This paper searches for stochastic trends and returns predictability in key energy asset markets in Europe over the last decade. The financial assets include Intercontinental Exchange Futures Europe (ICE-ECX) carbon emission allowances (the main driver of interest), European Energy Exchange (EEX) Coal ARA futures and ICE Brent oil futures (reflecting the two largest energy sources in Europe), Stoxx600 Europe Oil and Gas Index (the main energy stock index in Europe), EEX Power Futures (representing electricity), and Stoxx600 Europe Renewable Energy index (representing the sunrise energy industry). This paper finds that the Moving Average (MA) technique beats random timing for carbon emission allowances, coal, and renewable energy. In these asset markets, there seems to be significant returns predictability of stochastic trends in prices. The results are mixed for Brent oil, and there are no predictable trends for the Oil and Gas index. Stochastic trends are also missing in the electricity market as there is an ARFIMA-FIGARCH process in the day-ahead power prices. The empirical results are interesting for several reasons. We identified the data generating process in EU electricity prices as fractionally integrated (0.5), with a fractionally integrated Generalized AutoRegressive Conditional Heteroscedasticity (GARCH) process in the residual. This is a novel finding. The order of integration of order 0.5 implies that the process is not stationary but less non-stationary than the non-stationary I(1) process, and that the process has long memory. This is probably because electricity cannot be stored. Returns predictability with MA rules requires stochastic trends in price series, indicating that the asset prices should obey the I(1) process, that is, to facilitate long run returns predictability. However, all the other price series tested in the paper are I(1)-processes, so that their returns series are stationary. The empirical results are important because they give a simple answer to the following question: When are MA rules useful? The answer is that, if significant stochastic trends develop in prices, long run returns are predictable, and market timing performs better than does random timing.