An Extended Class of Instrumental Variables for the Estimation of Causal Effects
This paper examines the ways in which structural systems can yield observed variables, other than the cause or treatment of interest, that can play an instrumental role in identifying and estimating causal effects. We focus speciÖcally on the ways in which structures determine exclusion restrictions and conditional exogeneity relations that act to ensure identification. We show that by carefully specifying the structural equations and by extending the standard notion of instrumental variables, one can identify and estimate causal effects in the endogenous regressor case for a broad range of economically relevant structures. Some of these have not previously been recognized. Our results there create new opportunities for identifying and estimating causal effects in non-experimental situations. Our results for more familiar structures provide new insights. For example, we extend results of Angrist, Imbens, and Rubin (1996) by taking into account an important distinction between cases where Z is an observed exogenous instrument and those where it is a proxy for an unobserved exogenous instrument. A main message emerging from our analysis is the central importance of sufficiently specifying the causal relations governing the unobservables, as these play a crucial role in creating obstacles or opportunities for identification. Because our results exhaust the possibilities for identification, we ensure that there are no other opportunities for identification based on exclusion restrictions and conditional independence relations still to be discovered. To accomplish this characterization, we introduce notions of conditioning and conditional extended instrumental variables (EIVs). These are not proper instruments, as they are endogenous. They nevertheless permit identification and estimation of causal effects. We analyze methods using these EIVs either singly or jointly.
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