#
Atmospheric Temperature and CO_{2}: Hen-Or-Egg Causality?

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## Abstract

**:**

_{2}concentration plays a major role in enhancement of the greenhouse effect and contributes to global warming. The purpose of this study is to complement the conventional and established theory, that increased CO

_{2}concentration due to human emissions causes an increase in temperature, by considering the reverse causality. Since increased temperature causes an increase in CO

_{2}concentration, the relationship of atmospheric CO

_{2}and temperature may qualify as belonging to the category of “hen-or-egg” problems, where it is not always clear which of two interrelated events is the cause and which the effect. We examine the relationship of global temperature and atmospheric carbon dioxide concentration in monthly time steps, covering the time interval 1980–2019 during which reliable instrumental measurements are available. While both causality directions exist, the results of our study support the hypothesis that the dominant direction is T → CO

_{2}. Changes in CO

_{2}follow changes in T by about six months on a monthly scale, or about one year on an annual scale. We attempt to interpret this mechanism by involving biochemical reactions as at higher temperatures, soil respiration and, hence, CO

_{2}emissions, are increasing.

Πότερον ἡ ὄρνις πρότερον ἢ τὸ ᾠὸν ἐγένετο (Which of the two came first, the hen or the egg?).

Πλούταρχος, Hθικά, Συμποσιακὰ Β, Πρόβλημα Γ (Plutarch, Moralia, Quaestiones convivales, B, Question III).

## 1. Introduction

_{2}and temperature may qualify as belonging to the category of “hen-or-egg” problems. First, we discuss the relationship between temperature and CO

_{2}concentration by revisiting intriguing results from proxy data-based palaeoclimatic studies, where the change in temperature leads and the change in CO

_{2}concentration follows. Next, we discuss the databases of modern (instrumental) measurements related to global temperature and atmospheric CO

_{2}concentration and introduce a methodology to analyse them. We develop a stochastic framework, introducing useful notions of time irreversibility and system causality while we discuss the logical and technical complications in identifying causality, which prompt us to seek just necessary, rather than sufficient, causality conditions. In the Results section, we examine the relationship of these two quantities using the modern data, available at the monthly time step. We juxtapose time series of global temperature and atmospheric CO

_{2}concentration from several sources, covering the common time interval 1980–2019. In our methodology, it is the timing rather than the magnitude of changes that is important, being the determinant of causality. While logical, physically based arguments support the “hen-or-egg” hypothesis, indicating that both causality directions exist, interpretation of cross-correlations of time series of global temperature and atmospheric CO

_{2}suggests that the dominant direction is T → CO

_{2}, i.e., the change in temperature leads and the change in CO

_{2}concentration follows. We attempt to interpret this latter mechanism by noting the positive feedback loop—higher temperatures increase soil respiration and, hence, CO

_{2}emissions.

_{2}concentration, which followed a pattern similar to previous years (Figure 2). Indeed, according to the International Energy Agency (IEA) [3], global CO

_{2}emissions were over 5% lower in the first quarter of 2020 than in that of 2019, mainly due to an 8% decline in emissions from coal, 4.5% from oil, and 2.3% from natural gas. According to other estimates [4], the decrease is even higher: the daily global CO

_{2}emissions decreased by 17% by early April 2020 compared with the mean 2019 levels, while for the whole 2020, a decrease between 4% and 7% is predicted. Despite that, as seen in Figure 2, the normal pattern of atmospheric CO

_{2}concentration (increase until May and decrease in June and July) did not change. Similar was the behaviour after the 2008–2009 financial crisis, but the most recent situation is more characteristic because the COVID-19 decline in 2020 is the most severe ever, even when considering the periods corresponding to World Wars. It is also noteworthy that, as shown in Figure 1, there are three consecutive years in the 2010s where there are no major increases, in emissions while there was an increase in CO

_{2}concentration. (At first glance, this does not sound reasonable and we have therefore cross-checked the data with other sources, namely the Global Carbon Atlas [5], and the database of Our World In Data [6]; we found only slight differences.) Interestingly, Figure 1 also shows a rapid growth in emissions after the 2008–2009 global financial crisis, which is in agreement with Peters et al. [7].

## 2. Temperature and Carbon Dioxide—From Arrhenius and Palaeo-Proxies to Instrumental Data

_{2}) and temperature classify as a “hen-or-egg”-type causality? If we look at the first steps of studying the link between the two, the reply is clearly negative. Arrhenius (1896, [8]), the scientist most renowned for studying the causal relationship between two quantities, regarded the changes in atmospheric carbon dioxide concentration as the cause and the changes in temperature as the effect. Specifically, he stated:

Conversations with my friend and colleague Professor Högbom together with the discussions above referred to, led me to make a preliminary estimate of the probable effect of a variation of the atmospheric carbonic acid [meaning CO_{2}] on the temperature of the earth. As this estimation led to the belief that one might in this way probably find an explanation for temperature variations of 5–10 °C, I worked out the calculation more in detail and lay it now before the public and the critics.

