  Mendelian randomization (MR) is a powerful method that uses genetic variants
as instrumental variables (IVs) to infer the causal effect of a modifiable
exposure on an outcome. Although recent years have seen many extensions of
basic MR methods to be robust to certain violations of assumptions, few methods
were proposed to infer bi-directional causal relationships, especially for
phenotypes with limited biological understandings. The presence of horizontal
pleiotropy adds another layer of complexity. In this article, we show that
assumptions for common MR methods are often impossible or too stringent in the
existence of bi-directional relationships. We then propose a new focusing
framework for testing bi-directional causal effects between two traits with
possibly pleiotropic genetic variants. Our proposal can be coupled with many
state-of-art MR methods. We provide theoretical guarantees on the Type I error
and power of the proposed methods. We demonstrate the robustness of the
proposed methods using several simulated and real datasets.
