Statistical tests of a simple energy balance equation in a synthetic model of cotrending and cointegration
Introduction
Professor Pierre Perron has made foundational contributions in time series econometrics. It is lesser-known that he has also made significant contributions to climate science. He is one of the pioneers presenting empirical evidence of anthropogenic global warming based on cotrending analysis. This paper is closely related to his contributions in climate science.
One of the most studied implications from a simple energy balance model is the linear relationship between temperature and radiative forcing. The goal of this paper is to statistically test for the validity of this relationship using historical values of temperature and radiative forcing from the instrumental period, which is about after 1850.
The Earth absorbs energy from the Sun and emits energy back to space via blackbody radiation. The energy balance for the Earth () is the net incoming energy, which depends on temperature (), and thus . The energy balance is zero in equilibrium, . Suppose that a positive perturbation (), for example greenhouse gases, is imposed on the climate system in an equilibrium state. The climate system then responds by increasing temperature to restore equilibrium. This increase in temperature () also alters the energy balance via various physical processes. An externally imposed perturbation is called “radiative forcing” while a change induced by temperature responses is called “feedback”. The imposition of a radiative forcing and subsequent feedbacks can be expressed using the Taylor expansion of as where the first derivative stands for the feedback strength and collects higher order terms (). Assuming that the perturbation is small, there exists proportionality between and : with . A further assumption that the deep ocean is a heatsink into which the climate system transports heat gives , where is the ocean heat uptake coefficient. Then, and are proportional: where . The coefficients, and are directly linked to the two iconic numbers in climate science, the equilibrium climate sensitivity and the transient climate sensitivity.2 A number of studies estimate and from (2), (3) using historical values of , , and . However, the linear approximation in (1) is also criticized for a number of reasons.3 First, the linear approximation holds only for a small perturbation. If the imposed radiative forcing is not small, the higher order terms in (1) will not be negligible, making the proportionality in (2) invalid. Second, the first derivative may not be a constant. Feedbacks have different timescales and some of them even have a tipping point because they involve different physical processes. Thus, the constancy of the feedback strength depends on the time span and data frequency. While diverse data have been used in the literature, time series data from the instrumental period are among the most frequently used ones. In this paper, we aim at empirically investigating the validity of the linear approximation for the instrumental period. We focus on the relationship in (3) since it does not require data for .
Our testing strategy is based on the statistical characteristics of temperature and radiative forcing during the instrumental period. We use the global mean surface temperature (GMST) and the total radiative forcing (TRF). GMST is the average temperature across the surface of the sea and over the land. TRF is the sum of radiative forcing imposed by all forcing agents such as greenhouse gases, land use, albedo, etc. See Fig. 1, Fig. 2 for the time plots of these series. The most notable characteristic of both GMST and TRF is an upward trend of which slope has steepened from the 1960s and somewhat flattened since the 2000s. If the relationship in (3) is valid, GMST and TRF should share a common trend so that there exists a linear combination of GMST and TRF that is not trending. This has led to a strand of studies testing for common trends in GMST and TRF, to which econometricians have made significant contributions.
The suitability of a common trend test crucially depends on how trends are modeled statistically. In the literature, there are largely two approaches, one that uses a stochastic trend and the other a linear trend with changing slopes (broken linear trend, hereafter). In the presence of stochastic trends, cointegration tests are appropriate methods of testing. For example, see Kaufmann and Stern (2002), Kaufmann et al., 2010, Kaufmann et al., 2013, and Pretis (2020). In the presence of broken linear trends, cotrending or cobreaking tests are proper. For example, see Estrada et al., 2010, Estrada et al., 2013, Estrada and Perron (2014) and Kim et al. (2020).
Although cointegration and cotrending approaches differ in describing trends, they have the same purpose of showing the commonality of trends and deliver the same implication, the linear relationship between GMST and TRF. In fact, the commonality goes beyond being empirical evidence of an energy balance equation. Among a number of forcing agents, well-mixed-green-house-gases forcing (WMGHG) exhibits a very clear trend with almost no other variations, which also shapes the trend in TRF. WMGHG is believed to have been driven mostly by human activities. The commonality of trends in GMST and TRF implies that the anthropogenic trend created by WMGHG is imparted from TRF to GMST and thereby is causing global warming. Nonetheless, both cointegration and cotrending analyses have the same limitation that their methodological validity depends on the correctness of their modeling for the trend. When one is correct, the other is deemed to be faulty.
Our model synthesizes the stochastic trend model and the broken linear trend model by allowing both types of trends to be present. In addition, the time path of WMGHG is included as an observed trend function. This generality makes it possible to test for the commonality of trends without specifying the type of trends in TRF and GMST. More specifically, TRF is modeled to be the sum of a linear function in WMGHG and a stochastic term. While WMGHG is the most dominant component of TRF, other forcing agents under human influences exhibit a trend similar to WMGHG. The linear function in WMGHG is meant to capture all such human influences existing in TRF. The stochastic term is assumed to be either stationary or integrated of order one. GMST is allowed to have a broken linear trend, a linear function in WMGHG, or both, while its stochastic part is either stationary or integrated of order one. A broken linear trend is not allowed in TRF because once WMGHG is projected out from TRF, the remainder does not have much room for a deterministic trend.