_{2}in the greenhouse effect of the Earth’s atmosphere. He calculated the relative weights of absorption of CO

_{2}and water vapour as 1.5 and 0.88, respectively, or a ratio of 1:0.6.

_{2}] denote temperature and CO

_{2}concentration, respectively, T

_{0}and [CO

_{2}]

_{0}represent reference states, and α is a constant.

- Indeed, CO
_{2}plays a substantial role as a greenhouse gas. However, modern estimates of the contribution of CO_{2}to the greenhouse effect differ largely from Arrhenius’s results, attributing 19% of the long-wave radiation absorption to CO_{2}against 75% of water vapour and clouds (Schmidt et al. [15]), or a ratio of 1:4. - During the Phanerozoic Eon, Earth’s temperature varied by even more than 5–10 °C, which was postulated by Arrhenius—see Figure 3. Even though the link of temperature and CO
_{2}is beyond doubt, this is not clear in Figure 3, where it is seen that the CO_{2}concentration has varied by about two orders of magnitude and does not always synchronize with the temperature variation. Other factors may become more important at such huge time scales. Thus, an alternative hypothesis of the galactic cosmic ray flux as a climate driver via solar wind modulation has been suggested [16,17], which has triggered discussion or dispute [14,18,19,20,21,22,23]. The T–CO_{2}relationship becomes more legible and rather indisputable in proxy data of the Quaternary (see Figure 4). It has been demonstrated in a persuasive manner by Roe [24] that in the Quaternary, it is the effect of Milanković cycles (variations in eccentricity, axial tilt, and precession of Earth’s orbit), rather than of atmospheric CO_{2}concentration, that explains the glaciation process. Specifically (quoting Roe [24]),variations in atmospheric CO_{2}appear to lag the rate of change of global ice volume. This implies only a secondary role for CO_{2}—variations in which produce a weaker radiative forcing than the orbitally-induced changes in summertime insolation—in driving changes in global ice volume.

_{2}concentration as the effect. Such claims have been based on the fact that temperature change leads and CO

_{2}concentration change follows. In agreement with Roe [24], several papers have found the time lag to be positive, with estimates varying from 50 to 1000 years or more, depending on the time period and the particular study [27,28,29,30,31,32]. Claims that CO

_{2}concentration leads (i.e., a negative lag) have not generally been made in these studies. At most, a synchrony claim has been sought on the basis that the estimated positive lags are often within the 95% uncertainty range [31], while in one publication [29], it has been asserted that a “short lead of CO

_{2}over temperature cannot be excluded”. With respect to the last deglacial warming, Liu et al. [32], using breakpoint lead–lag analysis, again find positive lags and conclude that the CO

_{2}is an internal feedback in Earth’s climate system rather than an initial trigger.

**Figure 3.**Proxy-based reconstructions of global mean temperature and CO

_{2}concentration during the Phanerozoic. The temperature reconstruction by Scotese [33] was mainly based on proxies from [21,34,35,36], while the CO

_{2}concentration proxies have been taken from Davis [37], Berner [38], and Ekart et al. [39]; all original figures were digitized in this study. The chronologies of geologic eras shown in the bottom of the figure have been taken from the International Commission on Stratigraphy [40].

**Figure 4.**(

**upper**) Time series of temperature and CO

_{2}concentration from the Vostok ice core, covering part of the Quaternary (420,000 years) with time step of 1000 years. (

**lower**) Auto- and cross-correlograms of the two time series. The maximum value of the cross-correlation coefficient, marked as ◆, is 0.88 and appears at lag 1 (thousand years) (adapted from Koutsoyiannis [30]).

_{2}by one time step (1000 years), as illustrated in Figure 4. He also noted that this causality condition holds for a wide range of time lags, up to 26,000 years, and, hence, the time lag is positive and most likely real. He asserted that the problem is obviously more complex than that of exclusive roles of cause and effect, classifying it as a “hen-or-egg” causality problem. Obviously, however, the proxy character of these data and the overly large time step of the time series reduce the reliability and accuracy of the results.

show that properly specified tests of Ganger [sic] causality validate the consensus that human activity is partially responsible for the observed rise in global temperature and that this rise in temperature also has an effect on the global carbon cycle.

study unambiguously shows one-way causality between the total Greenhouse Gases and GMTA [global mean surface temperature anomalies]. Specifically, it is confirmed that the former, especially CO_{2}, are the main causal drivers of the recent warming.

