arXiv subjects

Eric Hillebrand

Publications and source records attributed to Eric Hillebrand.

12 recordsLinked to original sources

arXiv2026

Auditing the Global Carbon Budget: Exploring the 2024--2025 Vintage Shift

The Global Carbon Budget (GCB), the community reference dataset for the carbon cycle, is reissued annually. The 2025 release introduces several adjustments to the published series that we compare with prior releases starting in 2017. On a common $1959$--$2016$ sample, the mean of the GCB budget imbalance jumps from within $\pm 0.17$~GtC/yr of zero for every vintage $2017$--$2024$ to $+0.61$~GtC/yr in 2025, the only vintage whose $95\%$ confidence interval for the imbalance mean excludes zero. We document and explore this shift in two ways. First, we conduct a model-free analysis, where we attribute the shift to a new adjustment that places the published land sink $0.40$~GtC/yr below its ensemble mean (the average of the underlying models), a smaller adjustment in the ocean sink in the opposite direction, and the removal of one model from the bookkeeping ensemble. Second, we consider the dynamic statistical GCB model of \cite{BHK2023}, augmented with climate covariates. Its parameters are estimated for every GCB vintage 2017--2025. The coefficients of atmospheric concentrations in the sink equations shift sharply on the 2025 issue in opposite directions, mirroring the model-free findings. There is a persistent drifting imbalance across the entire sample in the budget equation. In case the intended effect of the adjustments to the 2025 vintage is to reduce the mean of the budget imbalance on the window of the last ten years, our results show that this comes at the cost of increased budget imbalance over the whole sample and inconsistency of the data record. We argue that the costs of the adjustments outweigh the benefits of a narrow view on the last ten years and are detrimental to statistical analysis of the full GCB sample.

stat.AP↗

arXiv2026

Continuous-time state space analysis of d18O, d13C, and CO2 in the Cenozoic Era

We develop a continuous-time state-space framework for the joint reconstruction of three Cenozoic climate proxies, benthic foraminiferal d18O and d13C and atmospheric CO2, from irregularly and unevenly sampled multi-site, multi-method data spanning the last 67 million years. The latent signals follow a trivariate random walk in continuous time; the measurement equation differentiates the error variance by drill site for the isotopes and by proxy group for CO2, with bias intercepts placing all sources on a common scale, and the transition equation lets the innovation covariance and a deterministic La2004 Milankovitch forcing depend on the prevailing climate state. All parameters are estimated by maximum likelihood through the Kalman filter with diffuse initialization. The estimated cross-proxy correlations reverse sign between the early Cenozoic greenhouse and the icehouse, the orbital sensitivity of the isotopes strengthens as continental ice sheets grow, and the reconstructed CO2 path, reported with calibrated confidence bands, places the atmospheric CO$_2$ thresholds of the major Cenozoic glaciations in relation to present-day concentrations.

stat.AP↗

arXiv2026

Property, Interest, and Money: Is Heinsohn and Steiger's Property Premium a Determinant of Interest?

Heinsohn and Steiger's "Eigentum, Zins und Geld" (1996) proposes the property premium as the foundational determinant of interest, replacing time preference. This paper examines whether the replacement succeeds. It does not. The two arguments against time preference, the savings-inelasticity claim after Hahn and the portfolio-shift claim after Keynes, both fail on standard microeconomic grounds. With time preference intact, the property premium sits within the standard decomposition of the interest rate. In ordinary collateralized credit it coincides with the risk premium. Only when the lender is a money-issuing bank with a real redemption obligation does a third term enter the decomposition that standard asset-pricing theory does not articulate. That third term is Heinsohn and Steiger's genuine contribution. The paper discusses its apparent disappearance or disguised operation after 2008, and the circularity of a property anchor measured in money.

econ.GN↗

arXiv2025

Teaching Mathematics for Economists

In this chapter, I discuss teaching mathematical tools specifically tailored for economics students. A typical one-semester course in this area seeks to blend a range of topics: from foundational elements of subjects such as linear algebra and multivariate calculus to intermediate areas like real and convex analysis and further into advanced topics such as dynamic optimization in both continuous and discrete time. This breadth of coverage corresponds to material usually spread across multiple years in traditional mathematics programs. Given the comprehensive nature of these courses, careful selection of topics is essential, balancing numerous trade-offs. I discuss potential course sequences and instructional design choices. I then focus on conceptualizing and explaining mathematical modeling in economics. I reflect on three years of teaching an advanced undergraduate course in mathematical methods online. The latter part of the chapter offers examples and visualizations I have found particularly beneficial for imparting intuition to economics students. They cover a range of topics at different degrees of difficulty and are meant as a resource for instructors in Mathematics for Economists. Among these, I use the Ramsey model as a recurring example, especially relevant when designing a mathematical tools course with an orientation towards preparing students for macroeconomic analysis.

math.HO↗

arXiv2024

The Global Carbon Budget as a cointegrated system

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.

