We show that the application of flexible semi-parametric statistical techniques enables significant improvements in model fitting of macroeconomic models. As applied to the explanation of the past economic growth (since 1900) in US, UK and Japan, the new results demonstrate quite conclusively the non-linear relationships between capital, labour and useful energy with economic growth. They also indicate that output elasticities of capital, labour and useful energy are extremely variable over time. We suggest that these results confirm the economic intuition that growth since the industrial revolution has been driven largely by declining energy costs due to the discovery and exploitation of relatively inexpensive fossil fuel resources. Implications for the 21st century, which are also discussed briefly by exploring the implications of an ACEGES-based scenario of oil production, are as follows: (a) the provision of adequate and affordable quantities of useful energy as a pre-condition for economic growth and (b) the design of energy systems as 'technology incubators' for a prosperous 21st century.