_{2}concentration with the emphasis given on the exploratory and explanatory aspect of our analyses. While we occasionally use the Granger statistical test, this is not central in our approach. Rather, we place the emphasis on time directionality in the relationship, which we try to identify in the simplest possible manner, i.e., by finding the lag, positive or negative, which maximizes the cross-correlation between the two processes (see Section 4.1). We visualize our results by plots, so as to be simple, transparent, intuitive, readily understandable by the reader, and hopefully persuading. For the algorithmic-friendly reader, we also provide statistical testing results which just confirm what is directly seen in the graphs.

- The short-term effects deserve to be studied, as well as the long-term ones.
- The modern instrumental records are short themselves and only allow the short-term effects to be studied.
- For the long-term effects, the palaeo-proxies provide better indications, as already discussed above.

## 3. Data

_{2}dataset is that of Mauna Loa Observatory [51]. The Observatory, located on the north flank of Mauna Loa Volcano on the Big Island of Hawaii, USA, at an elevation of 3397 m above sea level, is a premier atmospheric research facility that has been continuously monitoring and collecting data related to the atmosphere since the 1950s. The NOAA also has other stations that systematically measure atmospheric CO

_{2}concentration, namely at Barrow, Alaska, USA and at South Pole. The NOAA’s Global Monitoring Laboratory Carbon Cycle Group also computes global mean surface values of CO

_{2}concentration using measurements of weekly air samples from the Cooperative Global Air Sampling Network. The global estimate is based on measurements from a subset of network sites. Only sites where samples are predominantly of well-mixed marine boundary layer air, representative of a large volume of the atmosphere, are considered (typically at remote marine sea level locations with prevailing onshore winds). Measurements from sites at high altitude (such as Mauna Loa) and from sites close to anthropogenic and natural sources and sinks are excluded from the global estimate. (Details about this dataset are provided in [52]).

_{2}time series used in the study are depicted in Figure 6. They show a superposition of increasing trends and annual cycles whose amplitudes increase as we head from the South to the North Pole. The South Pole series has opposite phase of oscillation compared to the other three.

_{2}concentration) will become evident in Section 5. In the right panel of Figure 7, which depicts monthly statistics of the time series of the left panel, it is seen that in all sites but the South Pole, the annual maximum occurs in May; that of the South Pole occurs in September.

## 4. Methods

#### 4.1. Stochastic Framework

- A time reversible process is also stationary (Lawrance [53]).
- If a scalar process $\underset{\_}{x}\left(t\right)$ is Gaussian (i.e., all its finite dimensional distributions are multivariate normal) then it is reversible (Weiss [54]). The consequences are (a) a directional process cannot be Gaussian; (b) a discrete-time ARMA process (and a continuous-time Markov process) is reversible if and only if it is Gaussian.
- However, a vector (multivariate) process can be Gaussian and irreversible at the same time. A multivariate Gaussian linear process is reversible if and only if its autocovariance matrices are all symmetric (Tong and Zhang [55]).

_{t′}[occurring at time t′] is a prima facie cause of the event A

_{t}[occurring at time t] if and only if (i) ${t}^{\prime}<t,\left(\mathrm{ii}\right)P\left\{{B}_{{t}^{\prime}}\right\}0$, (iii) $P({A}_{t}|{B}_{{t}^{\prime}})>P\left({A}_{t}\right)$″. In addition, Granger’s [62] first axiom in defining causality reads, “The past and present may cause the future, but the future cannot”.

- If η
_{1}= 0, then there is no dominant direction. - If ${\eta}_{1}>0$, then the dominant direction is ${\underset{\_}{x}}_{\tau}\to {\underset{\_}{y}}_{\tau}$.
- If ${\eta}_{1}<0$, then the dominant direction is ${\underset{\_}{y}}_{\tau}\to {\underset{\_}{x}}_{\tau}$.

#### 4.2. Complications in Seeking Causality

when discussing the interpretation of a correlation coefficient or a regression, most textbooks warn that an observed relationship does not allow one to say anything about causation between the variables.

Determining true causality requires not only the establishment of a relationship between two variables, but also the far more difficult task of determining a direction of causality.

