Empathetic conversational models have been shown to improve user satisfaction and task outcomes in numerous domains. In Psychology, persona has been shown to be highly correlated to personality, which in turn influences empathy. In addition, our empirical analysis also suggests that persona plays an important role in empathetic conversations. To this end, we propose a new task towards persona-based empathetic conversations and present the first empirical study on the impact of persona on empathetic responding. Specifically, we first present a novel large-scale multi-domain dataset for persona-based empathetic conversations. We then propose CoBERT, an efficient BERT-based response selection model that obtains the state-of-the-art performance on our dataset. Finally, we conduct extensive experiments to investigate the impact of persona on empathetic responding. Notably, our results show that persona improves empathetic responding more when CoBERT is trained on empathetic conversations than non-empathetic ones, establishing an empirical link between persona and empathy in human conversations.