This study combines panel data techniques with wavelet spectral analysis to investigate the complex and controversial relationship between energy consumption and GDP. The approach in this study makes it possible to differentiate the relationship over time horizons, in addition to taking into account heterogeneity and dependence across cross-sectional units. This leads to more precise conclusions for decision-making, overcoming the limited understanding that previous studies offer when considering aggregated data for a single time horizon. Our analysis is applied to the 50 states of the U.S.A. for the period 1963 to 2017; it is then replicated for a broad set of 25 subgroups of states, based on geography, income, energy intensity, energy price and the predominant sector in the economy. Results from the Dumitrescu-Hurlin causality test show that in the short-run, there is mixed evidence on the direction of causality between energy consumption and GDP, while in the medium- and long-run there is bidirectional causality for most of the subgroups. Although there is a positive co-movement relationship between the cyclical components of these series, the existence and sign of a relationship in the very long run depends substantially on the characteristics of the states. These findings provide important implications for the analysis, formulation and implementation of economic, energy and environmental policies.