Results from Granger causality analyses neither establish nor require causality. Granger causality results do not reveal causal interactions, although they can provide evidence in support of a hypothesis about causal interactions.

coupled chaotic dynamical systems violate the first principle of Granger causality that the cause precedes the effect.[68]

#### 4.3. Additional Clarifications of Our Approach

- To make our assertions and, in particular, to use the “hen-or-egg” metaphor, we do not rely on merely statistical arguments. If we did that, based on our results presented in the next section, we would conclude that only the causality direction T → [CO
_{2}] exists. However, one may perform a thought experiment of instantly adding a big quantity of CO_{2}to the atmosphere. Would the temperature not increase? We believe it would, as CO_{2}is known to be a greenhouse gas. The causation in the opposite direction is also valid, as will be discussed in Section 6, “Physical Interpretation”. Therefore, we assert that both causality directions exist, and we are looking for the dominant one under the current climate conditions (those manifest in the datasets we use) instead of trying to make assertions of an exclusive causality direction. - While we occasionally use statistical tests (namely, the Granger test, Equations (14) and (15)), we opt to use, as the central point of our analyses, Equation (13) (and the conditions below it) because it is more intuitive and robust, fully reflects the basic causality axiom of time precedence, and is more straightforward, transparent (free of algorithmic manipulations), and easily reproducible (without the need for specialized software).
- For simplicity, we do not use any statistic other than correlation here. We stress that the system we are examining is indeed classified as Gaussian and, thus, it is totally unnecessary to examine any statistic in addition to correlation. The evidence of Gaussianity is provided by Figure A1 and Figure A2 in Appendix A.5, in terms of marginal distributions of the processes examined and in terms of their relationship. In particular, Figure A2 suggests a typical linear relationship for the bivariate process. We note that the linearity here is not a simplifying assumption or a coincidence as there are theoretical reasons implying it, which are related to the principle of maximum entropy [67,69].
- All in all, we adhere to simplicity and transparency and, in this respect, we illustrate our results graphically, so they are easily understandable, intuitive, and persuasive. Indeed, our findings are easily verifiable even from simple synchronous plots of time series, yet we also include plots of autocorrelations and lagged cross-correlation, which are also most informative in terms of time directionality.

## 5. Results

#### 5.1. Original Time Series

_{2}concentration while we keep T untransformed. Such a transformation has also been performed in previous studies, which consider the logarithm of CO

_{2}concentration as a proxy of total radiative forcing (e.g., [41]). However, by calling this quantity “forcing”, we indirectly give it, a priori (i.e., before investigating causation), the role of being the cause. Therefore, here, we avoid such interpretations; we simply call this variable the logarithm of carbon dioxide concentration and denote it as $\mathrm{ln}\left[{\mathrm{CO}}_{2}\right]$.

_{2}] → T. However, we deem that the entire picture is spurious as it is heavily affected by the fact that the autocorrelations are very high and, in particular, those of $\mathrm{ln}\left[{\mathrm{CO}}_{2}\right]$ are very close to 1 for all lags shown in the figure.

_{2}] → T, the null hypothesis is rejected at all usual significance levels. The attained p-value of the test is 1.8 × 10

^{−7}for one regression lag (η = 1), 1.8 × 10

^{−4}for η = 2, and remains below 0.01 for subsequent η. By contrast, in the direction T → [CO

_{2}], the null hypothesis is not rejected at all usual significance levels. The attained p-value of the test is 0.25 for η = 1, 0.22 for η = 2, and remains above 0.1 for subsequent η.

_{2}] dominates over [CO

_{2}] → T for 58% of the time. The attained p-value for direction T → [CO

_{2}] is lower than 1% for 1.4% of the time, much higher than in the opposite direction (0.3% of the time). All of these observations favour the T → [CO

_{2}] direction.

_{2}] dominates, attaining p-values as low as in the order of 10

^{−33}. However, we will avoid interpreting these results as unambiguous evidence that the consensus view (i.e., human activity is responsible for the observed warming) is wrong. Rather, what we want to stress is that it is inappropriate to draw conclusions from a methodology which is demonstrated to be so sensitive to the used time windows and data processing assumptions. In this respect, we have included this analyses in our study only (a) to show its weaknesses (which, for the reasons we explained in Section 4.2, we believe would not change if we used different statistics or different time series) and (b) to connect our study to earlier ones. For the sake of drawing conclusions, we contend that our full methodology in Section 4.1 and Section 4.3 is more appropriate. We apply this methodology in Section 5.2.

#### 5.2. Differenced Time Series

_{2}concentration and temperature (equivalent to thermal energy) represent “stocks”, i.e., stored quantities and, thus, the mass and energy fluxes are indeed represented by differences.

_{2}concentration at monthly scale; the symbols $\Delta T$ and $\Delta \mathrm{ln}\left[{\mathrm{CO}}_{2}\right]$ are used interchangeably with ${\underset{\_}{\tilde{x}}}_{\tau ,12}$ and ${\underset{\_}{\tilde{y}}}_{\tau ,12}$, respectively.

_{2}follows. However, there are cases where the changes in the two processes synchronize in time or even become decoupled.

_{2}follows.

_{2}concentration. This is not examined here (except a short note in the end of the section) as, given the focus in examining just the connection of the latter two processes, it lies out of our present scope.

