According to common judicial standard, judgment in favor of plaintiff should be made if and only if it is "more probable than not" that the defendant's action was the cause for the plaintiff's damage (or death). This paper provides formal semantics, based on structural models of counterfactuals, for the probability that event x was a necessary or sufficient cause (or both) of another event y. The paper then explicates conditions under which the probability of necessary (or sufficient) causation can be learned from statistical data, and shows how data from both experimental and nonexperimental studies can be combined to yield information that neither study alone can provide. Finally,weshow that necessity and sufficiency are two independent aspects of causation, and that both should be invoked in the construction of causal explanations for specific scenarios.

This report is based on lecture notes written for CS 262C, Spring 1998, and is organized as follows. Following a review of the SL framework (Section 2) Section 3 provides a comparison to other approaches to causation and suggests an explanation of why the notion of actual cause has encountered difficulties in those approaches. Section 3 defines "actual cause" and illustrates, through examples, how the "probability that event X = x actually caused event

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.

distribution. One of the great virtues of structural equation models is that they permit the quantification of causal and noncausal sources of statistical relationship. The present article discusses efficient matrix methods of computation for effect decomposition and extends these methods to models with unstandardized variables and to nonrecursive models. An appendix includes a computer program, written in APL, which implements the techniques described in the article.

In this paper we study the identifiability of nonparametric models, that is, models in which both the functional forms of the equations and the probability distributions of the disturbances remain unspecified. Identifiability in such models does not mean uniqueness of parameters but rather uniqueness of the set of predictions of interest to the investigator. For example, predicting the effects of changes, interventions, and control. We provide sufficient and necessary conditions for identifying a set of causal predictions of the type: "Find the distribution of Y , assuming that X is controlled by external intervention", where Y and X are arbitrary variables of interest. Whenever identifiable, such predictions can be expressed in closed algebraic form, in terms of observed distributions. We also show how the identifying criteria can be verified qualitatively, by inspection, using the graphical representation of the structural model. When compared to standard identifiability tests of lin...

In recursive linear models, the multivariate normal joint distribution of all variables exhibits a dependence structure induced by recursive systems of linear structural equations. Such models appear in particular in seemingly unrelated regressions, structural equation modelling, simultaneous equation systems, and in Gaussian graphical modelling. We show that recursive linear models that are ‘bow-free’ are well-behaved statistical models, namely, they are everywhere identifiable and form curved exponential families. Here, ‘bow-free ’ refers to models satisfying the condition that if a variable x occurs in the structural equation for y, then the errors for x and y are uncorrelated. For the computation of maximum likelihood estimates in ‘bow-free ’ recursive linear models we introduce the Residual Iterative Conditional Fitting (RICF) algorithm. Compared to existing algorithms RICF is easily implemented requiring only least squares computations, has clear convergence properties, and finds parameter estimates in closed form whenever possible. KEY WORDS: Linear structural equation model; curved exponential family; maximum likelihood estimation; residual iterative conditional fitting; bow-free acyclic path diagrams; BAP. 1

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