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On the invalidity of the ordinary least squares estimate of the equilibrium climate sensitivity

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The equilibrium climate sensitivity is often estimated by the ordinary least squares applied to annual data of observed/calculated temperature and forcing series. One of the conditions under which the ordinary least squares estimator is consistent is the uncorrelatedness of the regressor and regression error. However, this condition can fail in a regression using historical data of temperature and forcing. Alternative estimators established in econometrics are shown to mitigate the impact of the correlated regressor and regression error and deliver a more reliable estimate of the equilibrium climate sensitivity.

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The data used in this study are available from the author upon reasonable request.


  1. See a review article by Forster (2016) for more of related studies.

  2. Some early contributions statistically analyzing the existence of an increasing trend in GMST include (Bloomfield 1992; Fomby and Vogelsang 2002) among others. Since GMST in a steady state would not exhibit secular variations, the existence of an increasing trend implies that the climate system is in transition to a new steady state.

  3. If \({u_{t}^{y}}\) is not stationary due to its own stochastic trend, yt and xt cannot be linearly related, violating the energy balance equation.

  4. Only one slope change is considered since the sample starts at 1960.

  5. Climate at a Glance: Global Time Series, published April 2019, retrieved on May 13, 2019, from

  6. Hansen et al. (2010). Dataset accessed 2019-5-13 at

  7. Morice et al. (2012). Dataset accessed 2019-5-13 at

  8. Miller et al. (2014). Dataset accessed 2019-5-13 at

  9. TRF series smoothed by exponentially decaying weights and TRF without stratospheric aerosols are also analyzed. The results remain qualitatively the same and thus are not reported.

  10. World pentadal time series for 0\(\sim \)2000 m from the NOAA National Centers for Environmental information at

  11. The reported confidence intervals are obtained by applying the delta method to the asymptotic distribution of each estimator.

  12. See Table 1 in Forster (2016).


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An earlier draft of this paper was presented at the EGU 2019 in Vienna, Austria. Special thanks go to the conference participants who kindly provided the author with valuable comments. Two anonymous referees and the editors gave insightful comments, which helped improve the paper.


This research is supported by the Korea University Grant (K1911951).

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Correspondence to Dukpa Kim.

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Kim, D. On the invalidity of the ordinary least squares estimate of the equilibrium climate sensitivity. Theor Appl Climatol 146, 21–27 (2021).

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