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On the Relationship Between GHGs and Global Temperature Anomalies: Multi-level Rolling Analysis and Copula Calibration


The relationship between GHG emissions and global warming is studied through multi-level rolling analysis to assess whether or not there are increasing rates in global climate change as a result of higher levels of anthropogenic emissions, as we move forward in time. Furthermore, in order to assess whether we observe tail dependence, representing simultaneous occurrence of extreme events, we employ copula methods. Our main findings suggest a constant effect of emissions on temperature anomalies especially in the last decades. On the other hand we observe positive upper tail dependence in our copula analyses. This implies a comparably high probability of joint extreme large values (i.e., high temperatures and emission concentrations). As a guide to policy, it suggests to keep down extreme events in emissions to prevent possibilities of extreme warmings.

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  1. 1.

    The US government now seems to embrace conservative views about climate change and plans its environmental agenda with less sensitivity on the matter.

  2. 2.

    Several world-wide organizations appealed to a precautionary principle in policy measures. For example, the Rio Declaration on Environment and Development states: “In order to protect the environment, the precautionary approach shall be widely applied by states according to their capabilities. Where there are threats of serious or irreversible damage, lack of full scientific certainty shall not be used as a reason for postponing cost-effective measures to prevent environmental degradation” (UNFCCC 1992).

  3. 3.

    The main objective of climate policies is to reduce the amount of anthropogenic GHG emissions to achieve “stabilization of GHG concentrations in the atmosphere at a level that would prevent “dangerous anthropogenic interference” with the climate system” (UNFCCC 1992, art.2). Most of the ‘unburnable carbon’ debate has recently recognized the need of limiting the global mean temperature increase to 2 °C relative to preindustrial times—and even a relatively more ambitious target of 1.5 °C was reached in the Paris Agreement in December 2015 and is now integrated in the forthcoming sixth assessment report of the IPCC.

  4. 4.

    See, e.g., Allen et al. (2009), Zickfeld et al. (2009), Goodwin et al. (2015), MacDougall and Friedlingstein (2015), and MacDougall (2016). Matthews et al. (2009) defined CCR, which is now thought of as TCRE (transient climate response to cumulative CO2 emissions), combining the concepts of carbon sensitivity (i.e., the increase in atmospheric CO2 concentration from the emission of CO2), climate sensitivity (i.e., the physical response of the climate to an increase in atmospheric CO2 concentration) and the feedbacks between these two processes into a single metric. Allen et al. (2009) develop a similar metric to TCRE relating cumulative emissions to peak temperature following cessation of emissions (see MacDougall 2016, for a review).

  5. 5.

    For example, Allen et al. (2000) claim that the predicted response of climate change to a given emission scenario is inevitably uncertain. Recent observed changes appear to be attributable to human influence. Substantial changes in the current balance of GHG warming and sulphate aerosol cooling could increase the uncertainty. Allen et al. (2000) try to assess the range of warming rates that are consistent with the observed temperature as well as with the overall patterns of response, which is relatively robust to errors in models’ climate sensitivity, global response to sulphate aerosols, etc.

  6. 6.

    For a discussion of various types of uncertainty in climate sensitivity see, e.g., Pindyck (2017), Millner et al. (2013), and Heal and Millner (2014).

  7. 7.

    Alternatively, we could use the global land–ocean temperature index, which still employs the GISS analysis, combining available sea surface temperature records with meteorological station measurements. It is shown that, on average, warming in the recent four decades was larger over land than over ocean, in part because of the ocean’s larger thermal inertia.

  8. 8.

    Methane is expressed as a mole fraction in dry air (nanomol/mol) abbreviated as ppb ( and nitrous oxide emissions are combined data in ppb from the NOAA/ESRL Global Monitoring Division (see

  9. 9.

    In our analysis we have used the historical data in Meinshausen et al. (2011), up to 2005. After 2005, we have used for cumulative CO2 emissions data from scenario RCP 8.5 and for concentrations from scenario RCP6, since these two are closer to the available observed values until now, as we verified from the data elaborated by CDIAC and NOAA and reported in the World Bank’s World Development Indicators (

  10. 10.

    More details for each scenario can be found in Clarke et al.(2007), Smith and Wigley (2006), Wise et al. (2009), Fujino et al. (2006) Hijioka et al. (2008), and Riahi et al. (2007).

  11. 11.

    We are grateful to an anonymous referee who suggested both the specification as in the model by Castruccio et al. (2014), and the models of Eq. (2).

  12. 12.

    Correlograms of the residuals are available upon request.

  13. 13.

    Florides and Christodoulides (2008), Allan et al. (2014), Brown et al. (2014), Boykoff (2014), Boykoff and Boykoff (2004), Laepple and Huybers (2014), and Tollefson (2014).

  14. 14.

    We further calibrate Clayton and Gumbel copulas rotated by 90° (\( C^{ + - } \)) and 270° (\( C^{ - + } \)) in order to be able to model negative dependence. Rotation is not necessary for the Gaussian, the Student t and the Frank copula. They allow to model negative dependence in their standard versions and their survival copulas correspond to the original copulas. Copula rotation is only possible for bivariate copulas. Copula rotation is shortly explained in Brechmann and Schepsmeier (2013, p. 7f).

  15. 15.

    In their article, Genest and Rivest (1993) divide by \( T \) rather than by \( \left( {T + 1} \right) \). Dividing by \( \left( {T + 1} \right) \) has the advantage that it keeps the pseudo-observations away from the boundaries of the unit cube where the densities of many copulas take infinite values.

  16. 16.

    Here only for the Student t copula two parameters have to be estimated, while the other bivariate copulas have only one parameter.

  17. 17.

    The reason for the identical results of the log-likelihood for the copula calibration to temperatures and concentrations of N2O and GHGs respectively, and cumulative CO2 emissions, stems from the fact that all three time series are strictly monotonically increasing. This means that the first observations in all three time series are the ones with the lowest value and that all following observations are increasing year by year. Hence, the ranks for the observations of all three time series are \( \left\{ {1, 2, 3, \ldots ,n = 138} \right\} \). As the copula calibration with the pseudo-log-likelihood method is based on ranks, the results for the three time series pairs are identical.

  18. 18.

    For changes in emission levels (and cumulative CO2 emissions) we use relative changes (log-differences) while for changes in temperature we use the absolute differences of the temperature anomalies (in °C) and not the relative differences in Kelvin. As the copula calibration is based on ranks rather than on the actual values of the time series, we do not think that using the absolute differences for temperature changes has any impact on the results reported here.

  19. 19.

    Also for all other data pairs we cannot reject the null hypothesis of a Gaussian copula in favour of a Student t copula in any of the cases.

  20. 20.

    IPCC (2014), Latif (2010), Barr et al. (2011), Nazarenko et al. (2015), and Hansen et al. (2005). The recent paper by Howard and Sterner (2017) provide a meta-analysis of climate change estimates as a key tool for determining the relationship between temperature and climate damages.


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Correspondence to Thomas Alexopoulos.

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Agliardi, E., Alexopoulos, T. & Cech, C. On the Relationship Between GHGs and Global Temperature Anomalies: Multi-level Rolling Analysis and Copula Calibration. Environ Resource Econ 72, 109–133 (2019).

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  • GHGs
  • Global temperature anomalies
  • Rolling analysis
  • Copulas

JEL Classification

  • Q54
  • Q51
  • C53
  • C69