Over the past few decades, the observed global mean surface temperature has increased. With such an evident fact, global warming has received rapidly increasing attention during the past decades. Since all countries are involved in both the causes and consequences of this issue in a variety of complex ways, there have been worldwide debates on global warming among scientists and policymakers. To identify whether human activities are causing the recent rise in global mean temperature or whether their effects will have serious effects on the Earth, the detection and attribution of an anthropogenic influence on climate change has been studied extensively.
Climate sensitivity measures how much global warming will occur in response to a doubling of atmospheric concentration of carbon dioxide. Ultraviolet light from the Sun passes through greenhouse gases (GHG) such as carbon dioxide, water vapor, methane, nitrous oxide, and chlorofluorocarbons, and is absorbed by objects on the ground. Since GHG absorb the infrared radiation released by the objects and then reradiates it back to the surface of the Earth, global temperature is increasing as a result. We refer to this as the “Greenhouse Effect” (Hansen et al. 2011
). It is well-known that GHG in the atmosphere have been consistently increasing due to human activity. Note that the atmospheric lifetime of carbon dioxide is currently estimated at 5–200 years (IPCC 2007
). As a result of such accumulation, the stock of carbon dioxide in the atmosphere has increased by approximately 130 ppm over the last 270 years, from a range of between 275 and 285 ppm in the pre-industrial era to 410 ppm in 2018.
According to IPCC
), the global mean temperature on the surface of the Earth has increased by approximately 0.85
C since 1880 and “most of the observed warming over the last 50 years is likely to have been due to the increase in GHG emissions.” Scientists have attempted to estimate the effect of GHG, according to which doubling carbon dioxide concentration in the atmosphere (a forcing of 4 W/m
) may increase the average global temperature by 1.5 to 4.5
C. To stress the serious impact of increasing global temperature, Stern
) states that, “around 10,000–12,000 years ago, temperatures were approximately 5
C lower than today, and ice sheets came down to latitudes just north of London and south of New York. As the ice melted and sea levels rose, England separated from the continent, rerouting much of the river flow. These magnitudes of temperature changes transform the planet”.
A time series analysis has been employed to test the anthropogenic global warming hypothesis. This hypothesis test has resulted in extensive controversy over the last two decades. The main argument relates to whether observing trends in temperature series and radiative forcing contain stochastic trends or deterministic trends with a structural break. During early years, Kaufmann et al.
) and Kaufmann and Stern
) had a breakthrough on the linear cointegration analysis between temperature series and radiative forcing variables by assuming they are integrated processes or difference stationary processes. Using Dynamic Ordinary Least Squares estimation, they concluded that the increase in global mean temperature can be associated with the change in radiative forcing variables. Such a linear cointegration analysis has also been investigated by Pretis
). He linked a two-component energy balance climate model of global mean temperature with a testable cointegrated Vector Autoregressive model.
Some econometricians cast doubt on their statistical rigor and challenged their empirical results (Gay-Garcia et al. 2009
; Perron and Estrada 2012
; Estrada et al. 2013a
; Estrada and Perron 2014
). They first argued that temperatures and radiative forcing variables are described more effectively as trend stationary processes rather than difference stationary processes (or random walk with drift). By defining variables of interest as stationary processes fluctuating around a common breaking deterministic trend, they claimed that the conventional Least Squares (LS) method on the regression may negate a common feature as in cointegration analysis. Moreover, they argued that the residual-based ADF test (or formally nonparametric nonlinear co-trending test of Bierens 2000
) may identify the existence of a long-run relationship.
Recently, Chang et al.
) analyzed the global warming issue under a novel time series framework. By using Global, Northern Hemisphere, and Southern Hemisphere temperature anomaly data from 1850 to 2012, they generated the distributions of temperature anomalies for each year. Instead of only analyzing mean temperature anomalies, they analyzed and tested the persistent features of distributions of temperature anomalies by regarding them as functional time series observations on the Hilbert space. More importantly, they distinguished unit-root nonstationarity from deterministic or explosive nonstationarity in their testing procedure Chang et al.
