The BOLD response to a hypercapnic challenge, i.e. cerebrovascular reactivity (CVR), may vary between individuals and tissue types. Linear regression (GLM) between the BOLD signal and the end-tidal CO2 is the most common CVR processing method but does not allow for different haemodynamic responses across the brain. We propose to use shift-invariant dictionary learning (SIDL) as a promising method to enable data-driven extraction of BOLD response(s). We show that CVR and delay estimates from SIDL are comparable to estimates from GLM. Future work will focus on reducing the effect of drift on SIDL estimates.