Mendelian randomization (MR) is a valuable tool for detecting causal effects by using genetic variant associations. Opportunities to apply MR are growing rapidly with the increasing number of genome-wide association studies (GWAS). However, existing MR methods rely on strong assumptions that are often violated, leading to false positives. Correlated horizontal pleiotropy, which arises when variants affect both traits through a heritable shared factor, remains a particularly challenging problem. We propose a new MR method, Causal Analysis Using Summary Effect estimates (CAUSE), that accounts for correlated and uncorrelated horizontal pleiotropic effects. We demonstrate, in simulations, that CAUSE avoids more false positives induced by correlated horizontal pleiotropy than other methods. Applied to traits studied in recent GWAS studies, we find that CAUSE detects causal relationships that have strong literature support and avoids identifying most unlikely relationships. Our results suggest that shared heritable factors are common and may lead to many false positives using alternative methods.