stat.AP↗

arXiv2024

Estimating breakpoints between climate states in the Cenozoic Era

This study presents a statistical time-domain approach for identifying transitions between climate states, referred to as breakpoints, using well-established econometric tools. We analyze a 67.1 million year record of the oxygen isotope ratio delta-O-18 derived from benthic foraminifera. The dataset is presented in Westerhold et al. (2020), where the authors use recurrence analysis to identify six climate states. Fixing the number of breakpoints to five, our procedure results in breakpoint estimates that closely align with those identified by Westerhold et al. (2020). By treating the number of breakpoints as a parameter to be estimated, we provide the statistical justification for more than five breakpoints in the time series. Further, our approach offers the advantage of constructing confidence intervals for the breakpoints, and it allows for testing the number of breakpoints present in the time series.

stat.AP↗

arXiv2024

A Regression-Based Approach to the CO2 Airborne Fraction: Enhancing Statistical Precision and Tackling Zero Emissions

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 $47.0\%$ ($\pm 1.1\%$; $1 σ$), 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.

stat.AP↗

arXiv2024

A Statistical Reduced Complexity Climate Model for Probabilistic Analyses and Projections

We propose a new statistical reduced complexity climate model. The centerpiece of the model consists of a set of physical equations for the global climate system which we show how to cast in non-linear state space form. The parameters in the model are estimated using the method of maximum likelihood with the likelihood function being evaluated by the extended Kalman filter. Our statistical framework is based on well-established methodology and is computationally feasible. In an empirical analysis, we estimate the parameters for a data set comprising the period 1959-2022. A likelihood ratio test sheds light on the most appropriate equation for converting the level of atmospheric concentration of carbon dioxide into radiative forcing. Using the estimated model, and different future paths of greenhouse gas emissions, we project global mean surface temperature until the year 2100. Our results illustrate the potential of combining statistical modelling with physical insights to arrive at rigorous statistical analyses of the climate system.

stat.AP↗

arXiv2024

Continuous-time state-space methods for d18O and d13C in the Cenozoic Era

Time series analysis of d18O and d13C from benthic foraminifera for paleoclimatology poses significant challenges. The data span tens of millions of years, with sparse early records, dense later ones, uneven time stamps, and occasional multiples. These time series are largely non-stationary, exhibiting temporary, varying trends. We propose a continuous-time state space framework that handles these irregularities effectively. Univariate signal-plus-noise models are specified for d18O and d13C, with parameters estimated via maximum likelihood using Kalman filter recursions for signal extraction and likelihood evaluation. The framework interprets state space models as time-domain Butterworth filters. Measurement-error variances are differentiated by deep-sea drill site, including site-specific level offsets, and the record is partitioned into sub-periods reflecting the distinct climate states that drive the transition variance. Two extensions of the univariate model are explored: (i) modifying the signal specification for the Kalman filter to approximate a Butterworth filter of any order, and (ii) specifying a bivariate signal-plus-noise model for joint analysis. Results reveal substantial signal changes during the ``icehouse'' period (3.3 to 0.0006 Ma); the correlation between d18O and d13C signals is generally positive but turns negative during this period.

stat.AP↗

arXiv2024

Energy, economy, and emissions: A non-linear state space approach to projections

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.

stat.AP↗

arXiv2023

Is there evidence of a trend in the CO2 airborne fraction?

In a paper recently published in this journal, van Marle et al. (van Marle et al., 2022) introduce an interesting new data set for land use and land cover change CO2 emissions (LULCC) that they use to study whether a trend is present in the airborne fraction (AF), defined as the fraction of CO2 emissions remaining in the atmosphere. Testing the hypothesis of a trend in the AF has attracted much attention, with the overall consensus that no statistical evidence is found for a trend in the data (Knorr, 2009; Gloor et al., 2010; Raupach et al., 2014; Bennedsen et al., 2019). In their paper, van Marle et al. analyze the AF as implied by three different LULCC emissions time series (GCP, H&N, and their new data series). In a Monte Carlo simulation study based on their new LULCC emissions data, van Marle et al. find evidence of a declining trend in the AF. In this note, we argue that the statistical analysis presented in van Marle et al. can be improved in several respects. Specifically, the Monte Carlo study presented in van Marle et al. is not conducive to determine whether there is a trend in the AF. Further, we re-examine the evidence for a trend in the AF by using a variety of different statistical tests. The statistical evidence for an uninterrupted (positive or negative) trend in the airborne fraction remains mixed at best. When allowing for a break in the trend, there is some evidence for upward trends in both subsamples.

stat.AP↗

arXiv2022

Global temperature projections from a statistical energy balance model using multiple sources of historical data

This paper estimates the two-component energy balance model as a linear state space system (EBM-SS model) using historical data. It is a joint model for the temperature in the mixed layer, the temperature in the deep ocean layer, and radiative forcing. The EBM-SS model allows for the modeling of non-stationarity in forcing, the incorporation of multiple data sources for the latent processes, and the handling of missing observations. We estimate the EBM-SS model using observational datasets at the global level for the period 1955 - 2020 by maximum likelihood. We show in the empirical estimation and in simulations that using multiple data sources for the latent processes reduces parameter estimation uncertainty. When fitting the EBM-SS model to eight observational global mean surface temperature (GMST) anomaly series, the physical parameter estimates and the GMST projection under Representative Concentration Pathway (RCP) scenarios are comparable to those from Coupled Model Intercomparison Project 5 (CMIP5) models and the climate emulator Model for the Assessment of Greenhouse Gas Induced Climate Change (MAGICC) 7.5. This provides evidence that utilizing a simple climate model and historical records alone can produce meaningful GMST projections.

stat.AP↗