Is the influence of such fundamental drivers as electricity demand and fossil fuel prices on electricity price time- and scale-dependent? At which time-scales is the influence of electricity demand on electricity price the strongest? Starting at which time-scale do changes in the fuel markets impact the electricity price? To answer these questions, the paper proposes a novel methodology—a multi-scale adaptive regression analysis based on the time-dependent intrinsic regression. Its application is demonstrated both on simulated data and on actual data from three electricity markets: the Europe-Ural price area (ATS EU) and the Siberia price area (ATS SI) in Russia, and the APX power spot exchange in the UK. The obtained results show that the influence of fundamental factors on electricity price does depend on time-scale and for the majority of the time-scales is time-dependent. The influence of electricity demand is most prominent both in the short and long terms for ATS EU and APX, while for ATS SI its impact is significant only in the long term. Finally, for all the electricity exchanges under study, the influence of fuel markets is absent in the short term and becomes prominent only in the medium or long terms.