion for stability in causal models. Our theorems suggest that caution should be exercised when applying causal reasoning to equilibrium models or to models learned from databases wherein features were not measured simultaneously. 1

Abstract. The term “changes in structure, ” originating from work in econometrics, refers to structural modifications invoked by actions on a causal model. In this paper we formalize the representation of reversibility of a mechanism in order to support modeling of changes in structure in systems that contain reversible mechanisms. Causal models built on our formalization can answer two new types of queries: (1) When manipulating a causal model (i.e., making an endogenous variable exogenous), which mechanisms are possibly invalidated and can be removed from the model? (2) Which variables may be manipulated in order to invalidate and, effectively, remove a mechanism from a model? 1

Abstract This paper demonstrates the extensive scope of an alternative to standard instrumental variables methods, namely covariate-based methods, for identifying and estimating e¤ects of interest in general structural systems. As we show, commonly used econometric methods, speci…cally parametric, semi-parametric, and nonparametric extremum or moment-based methods, can all exploit covariates to estimate well-identi…ed structural e¤ects. The systems we consider are general, in that they need not impose linearity, separability, or monotonicity restrictions on the structural relations. We consider e¤ects of multiple causes; these may be binary, categorical, or continuous. For continuous causes, we examine both marginal and non-marginal e¤ects. We analyze e¤ects on aspects of the response distribution generally, de…ned by explicit or implicit moments or as optimizers (e.g., quantiles). Key for identi…cation is a speci…c conditional exogeneity relation. We examine what happens in its absence and …nd that identi…cation generally fails. Nevertheless, local and near identi…cation results hold in its absence, as we show.

stimulated this work, and for the comments and suggestions of Karim Chalak, Douglas R. This paper examines the relations between the causal models of Pearl and the settable systems framework recently introduced by White and Chalak. We pay particular attention to the suitability of these two approaches for analyzing the behavior of optimizing, interacting agents, the central concern of economics. We show that Pearl’s causal model is nested in the settable system framework, and we describe a number of ways in which Pearl’s causal models are not well suited to the study of interacting, optimizing agents. In contrast, settable systems have been explicitly designed to facilitate this study, accommodating systems of agents that not only optimize and interact, but that may learn from experience. We illustrate using examples from microeconomics, option pricing, game theory, and recursive estimation. Among the features of settable systems that distinguish it from Pearl’s causal models and that help deliver its capabilities are its countable rather than …nite structure, its explicit use of attributes, and the introduction of partitions and partition-speci…c response