Varying-coefficient stochastic differential equations (SDEs) are a useful tool to uncover mechanistic relationships underlying time series. However, in practice, the relationship between the parameters of the process of interest and covariates is often subject to regime-switching over time, which cannot readily be accommodated in basic SDEs. In this paper, we propose state-switching varying-coefficient SDEs as an extension of basic SDEs that accounts for such regime-switching over time. The suggested approach is illustrated using bond yields and stock returns, where we investigate the theory that bonds serve as safe havens against stocks.