We propose a non-linear state-space model to examine the relationship between CO$_2$ emissions, energy sources, and macroeconomic activity, using data from 1971 to 2019. CO$_2$ emissions are modeled as a weighted sum of fossil fuel use, with emission conversion factors that evolve over time to reflect technological changes. GDP is expressed as the outcome of linearly increasing energy efficiency and total energy consumption. The model is estimated using CO$_2$ data from the Global Carbon Budget, GDP statistics from the World Bank, and energy data from the International Energy Agency (IEA). Projections for CO$_2$ emissions and GDP from 2020 to 2100 from the model are based on energy scenarios from the Shared Socioeconomic Pathways (SSP) and the IEA's Net Zero roadmap. Emissions projections from the model are consistent with these scenarios but predict lower GDP growth. An alternative model version, assuming exponential energy efficiency improvement, produces GDP growth rates more in line with the benchmark projections. Our results imply that if internationally agreed net-zero objectives are to be fulfilled and economic growth is to follow SSP or IEA scenarios, then drastic changes in energy efficiency, not consistent with historical trends, are needed.

翻译：我们提出一种非线性状态空间模型，利用1971年至2019年的数据，探究CO₂排放、能源来源与宏观经济活动之间的关系。CO₂排放被建模为化石燃料使用的加权总和，其排放转换因子随时间演变以反映技术变革。GDP则被表示为线性增长的能源效率与总能源消耗共同作用的结果。该模型使用全球碳预算项目的CO₂数据、世界银行的GDP统计以及国际能源署（IEA）的能源数据进行估计。基于共享社会经济路径（SSP）和IEA净零路线图的能源情景，模型生成了2020年至2100年CO₂排放与GDP的预测。模型给出的排放预测与这些情景一致，但预测的GDP增长较低。另一种模型版本假设能源效率呈指数增长，其产生的GDP增长率与基准投影更为吻合。我们的结果表明，若要实现国际社会约定的净零目标，且经济增长遵循SSP或IEA情景，则需要能源效率发生与历史趋势不符的剧烈变化。