[Paper Review] Estimating Earth's Temperature Response with Transformed and Augmented OLS

Justin Sun|arXiv (Cornell University)|Mar 14, 2026

Climate variability and models0 citations

TL;DR

The paper applies TAOLS to a multicointegration framework to estimate Equilibrium Climate Sensitivity (ECS) and finds a lower ECS range (2.12–2.49 °C) than the prevailing MLE-based estimate.

ABSTRACT

The long-term relationship between radiative forcing and surface temperature is imperative for predicting the impacts of climate change. This study employs multicointegration to characterize this relationship and uses Transformed and Augmented Ordinary Least Squares (TAOLS) to estimate the model. The main goal is to estimate the Equilibrium Climate Sensitivity (ECS), defined as the global mean surface air temperature increase following a doubling of atmospheric carbon dioxide. Our results show that the ECS lies between $2.12^{\circ}$C and $2.49^{\circ}$C, which is lower than the existing maximum likelihood estimate of $2.8^{\circ}$C. TAOLS offers a more robust and accessible tool for climate research, providing novel insights for ongoing debates about Earth's warming trajectory.

Motivation & Objective

• Motivate the need to estimate the long-run relationship between radiative forcing and surface temperature.
• Model this relationship within a multicointegrating framework to capture ocean heat uptake and delayed temperature response.
• Apply the Transformed and Augmented Ordinary Least Squares (TAOLS) method as a robust alternative to MLE-based approaches.
• Estimate the Equilibrium Climate Sensitivity (ECS) from empirical data and compare with existing estimates.

Proposed method

• Formulate a multicointegration model where radiative forcing f_t and surface temperature s_t are linked via a cointegration coefficient λ: f_t = λ s_t + q_t, with q_t stationary.
• Define cumulative series F_t, S_t, and Q_t, and derive the multicointegrating regression F_t = γ + μ t + S_t λ + s_t φ + v_t.
• Augment the model with the first difference Δs_t to reduce endogeneity: F_t = γ + μ t + S_t λ + s_t φ + Δs_t δ + e_t.
• Transform all variables using low-frequency basis functions (e.g., sine basis) to obtain transformed variables, then apply ordinary least squares (OLS) to estimate γ, μ, λ, φ, δ.
• Rely on asymptotic properties of transformed series for consistent, mixed-normal inference rather than full parametrization of short-run dynamics.

Experimental results

Research questions

• RQ1 What is the long-run relationship between radiative forcing and global mean surface temperature under a multicointegrating framework?
• RQ2 How does ocean heat uptake (captured via cumulative equilibrium errors) influence surface temperature in the long run?
• RQ3 What ECS range emerges when using TAOLS relative to MLE-based multicointegration estimates?
• RQ4 How robust are TAOLS estimates to short-run dynamics and high-frequency fluctuations in the data?

Key findings

• TAOLS yields a cointegrating coefficient λ in the range 1.488 to 1.750 (mean 1.709) for partial-efficacy forcing with Berkeley Earth temperature data, implying ECS between 2.12°C and 2.49°C (average 2.17°C).
• The TAOLS ECS range is lower than the MLE-based ECS of 2.80°C reported by Bruns, Csereklyei, and Stern (BCS 2020).
• The standard error of λ under TAOLS is generally smaller than that of the MLE, and the confidence intervals are narrower, indicating higher estimation efficiency.
• The multicointegration parameter φ suggests that approximately 1–3% of total heat content contributes to atmospheric warming, aligning with independent estimates of ocean heat uptake dominance.
• The results indicate that TAOLS is robust to short-term fluctuations (e.g., El Niño, volcanic events) and does not rely on strict normality assumptions as MLE does.