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Register a new userIdentification and Model Testing in Linear Structural Equation Models using Auxiliary Variables
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Bryant Chen
Publication date
2016
fields
Mathematical Statistics
and research's language is
English
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We developed a novel approach to identification and model testing in linear structural equation models (SEMs) based on auxiliary variables (AVs), which generalizes a widely-used family of methods known as instrumental variables. The identification problem is concerned with the conditions under which causal parameters can be uniquely estimated from an observational, non-causal covariance matrix. In this paper, we provide an algorithm for the identification of causal parameters in linear structural models that subsumes previous state-of-the-art methods. In other words, our algorithm identifies strictly more coefficients and models than methods previously known in the literature. Our algorithm builds on a graph-theoretic characterization of conditional independence relations between auxiliary and model variables, which is developed in this paper. Further, we leverage this new characterization for allowing identification when limited experimental data or new substantive knowledge about the domain is available. Lastly, we develop a new procedure for model testing using AVs.
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In this paper, we extend graph-based identification methods by allowing background knowledge in the form of non-zero parameter values. Such information could be obtained, for example, from a previously conducted randomized experiment, from substantive understanding of the domain, or even an identification technique. To incorporate such information systematically, we propose the addition of auxiliary variables to the model, which are constructed so that certain paths will be conveniently cancelled. This cancellation allows the auxiliary variables to help conventional methods of identification (e.g., single-door criterion, instrumental variables, half-trek criterion), as well as model testing (e.g., d-separation, over-identification). Moreover, by iteratively alternating steps of identification and adding auxiliary variables, we can improve the power of existing identification methods via a bootstrapping approach that does not require external knowledge. We operationalize this method for simple instrumental sets (a generalization of instrumental variables) and show that the resulting method is able to identify at least as many models as the most general identification method for linear systems known to date. We further discuss the application of auxiliary variables to the tasks of model testing and z-identification.
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Building on the theory of causal discovery from observational data, we study interactions between multiple (sets of) random variables in a linear structural equation model with non-Gaussian error terms. We give a correspondence between structure in the higher order cumulants and combinatorial structure in the causal graph. It has previously been shown that low rank of the covariance matrix corresponds to trek separation in the graph. Generalizing this criterion to multiple sets of vertices, we characterize when determinants of subtensors of the higher order cumulant tensors vanish. This criterion applies when hidden variables are present as well. For instance, it allows us to identify the presence of a hidden common cause of k of the observed variables.
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