This paper tests the hypothesis on market efficiency for returns on the euro against fifteen currencies while assuming predictability of returns, dependent on the sign and magnitude of endogenous shocks. Considering the properties of exchange rate returns, the quantile autoregression approach was selected in empirical analysis. Based on the research data sample, consisting of daily exchange rates between January first, 1999, and April thirty, 2020, the paper suggests profitable trading strategies depending on a currency pair. In the case of six out of fifteen currency pairs, exchange rate returns were found non-predictable or almost non-predictable. In the case of nine considered currency pairs, there was a significant linkage between current and past exchange rate returns, found as dependent on the sign and magnitude of endogenous shocks in exchange rate returns. Finally, the paper considered possible factors of inefficiency and suggested further research of the topic.