). They concluded that the first few moments of temperature anomaly distribution indicate stochastic trends, rather than linear/exponential/quadratic trends or explosive roots.
In particular, they reasoned that the seemingly structural break in the global mean temperature anomaly trend, as argued by many authors, are more likely inherited from unit-root type persistency (stochastic trend), than from higher order persistency associated with deterministic trends. Based on their analysis, global temperature anomaly distribution and radiative forcing variables can share common stochastic trends and their linear combination can produce a stationary process. In this context, the nonlinear cointegration analysis, which allows for bidirectional causality and postulates a long-run equilibrium relationship between mean temperature anomalies and total radiative forcing and possibly with nonlinear moments of temperature anomaly distributions, seems the most reasonable approach.
More importantly, the linear regression model, which postulates the linear relationship between global mean temperature and total radiative forcing, fails to consider the climatological mechanism regarding how to change the global mean temperature. Since atmospheric carbon dioxide increases the global mean temperature by generating an imbalanced energy equation, a channel for the greenhouse effect must be considered. As shown in Section 2
, the climate channel could be represented as temperature-dependent (or spatially heterogenous) net incoming absorbed radiation, implying that the relationship between total radiative forcing and the global mean temperature could be expressed as a temperature heterogeneous function. Evidently, ignoring the net incoming absorbed radiation term generates an endogeneity problem, invalidating the slope estimator of the linear model.
Throughout this study, I aim to develop a new statistical model to provide a more informative estimate for climate sensitivity. To do so, I propose a nonlinear cointegrating regression of the Earth’s surface mean temperature anomalies on total radiative forcing under technical backgrounds provided by Chang et al.
). Put differently, I analyze the cointegrating relationship between the time series of spatial distributions of temperature anomalies and the time series of total radiative forcing. In a climatological sense, the proposed statistical methodology provides two types of nonlinear climate sensitivity that map the total radiative forcing to mean temperature anomalies for the Globe, Northern Hemisphere, and Southern Hemisphere.
Specifically, I explicitly estimate the nonlinear effects by defining the nonlinear temperature term from the temperature anomaly distribution. I refer to the nonlinear response function as the temperature-dependent effect of total radiative forcing by assuming that net incoming absorbed radiation is hypothetically determined at some regional spaces that correspond to the temperature anomaly. I also define the misspecification error from the nonlinear cointegration model and identify the source of error in terms of temperature anomaly. Lastly, I conduct cointegration and specification tests that support the existence of nonlinear effects of total radiative forcing.
The statistical result of the study contributes an important insight to climate research. As is well-known, the popular term, “polar amplification” describes that the warming speed of the higher latitude area has been faster than that of the lower latitude area. The proposed nonlinear cointegration model provides a better understanding of such spatially heterogeneous warming effects by incorporating the higher-order moments of spatial distributions of temperature anomaly. Moreover, the notion that human forcing generates the probabilistic change of the temperature anomaly indicates that the true effect of human forcing on the mean temperature anomaly would be overestimated in the literature. In other words, ignoring the spatially heterogenous effects on the change in the global mean temperature anomaly would generate the upward bias when we attempt to identify the effect of human forcing.
The remainder of the paper is organized as follows. Section 2
provides the climatological background according to the global energy balance climate model. Section 3
presents the statistical model and methodology. Specifically, I employ the functional coefficient model for a nonlinear cointegrating regression model and apply it to the current climate model. In Section 4
, the details of the data are discussed. The empirical results and interpretation of the results are presented in Section 5
, while Section 6
The following data sources are employed for the Global, Northern Hemisphere, and Southern Hemisphere temperature anomalies and the TRF variable. As in Chang et al.
