  The global fraction of anthropogenically emitted carbon dioxide (CO$_2$) that
stays in the atmosphere, the CO$_2$ airborne fraction, has been fluctuating
around a constant value over the period 1959 to 2022. The consensus estimate of
the airborne fraction is around $44\%$; the remaining $56\%$ is absorbed by the
oceanic and terrestrials biospheres. In this study, we show that the
conventional estimator of the airborne fraction, based on a ratio of changes in
atmospheric CO$_2$ concentrations and CO$_2$ emissions, suffers from a number
of statistical deficiencies, such as non-existence of moments and a
non-Gaussian limiting distribution. We propose an alternative regression-based
estimator of the airborne fraction that does not suffer from these
deficiencies. We show that the regression-based estimator has a Gaussian
limiting distribution and reduces estimation uncertainty substantially. Our
empirical analysis leads to an estimate of the airborne fraction over
1959--2022 of $47.0\%$ ($\pm 1.1\%$; $1 \sigma$), implying a higher, and better
constrained, estimate than the current consensus. Using climate model output,
we show that a regression-based approach provides sensible estimates of the
airborne fraction, also in future scenarios where emissions are at or near
zero.