Three conditions are needed for the linear relationship between GMST and TRF to hold. First, GMST should not have a broken linear trend since TRF does not have one. Second, there should be a linear combination of the stochastic parts of GMST and TRF that is stationary. This condition holds either when the stochastic parts are cointegrated or when both of them are stationary. For expositional brevity, we will call the last condition cointegration-stationarity. Lastly, GMST should also have a linear function in WMGHG and the linear combination in the second condition should cancel out the two linear functions in WMGHG existing in TRF and GMST.
We consider two null hypotheses, one only for the first two conditions and the other for all three conditions. The first null hypothesis is not sufficient for the linear relationship between GMST and TRF, but it is still of interest as a necessary condition.
Our test statistic is devised as the likelihood ratio with an assumption of normality and then is modified to remove nuisance parameters in the asymptotic null distribution derived under more general assumptions. The asymptotic null distribution and the required modification differ as to the existence of a stochastic trend in TRF. An adjustment similar to Park’s (1992) canonical cointegration regression (CCR) is needed for the case of a stochastic trend and only a much simpler adjustment is necessary for the case of no stochastic trend.
To test for common trends without knowing the presence or absence of a stochastic trend a priori, we combine the statistics modified in two ways using the super-efficient estimator of the sum of autoregressive coefficients from Perron and Yabu (2009). The combined test applies the CCR type adjustment if the super-efficient estimate is one and applies the simpler adjustment otherwise. Given the consistency of the super-efficient estimator, a correct adjustment is applied with probability increasing to one. The asymptotic critical values from the two cases remain close and we use the bigger one to control size for the combined test.
The proposed tests are applied to four GMST series and a forcing series that are being widely used in climate science. Our results suggest that the linear relationship between GMST and TRF is not rejected with typical sizes.
The rest of the paper is organized as follows: Section 2 provides the model and tests. Section 3 offers the asymptotic distributions and asymptotic critical values. Section 4 contains some Monte Carlo simulation results. Section 5 reports the test results applied to climate datasets. Section 6 concludes. Mathematical derivations are collected in an Appendix.
Section snippets
Model and tests
Suppose that and represent observed values of GMST and TRF respectively. We consider the following model. For , where is the time path of WMGHG, for and elsewhere, and and are stochastic components. Let denote the fraction of relative to such that , where denotes the integer part of a given number.
TRF () is modeled to be the sum of a stochastic term () and a linear function in WMGHG (
Asymptotic distributions
Now, we derive the asymptotic distributions of the test statistics. Recall that we rewrite using local-to-unity parameter such that . To describe the asymptotic distributions, we need more notations.
Definition 1 We define the following functions in . All integrals are taken from zero to one unless otherwise specified. Let and be independent standard Wiener processes defined on . (i) and , where ; (ii)
Monte Carlo simulation
We use the following data generating process (DGP), where , , and . We also specify and to be independent autoregressive processes of order one, that is, We first decide those parameters that are not affected by the hypotheses using the time series of TRF in our application. We use since our tests are exactly
Application
We use four GMST series. The first one is the global land and ocean temperature anomalies series (1901–2000 base period) obtained from the National Centers for Environmental Information at the National Oceanic and Atmospheric Administration.11 The second one is the Global Combined Land–Surface Air and Sea–Surface Water
Conclusion
We develop new tests for a simple energy balance model in a bivariate system of GMST and TRF where the time path of well-mixed-greenhouse-gases forcing is included as a potential common trend function in addition to a stochastic trend and a broken linear trend. We apply the proposed tests to four temperature series and a forcing series that are widely used in climate science. The linear relationship between GMST and TRF is not rejected for conventional sizes. The inference bases on a quite
Acknowledgments
We thank the editor and anonymous referees for their helpful comments. We also thank the participants at the Pi-day econometrics conference held at Boston University in March 2019 and the econometric models of climate change conference (EMCC-IV) held at University of Milano-Bicocca in August 2019. Carrion-i-Silvestre gratefully acknowledges the financial support from the Spanish Ministerio de Ciencia y Tecnología, Agencia Española de Investigación (AEI) and European Regional Development Fund
References (23)
Point optimal tests of the null hypothesis of cointegration
J. Econometrics
(2005)- et al.
Inference related to common breaks in a multivariate system with joined segmented trends with applications to global and hemispheric temperatures
J. Econometrics
(2020) Econometric modelling of climate systems: The equivalence of energy balance models and cointegrated vector autoregressions
J. Econometrics
(2020)Heteroskedasticity and autocorrelation consistent covariance matrix estimation
Econometrica
(1991)- et al.
Tests for cointegration, cobreaking and cotrending in a system of trending variables
J. Mark. Econ.
(2018) - et al.
Quasi-likelihood ratio tests for cointegration, cobreaking, and cotrending
Econometric Rev.
(2019) - et al.
A reply to “Does temperature contain a stochastic trend? Evaluating conflicting statistical results”
Clim. Change
(2010) - et al.
Detection and attribution of climate change through econometric methods
Bol. Soc. Mat. Mex.
(2014) - et al.
Statistically derived contributions of diverse human influences to twentieth-century temperature changes
Nat. Geosci.
(2013) Inference of climate sensitivity from analysis of earth’s energy budget
Ann. Rev. Earth Planet. Sci.
(2016)
Small global-mean cooling due to volcanic radiative forcing
Clim. Dynam.
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AQR-IREA research group. Department of Econometrics, Statistics and Applied Economics, Faculty of Economics and Business, University of Barcelona, Av. Diagonal, 690, 08034 Barcelona.