) who provided the technical background for the nonstationarity of global temperature anomaly distributions, the HadCRUT4 temperature anomaly data from 1850 to 2015 are employed for the Global, Northern Hemisphere, and Southern Hemisphere temperature anomaly data (Morice et al. 2012
Basically, temperature anomaly data from HadCRUT4 and the Goddard Institute for Space Studies (GISS) are generated from the same raw dataset. However, their treatments on the same raw dataset are different. Specifically, the GISS dataset uses an interpolated sea-surface temperature analysis by filling the empty grid boxes in the sea-surface area, while HadCRUT4 does not attempt to calculate the values for empty grid boxes. In light of this, HadCRUT4 would understate the effect of Arctic temperature anomalies, in which the warming has been significant over the past decades.9
Since the temperatures in land stations are measured at various elevations and since different countries employ different methods, the Global, Northern Hemisphere, and Southern Hemisphere temperature data are expressed as anomalies in degrees Celsius from the monthly temperature average from 1961 to 1990, which is known as the “zero-base” (i.e., climatological normal temperature) period. Note that since the number of stations and the methods of temperature measurement are different across grid boxes, calculating deviation from the zero-base may eliminate the heterogeneity across grid boxes over the entire space of the Earth. In this context, the temperature anomaly data are directly exploited for each grid box instead of recovering the actual temperature dataset.
Further, 99 percent of the total probability mass is exploited as the support of the temperature anomaly distribution at each time t
because 0.5 percent of the probability mass at each end would be an adequate threshold to minimize the estimation errors induced by the boundary problem from the standard kernel density estimation technique. Figure 1
shows the time series of left and right ends of the chosen support of the global temperature anomaly distribution, indicating that non-common support would be necessary to effectively generate the distributions of temperature anomaly. Specifically, the right end of the support of the global temperature anomaly distribution reveals an increasing trend, implying the warming phenomena.
For the radiative forcing variable, TRF data from 1850 to 2015 (Hansen et al. 2017
) are employed, which represent the sum of anthropogenic forcing and natural variability. Specifically, TRF is the sum of well-mixed GHG (CO
O, and CFCs), ozone, surface albedo and tropospheric aerosols, and solar irradiance.10 Figure 2
, Figure 3
and Figure 4
provide detailed information of the temperature anomaly distribution for the Globe, Northern Hemisphere, and Southern Hemisphere and TRF. Specifically, the top-left panels of Figure 2
, Figure 3
and Figure 4
show the temperature anomaly distribution generated by the procedure of Chang et al.
), and the top-right panels of the figures illustrate the graphical comparison between the first moment estimated from the generated temperature anomaly distribution and the web-posted mean temperature anomalies from the GISS11
and HadCRUT4 (the median of the 100 ensemble member time series)12
websites for the Globe, Northern Hemisphere, and Southern Hemisphere.
Indeed, 99 percent of probability mass in estimating the temperature anomaly distribution provides a good approximation for estimating the first moment (i.e., mean temperature anomaly) at each time t, in the sense that estimated mean temperature anomalies are similar to mean temperature anomalies widely-used by climate scientists. Note that the GISS surface temperature data are expressed as an anomaly in degrees Celsius with the base period 1951–1980, which is available after the year 1880. As mentioned earlier, the differences in the mean temperature anomalies between the HadCRUT4 and GISS sources are greater for the Southern Hemisphere, in which the GISS team interpolates the sea-surface temperature data.
The bottom panels of Figure 2
, Figure 3
and Figure 4
provide the first four central moments of the generated temperature anomaly distribution. While the variances of the estimated temperature anomaly distribution have decreased, roughly similar to Chang et al.
), the means, the skewness, and the kurtosis of the estimated temperature anomaly distribution have increased. In particular, the skewness appears to have increased from negative to positive. These statistical facts imply that the temperature anomaly distributions have been concentrated around their increasing means, and the probabilities of extremely positive temperature anomalies have increased. Not surprisingly, the decreasing variances of the temperature anomaly distribution would be on the same lines with the movements of the other central moments. Lastly, the calculated mean temperature anomalies are compared with the TRF variable, clearly showing that they moved together for the last 165 years.
5. Empirical Analysis
Throughout this study, the statistical testing result of the unit-root type nonstationarity of Chang et al.
