**Kernel Instrumental Variable Regression**

Instrumental variable regression is a strategy for learning causal rela-tionships in observational data. If measurements of input X and output Y are confounded, the causal relationship can nonetheless be identiﬁed if an instrumental variable Z is available that only inﬂuences Y via X. The two-stage least squares algorithm (2SLS) simpliﬁes the estimation prob-lem by assuming linear relationships, and it is the most popular empirical method in economics. We propose kernel instrumental variable regres-sion (KIV), a reproducing kernel Hilbert space (RKHS) generalization of 2SLS that relaxes linearity in a tractable way. In particular, we formulate the ﬁrst stage as a regression problem in a vector-valued RKHS, and the second stage as a regression problem in a real-valued RKHS. We impose Tikhonov regularization in both stages. In simulations, KIV outperforms state of the art algorithms for nonparametric instrumental variable re-gression. It also runs in cubic time. We analyze its consistency and convergence rate.

Authors: Rahul Singh, Arthur Gretton (UCL Gatsby), Maneesh Sahani (UCL Gatsby)