  Recently, interest has grown in the use of proxy variables of unobserved
confounding for inferring the causal effect in the presence of unmeasured
confounders from observational data. One difficulty inhibiting the practical
use is finding valid proxy variables of unobserved confounding to a target
causal effect of interest. These proxy variables are typically justified by
background knowledge. In this paper, we investigate the estimation of causal
effects among multiple treatments and a single outcome, all of which are
affected by unmeasured confounders, within a linear causal model, without prior
knowledge of the validity of proxy variables. To be more specific, we first
extend the existing proxy variable estimator, originally addressing a single
unmeasured confounder, to accommodate scenarios where multiple unmeasured
confounders exist between the treatments and the outcome. Subsequently, we
present two different sets of precise identifiability conditions for selecting
valid proxy variables of unmeasured confounders, based on the second-order
statistics and higher-order statistics of the data, respectively. Moreover, we
propose two data-driven methods for the selection of proxy variables and for
the unbiased estimation of causal effects. Theoretical analysis demonstrates
the correctness of our proposed algorithms. Experimental results on both
synthetic and real-world data show the effectiveness of the proposed approach.
