The Global Carbon Budget as a cointegrated system

Eric Hillebrand; Mikkel Bennedsen; Morten {\O}rregaard Nielsen

arxiv: 2412.09226 · v4 · submitted 2024-12-12 · 📊 stat.AP · econ.EM

The Global Carbon Budget as a cointegrated system

Mikkel Bennedsen , Eric Hillebrand , Morten {\O}rregaard Nielsen This is my paper

Pith reviewed 2026-05-23 07:24 UTC · model grok-4.3

classification 📊 stat.AP econ.EM

keywords cointegrationglobal carbon budgetanthropogenic CO2 emissionsatmospheric CO2carbon sinkserror correction modelstochastic trendtime series

0 comments

The pith

Anthropogenic CO2 emissions act as the single stochastic trend driving a cointegrated system of four global carbon budget time series.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper treats the four annual time series of the global carbon budget as one multivariate statistical system. Tests establish that the series are cointegrated with rank three and that only anthropogenic emissions supply the common nonstationary trend. The three stable relations recovered from the data line up with the physical identities that link the land and ocean sinks linearly to atmospheric concentrations and that equate concentration changes to emissions minus combined uptake. Because a restricted error-correction model that imposes exactly these identities is not rejected by likelihood ratio tests, the same structure can be used to generate projections under socioeconomic scenarios.

Core claim

The four time series are cointegrated with rank three, with anthropogenic CO2 emissions identified as the single stochastic trend driving the nonstationary dynamics of the system. The three cointegrating relations correspond to the physical relations that the sinks are linearly related to atmospheric concentrations and that the change in concentrations equals emissions minus the combined uptake by land and ocean. Likelihood ratio tests show that a parametrically restricted error-correction model that embodies these physical relations cannot be rejected on the data.

What carries the argument

Johansen cointegration rank test and likelihood ratio comparison of a physically restricted vector error-correction model against its unrestricted counterpart, applied to the four annual series beginning in 1959.

If this is right

The system admits a representation in which emissions act as the sole exogenous driver of the nonstationary components.
The restricted model passes statistical checks and therefore remains compatible with the observed historical record.
Out-of-sample projections generated from the model under Shared Socioeconomic Pathways scenarios remain consistent with results from established climate models.
The three cointegrating relations recovered statistically match the accounting identities used in carbon-cycle accounting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Short-run deviations from the estimated relations could be monitored as indicators of temporary changes in sink strength.
The same error-correction structure might be extended to include temperature or other climate variables as additional endogenous series.
Scenario analysis could be performed inside the cointegrated system rather than by coupling separate carbon-cycle and economic modules.

Load-bearing premise

The four series are each integrated of order one and the cointegrating relations remain linear and constant over time.

What would settle it

Rejection of the restricted error-correction model by the likelihood ratio test at standard significance levels, or a cointegration rank estimate different from three when the same four series are re-tested.

read the original abstract

The Global Carbon Budget, maintained by the Global Carbon Project, summarizes Earth's global carbon cycle through four annual time series beginning in 1959: atmospheric CO$_2$ concentrations, anthropogenic CO$_2$ emissions, and CO$_2$ uptake by land and by ocean. We analyze these four time series as a multivariate (cointegrated) system. Statistical tests show that the four time series are cointegrated with rank three and identify anthropogenic CO$_2$ emissions as the single stochastic trend driving the nonstationary dynamics of the system. The three cointegrated relations correspond to the physical relations that the sinks are linearly related to atmospheric concentrations and that the change in concentrations equals emissions minus the combined uptake by land and ocean. Furthermore, likelihood ratio tests show that a parametrically restricted error-correction model that embodies these physical relations cannot be rejected on the data. The model can be used for both in-sample and out-of-sample analysis. In an application of the latter, we demonstrate that projections based on this model, using Shared Socioeconomic Pathways scenarios, yield results consistent with established climate science.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper runs Johansen cointegration on the four GCP series, finds rank 3 with emissions as the sole trend, and shows the physical restrictions are not rejected by LR test, but the results rest on unverified I(1) assumptions with only ~65 observations.