_{2}time series, particularly for the Mauna Loa time series, in which the Hurst parameter appears to be close to 1/2. Based on this, one would exclude stationarity for the Mauna Loa CO

_{2}time series. However, a simpler interpretation of the graph is that the data record is not long enough to reveal that H = 0 for the differenced process. Actually, all available data belong to a period in which [CO

_{2}] exhibits a monotonic increasing trend (as also verified by the fact that all values of $\Delta \mathrm{ln}\left[{\mathrm{CO}}_{2}\right]$ in Figure 11 and Figure 12 are positive, while stationarity entails a zero mean of the differenced process). Had the available database been broader, both positive and negative trends could appear. Indeed, a broader view of the [CO

_{2}] process based on palaeoclimatic data (Figure 3 and Figure 4) would justify a stationarity assumption.

_{2}change follows in the same direction. We note, though, that temperature changes alternate in sign while CO

_{2}changes are always positive.

_{2}concentration; the autocorrelograms of the two processes are also plotted for comparison. The fact that the cross-correlogram does not have values consistently close to zero at any of the semi-axes eliminates the possibility of an exclusive (unidirectional) causality and suggests consistency with “hen-or-egg” causality.

_{2}], rather than [CO

_{2}] → T, as dominant causality direction. Similar are the graphs of the other combinations of temperature and CO

_{2}datasets, which are shown in Appendix A.5 (Figure A3, Figure A4, Figure A5, Figure A6 and Figure A7). In all cases, ${\eta}_{1}$ is positive, ranging from 5 to 11 months.

_{2}concentration so as to find a specification that maximizes the cross-correlation at the annual scale. In Figure 14, maximization occurs when the year specification is February–January (of the next year), i.e., if the lag is 8 months. The maximum cross-correlation is 0.66. If we keep the specification of the year for CO

_{2}concentration the same as in temperature (July–June), then maximization occurs at lag one year (12 months) and the maximum cross-correlation is 0.52. Table 1 summarizes the results for all combinations examined. The lags are always positive. They vary between 8 and 14 months for a sliding window specification and are 12 months for the fixed window specification. Most interestingly, the opposite phase in the annual cycle of CO

_{2}concentration in the South Pole, with respect to the other three sites, does not produce any noteworthy differences in the shape of the cross-correlogram nor in the time lags maximizing the cross-correlations.

- For the monthly scale and the causality direction [CO
_{2}] → T, the null hypothesis is not rejected at all usual significance levels for lag η = 1 and is rejected for significance level 1% for η = 2–8, with minimum attained p-value 1.8 × 10^{−4}for η = 6. - For the monthly scale and the causality direction T → [CO
_{2}], the null hypothesis is rejected at all usual significance levels for all lags η, with minimum attained p-value 2.1 × 10^{−8}for η = 7. - For the monthly scale, the attained p-values in the direction T → [CO
_{2}] are always smaller than in direction [CO_{2}] → T by about 4 to 5 orders of magnitude, thus clearly supporting T → [CO_{2}] as dominant direction. - For the annual scale with fixed year specification and the causality direction [CO
_{2}] → T, the null hypothesis is not rejected at all usual significance levels for any lag η, thus indicating that this causality direction does not exist. - For the annual scale with fixed year specification and the causality direction T → [CO
_{2}], the null hypothesis is not rejected at significance level 1% for all lags η = 1–6, with minimum attained p-value 5% for lag η = 2, thus supporting this causality direction at this significance level. - For the annual scale with fixed year specification, the attained p-values in the direction T → [CO
_{2}] are always smaller than in direction [CO_{2}] → T, again clearly supporting T → [CO_{2}] as the dominant direction.

_{2}produced by ENSO can lead to a misdiagnosis of the long-term cause of the recent atmospheric CO

_{2}increase”. Inspired by this comment, we have made a preliminary three-variable investigation using differenced temperatures (UAH), logarithms of CO

_{2}concentrations (Mauna Loa), and Equatorial South Oscillation index (SOI) characterizing ENSO. The investigation has been made on a monthly scale. $\Delta \mathrm{ln}\left[{\mathrm{CO}}_{2}\right]$ has been linearly regressed with $\Delta T$ and the running average of SOI for the previous 12 months. At synchrony (without applying any time lag), the correlation of SOI with $\Delta T$ is 0.40, higher than that of $\Delta T$ and $\Delta \mathrm{ln}\left[{\mathrm{CO}}_{2}\right]$ (0.24, as seen in Figure 14 at lag 0). The highest determination coefficient for the three regressed quantities is obtained when the time lag between $\Delta \mathrm{ln}\left[{\mathrm{CO}}_{2}\right]$ and $\Delta T$ is again 5 months, as in the two-variable case (the optimal lag for SOI is 0, but the regression is virtually insensitive to the change of that lag). Its value is ${r}^{2}=0.23$, corresponding to $r=0.48$, i.e., only slightly higher than the maximum cross-correlation coefficient of the two variable-case (which is 0.47 as seen in Table 1). In other words, by including ENSO in the modelling framework, the results do not change.

