We estimate a large Bayesian time-varying parameter vector autoregressive (TVP-VAR) model of daily stock return volatilities for 35 U.S. and European financial institutions. Based on that model we extract a connectedness index in the spirit of Diebold and Yilmaz (2014) (DYCI). We show that the connectedness index from the TVP-VAR model captures abrupt turning points better than the one obtained from rolling-windows VAR estimates. As the TVP-VAR based DYCI shows more pronounced jumps during important crisis moments, it captures the intensification of tensions in financial markets more accurately and timely than the rolling-windows based DYCI. Finally, we show that the TVP-VAR based index performs better in forecasting systemic events in the American and European financial sectors as well.