Physical interaction of regulatory elements in three-dimensional space poses a challenge for studies of disease because non-coding risk variants may be great distances from the genes they regulate. Experimental methods to capture these interactions, such as chromosome conformation capture, usually cannot assign causal direction of effect between regulatory elements, an important component of fine-mapping studies. We developed a Bayesian hierarchical approach that uses two-stage least squares and applied it to an ATAC-seq (assay for transposase-accessible chromatin using sequencing) data set from 100 individuals, to identify over 15,000 high-confidence causal interactions. Most (60%) interactions occurred over <20 kb, where chromosome conformation capture-based methods perform poorly. For a fraction of loci, we identified a single variant that alters accessibility across multiple regions, and experimentally validated the BLK locus, which is associated with multiple autoimmune diseases, using CRISPR genome editing. Our study highlights how association genetics of chromatin state is a powerful approach for identifying interactions between regulatory elements.