read the letter

The core finding is that the atmospheric CO2, emissions, land sink, and ocean sink series are cointegrated with rank three, emissions supply the single stochastic trend, and the error-correction model that imposes the three physical accounting relations cannot be rejected. They then use the restricted model for SSP-based projections that match existing climate results. This is a straightforward application of the Johansen framework to the Global Carbon Project data, and the non-rejection of the restricted model is the part that actually connects the econometrics to the physical equations without assuming them by construction. The exercise is new on this particular dataset and the LR test supplies a data-driven check rather than a tautology. The main limitation is the sample. Annual observations from 1959 give roughly 65 points. Both the trace test for rank and the LR test for the restrictions have their usual distributions only when every series is exactly I(1), the cointegrating space is constant, and the lag order is adequate. The abstract supplies none of the usual unit-root diagnostics, break tests, or lag-robustness checks. If any series is I(0) or I(2), or if there are unmodeled shifts, the reported rank and non-rejection are not reliable. That concern from the stress-test note stands. Readers who already work with cointegrated systems on environmental time series will find the exercise useful as a check on whether the carbon-budget identities survive statistical scrutiny. It will not alter climate modeling practice. The paper deserves referee time so the authors can supply the missing integration-order evidence and any sensitivity results; the underlying thinking is clear and the application is direct.

Referee Report

3 major / 2 minor

Summary. The manuscript treats the four annual Global Carbon Budget time series (atmospheric CO2 concentration, anthropogenic emissions, land uptake, ocean uptake) from 1959 as a cointegrated system. Johansen trace tests are reported to establish cointegration rank three with anthropogenic emissions as the single stochastic trend; the three cointegrating relations are shown to match the physical accounting identities (sinks linear in atmospheric concentration; concentration change equals emissions minus combined sinks). Likelihood-ratio tests fail to reject a parametrically restricted VECM that imposes these relations, and the model is applied to generate out-of-sample projections under SSP scenarios that are stated to be consistent with established climate science.

Significance. If the rank-3 finding and non-rejection of the restricted VECM hold under the maintained assumptions, the work supplies a formal econometric confirmation that the physical carbon-budget identities are statistically compatible with the observed non-stationary dynamics and supplies a parsimonious model for both in-sample attribution and scenario-based projection. The explicit mapping of cointegrating vectors to physical relations and the use of LR tests to evaluate those restrictions are strengths that distinguish the analysis from purely descriptive accounting exercises.

major comments (3)

[Methods / Cointegration rank section] The validity of the reported trace-test rank-3 conclusion and the subsequent LR comparison of restricted versus unrestricted VECM rests on all four series being exactly I(1) with time-invariant linear cointegration; no unit-root diagnostics, integration-order tests, or I(2) analysis are described, leaving the limiting distributions of the test statistics unverified for the ~65-observation annual sample.
[VECM specification / Results] Lag-order selection, deterministic-term specification (constant, trend, seasonal), and outlier handling are not reported; these choices directly affect the finite-sample size and power of both the trace test and the LR test for the physical restrictions, yet no robustness tables or alternative specifications are provided.
[Empirical results / Robustness] With only ~65 annual observations, the paper does not report small-sample corrections, bootstrap p-values, or subsample stability checks for the rank-3 finding or the non-rejection of the restricted model; such checks are load-bearing because the central claim that the physical relations are data-supported depends on the tests having correct size.

minor comments (2)

[Abstract] The abstract states that the restricted model 'cannot be rejected' but does not report the numerical value of the LR statistic or its degrees of freedom, which would allow readers to assess the strength of the non-rejection.
[Model section] Notation for the cointegrating vectors and the loading matrix in the VECM could be aligned more explicitly with the physical identities to improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Methods / Cointegration rank section] The validity of the reported trace-test rank-3 conclusion and the subsequent LR comparison of restricted versus unrestricted VECM rests on all four series being exactly I(1) with time-invariant linear cointegration; no unit-root diagnostics, integration-order tests, or I(2) analysis are described, leaving the limiting distributions of the test statistics unverified for the ~65-observation annual sample.

Authors: We agree that explicit confirmation of integration order strengthens the analysis. Although the I(1) character of these series is standard in the carbon-cycle literature, we will add Augmented Dickey-Fuller and KPSS tests for each series, together with a brief I(2) analysis using the Johansen procedure, in the revised manuscript. revision: yes
Referee: [VECM specification / Results] Lag-order selection, deterministic-term specification (constant, trend, seasonal), and outlier handling are not reported; these choices directly affect the finite-sample size and power of both the trace test and the LR test for the physical restrictions, yet no robustness tables or alternative specifications are provided.

