Focusing on eight emerging markets from South Asia to South America, this paper analyzes three risk spillovers - flight to quality, flight from quality and financial contagion - between emerging market stocks and the U.S. bonds. In doing so, it employs Granger causality tests in moments developed by Chen (2016) which distinctly allow for examining causality from the left tail of one distribution to the right tail of another distribution, and vice versa. It has a sample of daily closing prices for a period of more than 14 years, from 01/01/2002 to 26/02/2016, but also uses an additional 403 observations up to 14/09/2017 for out-of-sample tests. Besides, it conducts the Balcilar et al. (2016, 2017) and the Hong (2001) Granger causality in mean, variance and quantiles tests and compare their results with those of Chen. Its findings suggest that Chen's test results outperform the others in terms of robustness and reveal that the U.S. monetary policy could indeed influence investors willing to park their money in emerging markets.