) was followed. Specifically, the estimated persistence of the global mean temperature anomaly was closer to a stochastic trend, but not high enough to a deterministic trend, implying that no linear deterministic trend has been detected. As some econometricians discover a broken deterministic trend from the global mean temperature anomaly (Gay-Garcia et al. 2009
; Estrada et al. 2013b
, inter alia), moreover, Gao and Hawthorne
) attempt to estimate a general deterministic trend by allowing the flexible nonlinearity in the deterministic trend component. However, Chang et al.
) argue that a deterministic trend with the excessive nonlinearity and variability could be better expressed as a stochastic trend. Subsequently, the failure of rejection of the cointegration test implies that the TRF variable shares a stochastic trend with the global temperature anomaly distribution.
In the literature, climate sensitivity for the Globe is estimated as the value 0.43
) and 0.35
) with AMO-unfiltered HadCRUT4 and NASA dataset, respectively (Estrada et al. 2013b
As expected, linear climate sensitivity is estimated as the value 0.435
), which is a similar value to the Global case.
provides the estimation result of Equation (13
) with a derivative of interest. Based on AIC and BIC criteria, the optimal models for the Globe, Northern Hemisphere, and Southern Hemisphere are
, respectively. Moreover, the first VAT test statistics (VAT
) for the Globe, Northern Hemisphere, and Southern Hemisphere cases as well as for linear/optimally chosen nonlinear models, indicate that all considered models are authentic, supporting the cointegration technique using the CCR methodology. However, the second VAT test statistics (VAT
) indicate that all regression models could be spurious if we assume the strict stationarity in the residual. Notice that the second VAT statistics decreased by considering the nonlinear temperature term for the Globe and Northern Hemisphere cases, indicating that the nonlinear temperature term
in Equation (18
) plays a role in explaining the oceanic multidecadal oscillation.
The nonlinear estimator represents climate sensitivity that considers all nonlinear effects across the observed temperature anomalies. The resulting climate sensitivity, , of the nonlinear cointegration model is provided as the value 0.380 for the Globe, indicating that the global mean temperature anomaly increases by C when TRF increases by 1 W/m. In the meantime, the misspecification error of the linear model is greatest for the Northern Hemisphere (0.1077 C/(W/m)), and lowest for the Southern Hemisphere (0.0402 C/(W/m)). The Wald test decisively rejects the null of no statistical significance of the nonlinear temperature term for the Globe, Northern Hemisphere, and Southern Hemisphere cases (p-values are less than 0.01).
As shown by Figure 2
, Figure 3
and Figure 4
, TRF has affected not only the mean temperature anomaly, but also variance, skewness, and kurtosis temperature anomalies for approximately 150 years. This implies that the Earth’s surface temperature has been affected by human and natural forcing in a spatially heterogenous manner. The nonlinear cointegration model enables us to estimate the true effect by including the spatial distributions of temperature anomalies in the model, showing that the change in the global mean temperature anomaly associated with TRF (0.380
)) would be less than what we have observed (0.435
In the literature, the transient climate response is often calculated as the global mean temperature response in
C to a doubling of atmospheric CO
from pre-industrial level by an increase of 1 percent per year. Schwartz
) estimates the value 3.71 W/m
of the TRF level at the time when the atmospheric CO
is doubled from the pre-industrial level. Subsequently, the transient climate response is calculated as the value 1.410
C (=3.71 × 0.380
)), which is 0.204
C lower than the estimated transient climate response from the linear model (1.614
C). Given that the global mean temperature anomaly was
C in 1850, the predicted global mean temperature anomaly associated with a doubled atmospheric CO
level is 1.01
Climatologically, the TRF response function at temperature anomaly
, indicates a change in the mean temperature anomaly, if net incoming absorbed radiation is solely determined by temperature anomaly
. As such, the nonlinear effect, in addition to the mean effect (or linear effect), would provide the temperature-dependent TRF effect on the mean temperature anomaly. The nonlinear effects of the estimated climate sensitivity,
, are calculated as values −0.018
), and −0.009
) for the Globe, Northern Hemisphere, and Southern Hemisphere (i.e., the term,
in Equation (10
)). Therefore, we may conclude that the nonlinearity of the relationship induced by the spatial heterogeneity was strongest for the Northern Hemisphere.