Authors: We accept that these modeling choices must be documented. In revision we will report the information criteria used for lag selection, the deterministic specification retained, and any outlier treatment, and we will add a robustness table comparing rank and restriction-test results across plausible lag lengths and deterministic-term choices. revision: yes
Referee: [Empirical results / Robustness] With only ~65 annual observations, the paper does not report small-sample corrections, bootstrap p-values, or subsample stability checks for the rank-3 finding or the non-rejection of the restricted model; such checks are load-bearing because the central claim that the physical relations are data-supported depends on the tests having correct size.

Authors: We acknowledge the finite-sample limitations inherent in an annual sample of this length. We will add bootstrap p-values for both the trace test and the LR test of the physical restrictions, and we will report a simple subsample stability check (pre- and post-1990). Full small-sample corrections beyond bootstrapping are not feasible without substantially altering the modeling framework. revision: partial

Circularity Check

0 steps flagged

No significant circularity; cointegration rank and LR tests are empirical outcomes on external data.

full rationale

The paper applies standard Johansen trace tests to four observed time series (atmospheric CO2, emissions, land and ocean uptake) to conclude cointegration rank 3 with emissions as the sole stochastic trend, then imposes parametric restrictions on the VECM that encode the physical accounting identities and shows via LR test that the restricted model is not rejected. These steps are data-driven statistical procedures whose results are not forced by definition, by renaming of inputs, or by load-bearing self-citation; the physical relations enter only as testable restrictions whose non-rejection is an empirical finding. No self-definitional loops, fitted-input-as-prediction, or ansatz smuggling via prior work by the same authors appear in the derivation chain. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard time-series assumptions rather than new free parameters or invented entities; the physical relations are treated as testable restrictions, not fitted constants.

axioms (2)

domain assumption The four time series are integrated of order one (I(1))
Prerequisite for applying cointegration rank tests and error-correction modeling as described.
domain assumption The cointegrating relations are linear and constant over the sample
Required for the Johansen procedure and the parametric restrictions to be correctly specified.

pith-pipeline@v0.9.0 · 5727 in / 1464 out tokens · 27306 ms · 2026-05-23T07:24:52.884328+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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and Keeling, C

Bacastow, R. and Keeling, C. D. (1973). Atmospheric carbon dioxide and radiocarbon in the natural cycle: II . changes from A. D. 1700 to 2070 as deduced from a geochemical model. In Carbon and the Biosphere Conference Proceedings , pages 86--135. Brookhaven Symposia in Biology, Upton, New York, USA

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[2] [2]

Bennedsen, M. (2021). Designing a statistical procedure for monitoring global carbon dioxide emissions. Climatic Change , 166(21):1--19

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[3] [3]

Bennedsen, M., Hillebrand, E., and Koopman, S. J. (2019). Trend analysis of the airborne fraction and sink rate of anthropogenically released CO _2 . Biogeosciences , 16(18):3651--3663

work page 2019

[4] [4]

Bennedsen, M., Hillebrand, E., and Koopman, S. J. (2021). Modeling, forecasting, and nowcasting U.S. CO _2 emissions using many macroeconomic predictors. Energy Economics , 96:105118

work page 2021

[5] [5]

Bennedsen, M., Hillebrand, E., and Koopman, S. J. (2023). A multivariate dynamic statistical model of the global carbon budget 1959–-2020. Journal of the Royal Statistical Society Series A: Statistics in Society , 186(1):20--42

work page 2023

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Bennedsen, M., Hillebrand, E., and Koopman, S. J. (2024). A regression-based approach to the CO _2 airborne fraction. Nature Communications , 15(8507):1--9

work page 2024

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Boswijk, H. P. and Doornik, J. A. (2004). Identifying, estimating and testing restricted cointegrated systems: An overview. Statistica Neerlandica , 58(4):440--465

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C., Cox, P., Eliseev, A., Henson, S., Ishii, M., Jaccard, S., Koven, C., Lohila, A., Patra, P., Piao, S., Rogelj, J., Syampungani, S., Zaehle, S., and Zickfeld, K

Canadell, J.G., Monteiro, P., Costa, M., da Cunha, L. C., Cox, P., Eliseev, A., Henson, S., Ishii, M., Jaccard, S., Koven, C., Lohila, A., Patra, P., Piao, S., Rogelj, J., Syampungani, S., Zaehle, S., and Zickfeld, K. (2021). Global carbon and other biogeochemical cycles and feedbacks. In Masson-Delmotte, V. et al., editors, Climate Change 2021: The Physi...

