This study complements the current literature, providing a thorough investigation of the lead-lag connection between stock indices and sovereign credit default swap (CDS) returns for 14 European countries and the US over the period 2004-2016. We use a rolling VAR framework that enables us to analyse the connection process over time covering both crisis and non-crisis periods. In addition, we analyse the relationship between stock market volatility and CDS returns. We find that the connection between the credit and equity markets does exist and that it is time variable and seems to be related to financial crises. We also observe that stock market returns anticipate sovereign CDS returns, and sovereign CDSs anticipate the conditional volatility of equity returns, closing a connectedness circle between markets. Contribution percentages in terms of returns are more intense in the US than in Europe and the opposite result is found with respect to volatilities. Within Europe, a greater impact in Eurozone countries compared to non-Eurozone countries is observed. Finally, an additional analysis is also carried out for the financial sector, obtaining results largely consistent with those found using sovereign data.