Identification with Conditioning Instruments in Causal Systems

We study the structural identification of causal effects with conditioning instruments within the settable system framework. In particular, we provide causal and predictive conditions sufficient for conditional exogeneity to hold. We provide two procedures based on “exclusive of A” (~A)-causality matrices and the direct causality matrix for inferring conditional causal isolation among vectors of settable variables and consequently conditional independence among corresponding vectors of random variables. Similarly, we provide sufficient conditions for conditional stochastic isolation in terms of the σ-algebras generated by the conditioning variables. We build on these results to study the structural identification of average effects and average marginal effects with conditioning instruments. We distinguish between structural proxies and predictive proxies. PRELIMINARY AND INCOMPLETE Acknowledgment: The authors thank Julian Betts, Graham Elliott, Clive Granger, Mark Machina, Dimitris Politis, and Ruth Williams for their helpful comments and suggestions. All errors and omissions are the authors’ responsibility. 1 Karim Chalak, Department of Economics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA 02467 (email: [email protected]). Halbert White, Department of Economics 0508, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0508 (email: [email protected]).