Directional Energy Emissions Responses Testing Proportionality and Symmetry in Provincial Carbon Accounting

Accurate energy-to-carbon conversion is essential for evaluating provincial energy transitions and emissions outcomes. This study examines whether annual changes in carbon emissions respond proportionally to energy consumption and whether the relationship differs between energy expansions and contractions. We combine data-consistency diagnostics with a directional two-way fixed-effects model for 30 Chinese provincial-level units over 2013–2022. The analytical sample comprises 255 annual changes, including 147 expansions and 108 contractions. The estimated expansion elasticity is 0.297 (province-clustered standard error = 0.107), rejecting proportionality under restricted wild-cluster bootstrap inference (p<0.001). This finding is sustained across alternative model specifications, weighting schemes, sample restrictions, and carbon-factor screening thresholds. The contraction elasticity is 0.129 (95% confidence interval: [−0.229,0.488]; p=0.443), while the expansion contraction difference is 0.168 (p=0.420). Sensitivity analyses characterize the variation in these latter estimates across screening choices. Regional interaction tests do not detect systematic differences across four predefined macro-regions. An external comparison with published CEADs accounts shows strong concordance in provincial emissions levels over the common 2013–2015 period. Together, the findings establish a stable sub-proportional association between energy expansion and emissions growth and provide an evidence-based framework for assessing directional energy-to-carbon relationships in provincial accounting.