In Figure 5
, both the net incoming absorbed radiation,
, and the derivative of the temperature anomaly distribution with respect to TRF,
, are presented for the Globe, Northern Hemisphere, and Southern Hemisphere. Note that the domain of the temperature anomaly is shortened below −1.0
C and above 2.0
C for a practical purpose. The left panels of Figure 5
illustrate that the nonlinear effect becomes more significant in the opposite direction as the temperature anomaly approaches the left and right ends. As mentioned in Section 3
, more specifically,
estimates the effect of net incoming absorbed radiation of the regions that are represented by the temperature anomaly r
Note that the popular term, “polar amplification,” states that the low (high) temperature anomaly region, which mainly represents higher (lower) latitude areas, is related to the mean temperature anomaly in a positive (negative) direction (Boer and Yu 2003
). Consistent with this notion, a positive (negative) value of the net incoming absorbed radiation term,
implies that the TRF effect under linearity would be underestimated (overestimated). In other words, the TRF effect under linearity should be amplified for the negative anomalies (high latitude areas) for the Globe and Northern Hemisphere, and be attenuated for the positive anomalies (low latitude areas) for the Globe, Northern Hemisphere, and Southern Hemisphere.
By considering the nonlinear temperature term with the derivative of the temperature anomaly distribution with respect to TRF,
, the nonlinear TRF effect would be obtained by amplifying or attenuating the linear TRF effect (see Equation (11
)). Since the changes in probability with respect to the change in TRF (i.e.,
) are very small at extreme temperature anomalies, the nonlinear effects at extreme temperature anomalies could be much less than those at low or high temperature anomalies.
illustrates the TRF response function for the Globe, Northern Hemisphere, and Southern Hemisphere, which is the estimator
in Equation (11
). The reference (dashed blue) line represents the linear TRF response function, postulating that the TRF effect on the mean temperature anomaly is constant across temperature anomalies. That is, it illustrates the TRF effect on the mean temperature anomaly without the nonlinear effect. For the Globe, the linear term provides an estimate of the constant climate sensitivity as the value 0.40
), and the nonlinear TRF response function intersects with the linear TRF response function at two anomalies, −0.14
C and 0.18
C. This implies that the linear TRF response function underestimates the true effect between these two anomalies.
In Figure 6
, more importantly, the nonlinear TRF response function (i.e., nonlinear climate sensitivity) provides an interpretation of the relative magnitude of the TRF effect on the mean temperature anomaly. In particular, the greatest temperature-dependent TRF effect on the global mean temperature anomaly is estimated as the value 0.398
), if net incoming absorbed radiation is solely determined by a temperature anomaly, 0.015
C. That is, the spatial contribution to the change in the global mean temperature anomaly is strongest for the areas in which their temperature levels correspond to
C. Such spatially heterogeneous contributions could be better understood by incorporating the higher-order moments of spatial distributions of temperature anomaly in the nonlinear cointegration model.
Moreover, the smoothed global mean temperature anomaly using a polynomial function was approximately 0.01
C in 1974, implying that the TRF effect on the global mean temperature anomaly would be strongest in 1974. Note that the sum of nonlinear effects of estimator
across temperature anomaly in Equation (11
) is equal to the nonlinear effect of estimator
in Equation (10
). In this light, the high degrees of nonlinearity could be estimated by adding all nonlinear effects, although the magnitude of nonlinear effect at each temperature anomaly is small. Further note that the nonlinear TRF effect for the Northern Hemisphere case shows a similar pattern to the Global case. However, its effect is greater than the Global case because the speed of global warming is faster in the Northern Hemisphere than in the Southern Hemisphere.