_{2}].

## 6. Physical Interpretation

_{s}, defined to be the flux of microbially and plant-respired CO

_{2}, clearly increases with temperature. It is known to have increased in the recent years [74,75]. Observational data of R

_{s}(e.g., [76,77]; see also [78]) show that the process intensity increases with temperature. Rate of chemical reactions, metabolic rate, as well as microorganism activity, generally increase with temperature. This has been known for more than 70 years (Pomeroy and Bowlus [79]) and is routinely used in engineering design.

_{2}in water decreases with increasing temperature [14,81]. In addition, photosynthesis must have increased, as in the 21st century the Earth has been greening, mostly due to CO

_{2}fertilization effects [82] and human land-use management [83]. Specifically, satellite data show a net increase in leaf area of 2.3% per decade [83]. The sums of carbon outflows from the atmosphere (terrestrial and maritime photosynthesis as well as maritime absorption) amount to 203 Gt C/year. The carbon inflows to the atmosphere amount to 207.4 Gt C/year and include natural terrestrial processes (respiration, decay, fire, freshwater outgassing as well as volcanism and weathering), natural maritime processes (respiration) as well as anthropogenic processes. The latter comprise human CO

_{2}emissions related to fossil fuels and cement production as well as land-use change, and amount to 7.7 and 1.1 Gt C/year, respectively. The change in carbon fluxes due to natural processes is likely to exceed the change due to anthropogenic CO

_{2}emissions, even though the latter are generally regarded as responsible for the imbalance of carbon in the atmosphere.

## 7. Conclusions

_{2}concentration plays a major role in enhancement of the greenhouse effect and contributes to global warming.

_{2}concentration due to anthropogenic emissions causes an increase of temperature, by considering the concept of reverse causality. The problem is obviously more complex than that of exclusive roles of cause and effect, qualifying as a “hen-or-egg” (“ὄρνις ἢ ᾠὸν”) causality problem, where it is not always clear which of two interrelated events is the cause and which the effect. Increased temperature causes an increase in CO

_{2}concentration and, hence, we propose the formulation of the entire process in terms of a “hen-or-egg” causality.

_{2}being the dominant, despite the fact that CO

_{2}→ T prevails in public, as well as in scientific, perception. Indeed, our results show that changes in CO

_{2}follow changes in T by about six months on a monthly scale, or about one year on an annual scale.

_{2}emission. Thus, the synchrony of rising temperature and CO

_{2}creates a positive feedback loop. This poses challenging scientific questions of interpretation and modelling for further studies. In this respect, we welcome the review by Connolly [14], which already proposes interesting interpretations within a wider epistemological framework and in connection with a recent study [84]. In our opinion, scientists of the 21st century should have been familiar with unanswered scientific questions as well as with the idea that complex systems resist simplistic explanations.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Data Availability

_{2}time series are readily available on monthly scale from http://climexp.knmi.nl. All NOAA CO

_{2}data are available from https://www.esrl.noaa.gov/gmd/ccgg/trends/gl_trend.html. The CO

_{2}data of Mauna Loa were retrieved from http://climexp.knmi.nl/data/imaunaloa_f.dat while the original measurements are in https://www.esrl.noaa.gov/gmd/dv/iadv/graph.php?code=MLO. The Barrow series is available (in irregular step) in https://www.esrl.noaa.gov/gmd/dv/iadv/graph.php?code=BRW, and the South Pole series in https://www.esrl.noaa.gov/gmd/dv/data/index.php?site=SPO. All these data were accessed (using the “Download data” link in the above sites) in June 2020. The global CO

_{2}series is accessed at https://www.esrl.noaa.gov/gmd/ccgg/trends/gl_data.html, of which the “Globally averaged marine surface monthly mean data” are used here. The palaeoclimatic data of Vostok CO

_{2}were retrieved from http://cdiac.ess-dive.lbl.gov/ftp/trends/co2/vostok.icecore.co2 (dated January 2003, accessed September 2018) and the temperature data from http://cdiac.ess-dive.lbl.gov/ftp/trends/temp/vostok/vostok.1999.temp.dat (dated January 2000, accessed September 2018).

## Appendix A

#### Appendix A.1. On Early Non-Systematic Measurements of CO_{2}

_{2}.

_{2}measurements. Indeed, it could be worthwhile to have a critical look at the historical data and to try to make order in them and utilize them. However, this would certainly warrant an individual paper with this particular aim.