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G., Pataki, D

Canadell, J. G., Pataki, D. E., Gifford, R., Houghton, R. A., Luo, Y., Raupach, M. R., Smith, P., and Steffen, W. (2007). Saturation of the terrestrial carbon sink. In Canadell, J. G., Pataki, D. E., and Pitelka, L. F., editors, Terrestrial Ecosystems in a Changing World , pages 59--78. Springer Berlin Heidelberg, Berlin, Heidelberg

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University of East Anglia

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work page 2024

[11] [11]

and Lassey, K

Enting, I. and Lassey, K. (1993). Projections of Future CO _2 . CSIRO Division of Atmospheric Research Technical Paper no. 27 , http://www.cmar.csiro.au/e-print/open/enting\_2000e.pdf

work page 1993

[12] [12]

A., Wanninkhof, R., Takahashi, T., and Tans, P

Feely, R. A., Wanninkhof, R., Takahashi, T., and Tans, P. (1999). Influence of E l N i\ no on the equatorial Pacific contribution to atmospheric CO _2 accumulation. Nature , 398(6728):597--601

work page 1999

[13] [13]

Friedlingstein, P. (2015). Carbon cycle feedbacks and future climate change. Philosophical Transactions of the Royal Society A , 373:20140421

work page 2015

[14] [14]

W., Andrew, R

Friedlingstein, P., O'Sullivan, M., Jones, M. W., Andrew, R. M., Bakker, D. C. E., Hauck, J., Landsch\"utzer, P., Le Qu\'er\'e, C., Luijkx, I. T., Peters, G. P., Peters, W., Pongratz, J., Schwingshackl, C., Sitch, S., Canadell, J. G., Ciais, P., Jackson, R. B., Alin, S. R., Anthoni, P., Barbero, L., Bates, N. R., Becker, M., Bellouin, N., Decharme, B., Bo...

work page 2023

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Gifford, R. (1993). Implications of CO _2 effects on vegetation for the global carbon budget. In Heimann, M., editor, The Global Carbon Cycle , pages 159--199. Springer

work page 1993

[16] [16]

L., and Gruber, N

Gloor, M., Sarmienti, J. L., and Gruber, N. (2010). What can be learned about carbon cycle climate feedbacks from the CO _2 airborne fraction? Atmospheric Chemistry and Physics , 10:7739--7751

work page 2010

[17] [17]

Haverd, V., Smith, B., Nieradzik, L., Briggs, P., Woodgate, W., Trudinger, C., Canadell, J., and Cuntz, M. (2018). A new version of the CABLE land surface model (subversion revision r4601) incorporating land use and land cover change, woody vegetation demography, and a novel optimisation-based approach to plant coordination of photosynthesis. Geoscientifi...

work page 2018

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Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models . Oxford University Press

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Juselius, K. (2006). The Cointegrated VAR Model: Methodology and Applications . Oxford University Press

work page 2006

[20] [20]

Knorr, W. (2009). Is the airborne fraction of anthropogenic CO _2 emissions increasing? Geophysical Research Letters , 36

work page 2009

[21] [21]

Le Qu \'e r \'e , C., Raupach, M., Canadell, J., Marland, G., Bopp, L., Ciais, P., Conway, T., Doney, S., Feely, R., Foster, P., Friedlingstein, P., Gurney, K., Houghton, R., House, J., Huntingford, C., Levy, P., Lomas, M., Majkut, J., Metzl, N., Ometto, J., Peters, G., Prentice, I., Randerson, J., Running, S., Sarmiento, J., Schuster, U., Sitch, S., Taka...

work page 2009

[22] [22]

T., Conway, T

Le Qu \'e r \'e , C., R \"o denbeck, C., Buitenhuis, E. T., Conway, T. J., Langenfelds, R., Gomez, A., Labuschagne, C., Ramonet, M., Nakazawa, T., Metzl, N., Gillett, N., and Heimann, M. (2007). Saturation of the southern ocean CO _2 sink due to recent climate change. Science , 316(5832):1735--1738

work page 2007

[23] [23]

Meinshausen, M., Raper, S. C. B., and Wigley, T. M. L. (2011). Emulating coupled atmosphere-ocean and carbon cycle models with a simpler model, MAGICC 6 -- Part 1: Model description and calibration. Atmospheric Chemistry and Physics , 11(4):1417--1456

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