This study examines the martingale difference hypothesis (MDH) for the carbon emission allowance market within the European Union Emission Trading Scheme (EU ETS) during the Phase I and the Phase II, using both daily and weekly data over the 2005-2009 period. We analyze the MDH for spot prices negotiated on BlueNext, European Energy Exchange and Nord Pool along with futures prices negotiated on BlueNext and European Climate Exchange, using the new variance ratio tests developed by Kim (2009) and the generalized spectral test proposed by Escanciano and Velasco (2006). For the Phase I, the results show that the spot price changes of these three markets are predictable, suggesting the possibility of abnormal returns through speculation, except during the period April 2006 to October 2006, namely after the compliance break and before the ECs of stricter NAP II. Finally, we find that the CO2 spot and futures price changes are unpredictable during the Phase II because we failed to reject the MDH based on both daily and weekly data. Thus, these markets are found to be weak-form efficient.