The radial velocity technique is one of the two main approaches for detecting planets outside our solar system, or exoplanets as they are known in astronomy. When a planet orbits a star it causes the star to move and this induces a Doppler shift (i.e. the star light appears redder or bluer than expected), and it is this effect that the radial velocity method attempts to detect. Unfortunately, these Doppler signals are typically contaminated by various "stellar activity" phenomena, such as dark spots on the star surface. A principled approach to recovering planet Doppler signals in the presence of stellar activity was proposed by Rajpaul et al. (2015), and involves the use of dependent Gaussian processes to jointly model the corrupted Doppler signal and multiple proxies for stellar activity. We build on this work in two ways: (i) we propose using dimension reduction techniques to construct more informative stellar activity proxies; (ii) we extend the Rajpaul et al. (2015) model to a larger class of models and use a model comparison procedure to select the best model for the particular stellar activity proxies at hand. Our approach results in substantially improved statistical power for planet detection than using existing stellar activity models in the astronomy literature. Future work will move beyond our current class of models by making use of kernel-learning methods.