A regression-based approach to the CO₂ airborne fraction

The study addresses how to robustly estimate the CO₂ airborne fraction (AF), defined as the proportion of anthropogenic CO₂ emissions that remains in the atmosphere. There has been debate over whether the AF has increased, decreased, or remained constant since 1959. While early studies suggested increases, more recent work indicates a roughly constant AF around 44%. Theory shows AF can be constant under exponential emissions and linear sink uptake. The conventional estimator, based on yearly ratios of atmospheric CO₂ growth to emissions, may yield invalid inference due to non-stationarity and trending behavior in both series, and it can fail when emissions approach zero—a crucial concern for future low- or net-zero-emission scenarios. The purpose is to develop and justify a regression-based estimator with sound statistical properties that can handle both historical and future conditions, including time-varying AF.

Earlier studies reported possible increases in AF but with notable uncertainty. Subsequent analyses converged on an approximately constant AF near 44%. Raupach provided a theoretical basis for constant AF under exponential emissions and linear sink response; Bennedsen et al. formalized such systems statistically and reported AF ≈ 0.44. The conventional literature commonly estimates AF as the sample mean of yearly ratios G/E, and an alternative cumulative airborne fraction (CAF) has been proposed but is less used and less tractable statistically. Climate model studies indicate AF may increase under high-emission futures and decrease under low-emission futures, motivating methods that can accommodate time-varying AF and handle near-zero emissions.

Data: Yearly changes in atmospheric CO₂ (G), fossil-fuel CO₂ emissions (EFF), land-use and land-cover change emissions (ELULCC), with total emissions Et = EFF + ELULCC, all in GtC/yr, from the Global Carbon Project for 1959–2022. ENSO and volcanic activity indices (VAI) are included as covariates. For future analysis, trajectories of Gt and Et are generated by the reduced-complexity model MAGICC under SSP scenarios (e.g., SSP1-2.6). To emulate observational variability, Gaussian noise with variances estimated from historical data is added to MAGICC outputs. Statistical framework: The authors test for non-stationarity with Dickey–Fuller tests, then examine cointegration between Gt and Et via Engle–Granger. Cointegration implies a stable long-run relationship Gt = αEt + ut with stationary residuals ut, supporting constant AF on 1959–2022. Estimators for constant AF: (1) Ratio-based model Gt/Et = α + ut, estimated by the sample mean or OLS intercept. (2) Regression-based model Gt = αEt + ut, estimated by OLS slope. Standard errors are computed using heteroskedasticity and autocorrelation consistent (HAC; Newey–West) covariance estimators. Models are extended with covariates: (3) Gt/Et = α + γ1 ENSO + γ2 VAI + ut; (4) Gt = αEt + γ1 ENSO + γ2 VAI + ut. Robustness is assessed with subsample (1992–2022) analysis and Deming regression to account for measurement error. Asymptotics: Under emissions following a random walk with drift Et = E0 + bt + xt, the regression-based estimator is shown to be consistent and converges at rate T^{3/2}, while the ratio-based estimator converges at rate T. If Et has positive density at zero, the ratio-based estimator lacks finite moments; additionally, a Gaussian error assumption is needed for its CLT, whereas the regression-based estimator attains a CLT without requiring Gaussianity of ut. Time-varying AF: AF is modeled as αt. A ratio-based time-varying estimate α1,t uses Gt/Et directly, while a regression-based state-space model with αt estimated by the Kalman filter/smoother provides smoothed trajectories and uncertainty bands. ENSO and VAI can be incorporated as regressors in the state-space framework.

• G and E are trending and non-stationary; Engle–Granger tests confirm cointegration of Gt and Et over 1959–2022, implying a statistically constant AF in the historical period. Residuals appear Gaussian (Jarque–Bera p ≈ 0.24).
• Asymptotic properties: Regression-based AF estimator converges at rate T^{3/2}; ratio-based at rate T. The ratio-based estimator may lack finite moments if Et can be zero and requires Gaussianity for standard CLT-based inference.
• Historical AF estimates (Global Carbon Project, 1959–2022): • Without covariates: ratio-based α1 = 0.4386 (SE 0.0159), regression-based α2 = 0.4478 (SE 0.0141), with α2 having ~11% lower SE than α1. • With ENSO and VAI: ratio-based α3 = 0.4716 (SE 0.0126), regression-based α4 = 0.4697 (SE 0.0105). The regression-based SE is ~16% lower than the ratio-based with covariates and ~34% lower than the conventional ratio without covariates. • Preferred estimate: AF = 47.0% ± 1.1% (1σ), 95% CI [44.9%, 49.0%], from regression with ENSO and VAI. • Excluding ENSO/VAI, regression-based AF ≈ 44.8% ± 1.4% (1σ), close to the consensus ~44%. • Subsample 1992–2022: AF ≈ 46% ± 1.0% (1σ); results consistent with full sample.
• Inclusion of ENSO and VAI substantially reduces residual variance and increases R², improving precision of AF estimates.
• Future scenarios (MAGICC SSP1-2.6 with perturbed noise): Ratio-based α1,t is highly noisy, especially when Et ≈ 0. Regression-based α2,t from a Kalman smoother yields stable, interpretable AF with confidence bands. Under SSP1-2.6, α̂t is roughly constant until ~2050, then declines toward zero; Gt turns negative around 2060 (implying negative AF), emissions turn negative by ~2077, after which AF estimates exceed one, reflecting continued sink uptake despite negative emissions. SSP1-1.9 shows similar early behavior but maintains α̂t < 1 after ~2080.

The findings support a statistically constant AF over 1959–2022 and indicate that the regression-based estimator offers clear advantages over the conventional ratio-based approach: faster convergence, valid CLT without assuming Gaussian errors, existence of moments even when emissions are zero, and improved finite-sample precision, especially when controlling for ENSO and volcanic activity. The preferred estimate, AF ≈ 47%, is slightly higher than the conventional consensus (~44%) and is better constrained. For future analyses where emissions may decline to or below zero, the regression-based state-space approach provides stable time-varying AF estimates and associated confidence intervals, overcoming the instability and inferential challenges of the ratio-based method. These results are robust across subsamples, alternative data sets, and measurement-error–robust (Deming) regressions.

This study proposes and validates a regression-based framework to estimate the CO₂ airborne fraction that corrects key statistical shortcomings of the traditional ratio-based estimator. Empirically, the regression-based approach yields a higher and more precise historical AF estimate (≈47% over 1959–2022 with ENSO and VAI controls) and demonstrates robustness across samples and data sets. The method extends naturally to time-varying AF using a state-space model with Kalman filtering/smoothing, enabling reliable analysis under future low- and net-zero-emission scenarios. Future research can apply this framework across diverse climate model outputs and scenarios, refine covariate treatments for internal climate variability and external forcings, and explore structural changes in carbon sink dynamics.