To elaborate on this discussion, Figure 7
provides the nonlinear TRF response function over time. Specifically, I postulate that the spatial regions where provide the biggest contribution on the change of mean temperature anomaly at time t
is located at the areas where have a record of the mean temperature anomaly at time t
. This is a reasonable situation, in the sense that the mean value of spatial distribution would be largely determined by the spatial areas where have a similar or closer mean value. Consequently, the value of the nonlinear TRF response function at time t
indicates the relative magnitude of the TRF effect when the biggest spatial contribution is evaluated at the mean temperature anomaly at time t
The TRF response functions presented in Figure 7
imply that the warming speed of the Globe and Northern Hemisphere has decreased since 1980. However, the speed would start to increase when the mean temperature anomaly reaches approximately 0.8
C. Notably, the presented TRF response function for the Southern Hemisphere has decreased since 1910. Similar to the Globe and Northern Hemisphere cases, however, the warming speed of the Southern Hemisphere would start to increase when the mean temperature anomaly reaches approximately 0.7
C, which is much earlier compared to other cases. Based on the information provided by the HadCRUT4 website,15
the mean temperature anomalies for the Globe, Northern Hemisphere, and Southern Hemisphere in 2019 were estimated at 0.736
C, and 0.502
C, respectively, indicating that the warming speed of the Northern Hemisphere has already started to increase.
In this paper, I proposed the nonlinear cointegration model based on the well-known EBCM. For this, the nonlinear cointegrating regression of the mean temperature anomaly for the Globe, Northern Hemisphere, and Southern Hemisphere, were estimated using the spatial distributions of temperature anomalies. Subsequently, the nonlinear TRF effect on the mean temperature anomaly was estimated, suggesting that the TRF effect on the mean temperature anomaly would be temperature-dependent for the Globe, Northern Hemisphere, and Southern Hemisphere. Graphically, the TRF response function has a flexible shape to represent the change in the mean temperature anomaly when net incoming absorbed radiation is hypothetically determined at some temperature anomaly.
Statistically, the linear model fails to take into account the net incoming absorbed radiation term, which would cause the slope estimator to be invalid. Considering the functional form of net incoming absorbed radiation, the proposed nonlinear cointegration model shows the reasonable nonlinear dependence structure between the mean temperature anomaly and TRF. Specifically, climate sensitivity is estimated to be temperature-dependent (or spatially heterogenous), providing that the estimated nonlinear climate sensitivity was highest in the mid-1970s for the Globe. Moreover, the TRF effect on the mean temperature anomaly, which considers all nonlinear effects, is less than the estimate of the linear climate sensitivity provided by the literature. Lastly, the statistical testing results indicate that the linear model possesses a significant misspecification error for the Globe, Northern Hemisphere, and Southern Hemisphere cases.
The next step regarding this research refers to climate variability. As emphasized by Brock et al.
), spatial complexity on Earth is a big challenge for analyzing temperature series at hemispheric scales. The complexity inherited from spatial diversity produces hardly explainable natural variability through the observed data. At least to date, the well-known inter-annual global variability or hemispheric climate variability is regarded as the El Niño/Southern Oscillation (ENSO), the North Atlantic Oscillation (NAO), and the Atlantic Multidecadal Oscillation (AMO). Among them, the most influential natural variation for the attribution study is the AMO, which could distort the long-term global warming trend. Specifically, the large ocean-atmosphere cycle over the North Atlantic, which is defined as approximately 60 to 90 years of low-frequency patterns of sea surface temperature variability, explains the larger variability in Northern Hemisphere temperatures and therefore, globally.
In particular, Estrada et al.
) identified its difficulty when they conducted an attribution study. They argued that the detrended Global and Northern Hemisphere temperatures with forcing variables could be further explained by the AMO and therefore the difference between the dates of a structural break could be explained as well. In light of this, Estrada et al.
) filtered the AMO information from the Global and Northern Hemisphere mean temperature anomalies to estimate constant climate sensitivity. To put these factors into perspective, it is worth estimating nonlinear climate sensitivity after extracting the major climate variabilities.
Moreover, the uncertainty of the estimator of the TRF response function needs to be evaluated. As the estimation is implemented via two-step approach, it is difficult to evaluate the bootstrapping confidence bands, in the sense that the estimator’s uncertainty comes from both steps. We may estimate the uncertainty by fixing the point estimate of the first step as a constant. Since the first step is the nonstationary nonlinear regression model, however, the uncertainty of the first step’s estimator would be larger than that of the second step’s estimator. I leave these tasks to future research.