We introduce in this paper a testing approach that allows checking whether two financial institutions are systemically equivalent, with systemic risk measured by CoVaR (Adrian and Brunnermeier, 2011). The test compares the difference in CoVaR forecasts for two financial institutions via a suitable loss function that has an economic content. Our testing approach differs from those in the literature in the sense that it is conditional, and helps evaluating in a forward-looking manner, the extent to which statistically significant differences in CoVaR forecasts can be attributed to lag values of market state variables. Moreover, the test can be used to identify systemically important financial institutions (SIFIs). Extensive Monte Carlo simulations show that the test has desirable small sample properties. With an application on a sample including 70 large U.S. financial institutions, our conditional test using market state variables such as VIX and various yield spreads, reveals more (resp. less) heterogeneity in the systemic profiles of these institutions compared to its unconditional version, in crisis (resp. non-crisis) period. It also emerges that the systemic ranking provided by our testing approach is a good forecast of a financial institution’s sensitivity to a crisis. This is in contrast to the ranking obtained directly using CoVaR forecasts which has less predictive power because of estimation uncertainty.