_{2}concentration increases from 340 to 550 ppm (much more than in Beck´s Figure 5 discussed by Keeling [90] and Meijer [91] as quoted above), with weird seasonal behaviour. Beck himself admitted that the results for Giessen “need to be adjusted downwards to take account of anthropogenic sources of CO

_{2}from nearby city, an influence that has been estimated as lying between 10 and 70 ppm […] by different authors”.

_{2}emissions from fossil fuel burning”. On the other hand, Meijer [91] wrote: “The author even accuses the pioneers Callendar and [Charles David] Keeling of selective data use, errors or even something close to data manipulation”. In addition, [R.F.] Keeling [90] noted: “Beck is […] wrong when he asserts that the earlier data have been discredited only because they don’t fit a preconceived hypothesis of CO

_{2}and climate. […] Instead, the data have been ignored because they cannot be accepted as representative without violating our understanding of how fast the atmosphere mixes”.

#### Appendix A.2. Some Notes on the Averaged Differenced Process

#### Appendix A.3. Some Notes on Time Directionality of Causal Systems

#### Appendix A.4. Some Notes on the Alternative Procedures on Causality

Our thoughts and enquiries are, therefore, every moment, employed about this relation: Yet so imperfect are the ideas which we form concerning it, that it is impossible to give any just definition of cause, except what is drawn from something extraneous and foreign to it.

The probability${p}_{1}=P(Y=1|{X}_{f}=1)$of the event occurring in the real world, with f present, is referred to as factual, while${p}_{0}=P(Y=1|{X}_{f}=0)$is referred to as counterfactual. Both terms will become clear in the light of what immediately follows. The so-called fraction of attributable risk (FAR) is then defined asThe FAR is interpreted as the fraction of the likelihood of an event that is attributable to the external forcing.$$\mathrm{FAR}=1-\frac{{p}_{0}}{{p}_{1}}$$

- x = 1: being hot above a threshold;
- y = 1: wearing clothes with weight below a threshold;
- z = 1: sweat quantity above a threshold;

- Cold, heavy clothes: $P\left\{\underset{\_}{z}=1|\underset{\_}{x}=0,\underset{\_}{y}=0\right\}=0.2$
- Cold, light clothes: $P\left\{\underset{\_}{z}=1|\underset{\_}{x}=0,\underset{\_}{y}=1\right\}=0.1$
- Hot, heavy clothes: $P\left\{\underset{\_}{z}=1|\underset{\_}{x}=1,\underset{\_}{y}=0\right\}=0.95$
- Hot, light clothes: $P\left\{\underset{\_}{z}=1|\underset{\_}{x}=1,\underset{\_}{y}=1\right\}=0.80$

**Table A1.**Joint probabilities $P\left\{\underset{\_}{x}=x,\underset{\_}{y}=y,\underset{\_}{z}=z\right\}$ for all triplets $\left\{x,y,z\right\}$

x | y | z = 0 | z = 1 |
---|---|---|---|

0 | 0 | 0.38 | 0.095 |

0 | 1 | 0.0225 | 0.0025 |

1 | 0 | 0.00125 | 0.02375 |

1 | 1 | 0.095 | 0.38 |

$P\left\{\underset{\_}{z}=z\right\}=$ | 0.49875 | 0.50125 |

_{2}) are of low dimensionality, and we deem it unnecessary to discuss the issue further. We only note the fact that global temperature and CO

_{2}virtually behave as Gaussian, which enables reliable estimation of standard correlations and dismiss the need to use the overly complex and uncertain correlation sums.

#### Appendix A.5. Additional Graphical Depictions

**Figure A1.**Normal probability plots of ΔT and Δln[CO

_{2}] where T is the UAH temperature and [CO

_{2}] is the CO

_{2}concentration at Mauna Loa at monthly scale.

**Figure A2.**Scatter plot of $\Delta T$ and $\Delta \mathrm{ln}\left[{\mathrm{CO}}_{2}\right]$ where T is the UAH temperature and [CO

_{2}] is the CO

_{2}concentration at Mauna Loa at monthly scale; the two quantities are lagged in time using the optimal lag of 5 months (Table 1). The two linear regression lines are also shown in the figure.

**Figure A3.**Auto- and cross-correlograms of the differenced time series of UAH temperature and Barrow CO

_{2}concentration.

**Figure A4.**Auto- and cross-correlograms of the differenced time series of UAH temperature and South Pole CO

_{2}concentration.

**Figure A5.**Auto- and cross-correlograms of the differenced time series of UAH temperature and global CO

_{2}concentration.

**Figure A6.**Auto- and cross-correlograms of the differenced time series of CRUTEM4 temperature and Mauna Loa CO

_{2}concentration.

