We study the relation between the BRENT and seventeen stock market indexes of important oil-dependent economies. We focus on connectedness between these markets and characterize the dynamics of transmission and reception. We use LASSO methods to shrink, select, and estimate the high dimensional network linking these markets between August, 1999 and March, 2018. This methodological innovation allows the inclusion of a significantly larger number of markets in the network, providing finer results regarding connectedness in the oil-stock market nexus. We show that transmission runs mainly from stock markets to the BRENT. Connectedness varies considerably over time, reaching peaks during times of financial distress. Dynamic predictive causality tests show evidence of time-varying bidirectional causality. Causality from stock markets to the BRENT is detected mostly for the last part of the sample period. This finding indicates that the impact of stock market developments on oil markets is growing over time.