**Figure A7.**Auto- and cross-correlograms of the differenced time series of CRUTEM4 temperature and global CO

_{2}concentration.

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**Figure 1.**Annual change in global energy-related CO

_{2}emissions (adapted from International Energy Agency (IEA) [3]).

**Figure 2.**Atmospheric CO

_{2}concentration measured in Mauna Loa, Hawaii, USA, in the last four years.

**Figure 5.**Plots of the data series of global temperature “anomalies” since 1980, as used in the study, from satellite measurements over the globe (UAH) and from ground measurements over land (CRUTEM4).

**Figure 6.**Plots of the data series of atmospheric CO

_{2}concentration measured in Mauna Loa (Hawaii, USA), Barrow (Alaska, USA), and South Pole, and the global average.

**Figure 7.**Plots of atmospheric CO

_{2}concentration after standardization: (

**left**) each monthly value is standardized by dividing with the geometric average of the 5-year period before it. (

**right**) Monthly statistics of the values of the left panel; for each month, the average is shown in continuous line and the minimum and maximum in thin dashed lines of the same colour as the average.

**Figure 8.**Synchronous plots of the time series of UAH temperature and logarithm of CO

_{2}concentration at Mauna Loa at monthly scale.

**Figure 9.**Auto- and cross-correlograms of the time series of UAH temperature and logarithm of CO

_{2}concentration at Mauna Loa.

**Figure 10.**Plots of p-values of the Granger test for 10-year-long moving windows for the monthly time series of UAH temperature and logarithm of CO

_{2}concentration at Mauna Loa for number of lags (

**left**) η = 1 and (

**right**) η = 2. The time series used are (

**upper**) the original and (

**lower**) that obtained after “removing” the periodicity by averaging over the previous 12 months.

**Figure 11.**Differenced time series of UAH temperature and logarithm of CO

_{2}concentration at Mauna Loa at monthly scale. The graph in the upper panel was constructed in the manner described in the text. The graph in the lower panel is given for comparison and was constructed differently by taking differences of the values of each month with the previous month and then averaging over the previous 12 months (to remove periodicity); in addition, the lower graph includes the CRUTEM4 land temperature series.

**Figure 12.**Annually averaged time series of differenced temperatures (UAH) and logarithms of CO

_{2}concentrations (Mauna Loa). Each dot represents the average of a one-year duration ending at the time of its abscissa.

**Figure 13.**Empirical climacograms of the indicated differenced time series; the characteristic slopes corresponding to values of the Hurst parameter H = 1/2 (large-scale randomness), 0 (full antipersistence) and 1 (full persistence) are also plotted (note, H = 1 + slope/2).

**Figure 14.**Auto- and cross-correlograms of the differenced time series of UAH temperature and Mauna Loa CO

_{2}concentration.

**Table 1.**Maximum cross-correlation coefficient (MCCC) and corresponding time lag in months. The annual window for temperature is July–June, while for CO

_{2}, it is either different (sliding), determined so as to maximize MCCC, or the same (fixed).

Temperature—CO_{2} Series | Monthly Time Series | Annual Time Series—Sliding Annual Window | Annual Time Series—Fixed Annual Window | |||
---|---|---|---|---|---|---|

MCCC | Lag | MCCC | Lag | MCCC | Lag | |

UAH—Mauna Loa | 0.47 | 5 | 0.66 | 8 | 0.52 | 12 |

UAH—Barrow | 0.31 | 11 | 0.70 | 14 | 0.59 | 12 |

UAH—South Pole | 0.37 | 6 | 0.54 | 10 | 0.38 | 12 |

UAH—Global | 0.47 | 6 | 0.60 | 11 | 0.60 | 12 |

CRUTEM4—Mauna Loa | 0.31 | 5 | 0.55 | 10 | 0.52 | 12 |

CRUTEM4—Global | 0.33 | 9 | 0.55 | 12 | 0.55 | 12 |

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**MDPI and ACS Style**

Koutsoyiannis, D.; Kundzewicz, Z.W.
Atmospheric Temperature and CO_{2}: Hen-Or-Egg Causality? *Sci* **2020**, *2*, 83.
https://doi.org/10.3390/sci2040083

**AMA Style**

Koutsoyiannis D, Kundzewicz ZW.
Atmospheric Temperature and CO_{2}: Hen-Or-Egg Causality? *Sci*. 2020; 2(4):83.
https://doi.org/10.3390/sci2040083

**Chicago/Turabian Style**

Koutsoyiannis, Demetris, and Zbigniew W. Kundzewicz.
2020. "Atmospheric Temperature and CO_{2}: Hen-Or-Egg Causality?" *Sci* 2, no. 4: 83.
https://doi.org/10.3390/sci2040083