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Article

Atmospheric Temperature and CO2:

Hen-Or-Egg Causality?

Demetris Koutsoyiannis 1, * and Zbigniew W. Kundzewicz 2

1Department of Water Resources and Environmental Engineering, School of Civil Engineering,

National Technical University of Athens, 157 80 Athens, Greece

2Institute for Agricultural and Forest Environment, Polish Academy of Sciences, 60-809 Pozna´n, Poland;

kundzewicz@yahoo.com

*Correspondence: dk@itia.ntua.gr

Received: 7 September 2020; Accepted: 16 November 2020; Published: 25 November 2020

Abstract:

It is common knowledge that increasing CO

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 Tby 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.

Keywords: temperature; global warming; greenhouse gases; atmospheric CO2concentration

Π

ó

τερ

o

νἡὄρνις πρ

ó

τερ

o

νἢτὸ ᾠὸνἐγένετ

o(Which of the two came ﬁrst, the hen or the egg?).

Πλ

o

ύταρχ

o

ς

, H

θικά

,

Συµπ

o

σιακὰ

B,

Πρ

ó

βληµα Γ

(Plutarch, Moralia, Quaestiones

convivales, B, Question III).

1. Introduction

The phrase “hen-or-egg” is a metaphor describing situations where it is not clear which of two

interrelated events or processes is the cause and which the eﬀect. Plutarch was the ﬁrst to pose this

type of causality as a philosophical problem using the example of the hen and the egg, as indicated

in the motto above. We note that in the original Greek text, “

ἡ ὄρνις

” is feminine (article and noun),

meaning the hen rather than the chicken. Therefore, here, we preferred the form “hen-or-egg” over

“chicken-or-egg”, which is more common in English (Very often, in online Greek texts, e.g., [

1

,

2

],

“

ἡ ὄρνις

” appears as “

ἡ ἄρνις

”. After extended searching, we contend that this must be an error, either

an old one in copying of manuscripts, e.g., by monks in monasteries, or a modern one, e.g., in OCR.

We are conﬁdent that the correct word is “ὄρνις”).

The objective of the paper is to demonstrate that the relationship of atmospheric CO

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

Sci 2020,2, 83; doi:10.3390/sci2040083 www.mdpi.com/journal/sci

Sci 2020,2, 83 2 of 33

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, CO2emissions.

The analysis reported in this paper was prompted by observation of an unexpected

(and unfortunate) real-world experiment: during the COVID-19 lockdown in 2020, despite the

unprecedented decrease in carbon emissions (Figure 1), there was an increase in atmospheric CO

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 1also shows a rapid growth in emissions after the 2008–2009 global ﬁnancial crisis, which is in

agreement with Peters et al. [7].

Sci 2020, 3, x FOR PEER REVIEW 2 of 33

data-based palaeoclimatic studies, where the change in temperature leads and the change in CO₂

concentration follows. Next, we discuss the databases of modern (instrumental) measurements

related to global temperature and atmospheric CO₂ 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 CO₂

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 CO₂ suggests that the dominant direction is T → CO₂,

i.e., the change in temperature leads and the change in CO₂ concentration follows. We attempt to

interpret this latter mechanism by noting the positive feedback loop—higher temperatures increase

soil respiration and, hence, CO₂ emissions.

The analysis reported in this paper was prompted by observation of an unexpected (and

unfortunate) real-world experiment: during the COVID-19 lockdown in 2020, despite the

unprecedented decrease in carbon emissions (Figure 1), there was an increase in atmospheric CO₂

concentration, which followed a pattern similar to previous years (Figure 2). Indeed, according to the

International Energy Agency (IEA) [3], global CO₂ 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

CO₂ 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 CO₂ 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 CO₂ 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].

Figure 1. Annual change in global energy-related CO₂ emissions (adapted from International Energy

Agency (IEA) [3]).

Figure 1.

Annual change in global energy-related CO

2

emissions (adapted from International Energy

Agency (IEA) [3]).

Sci 2020,2, 83 3 of 33

Sci 2020, 3, x FOR PEER REVIEW 3 of 33

Figure 2. Atmospheric CO₂ concentration measured in Mauna Loa, Hawaii, USA, in the last four

years.

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

Does the relationship of atmospheric carbon dioxide (CO₂) 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 CO₂] 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.

Furthermore, following the Italian meteorologist De Marchi (1895, [9]), whom he cited, he

rejected what we call today Milanković cycles as possible causes of the glacial periods. In addition, he

substantially overestimated the role of CO₂ in the greenhouse effect of the Earth’s atmosphere. He

calculated the relative weights of absorption of CO₂ and water vapour as 1.5 and 0.88, respectively,

or a ratio of 1:0.6.

Arrhenius [8] also stated that “if the quantity of carbonic acid increases in geometric progression,

the augmentation of the temperature will increase nearly in arithmetic progression”. This Arrhenius’s

“rule” (which is still in use today) is mathematically expressed as

𝑇−𝑇

=𝛼ln[CO]

[CO] (1)

where T and [CO] denote temperature and CO₂ concentration, respectively, T0 and [CO]

represent reference states, and α is a constant.

Here, it is useful to note that Arrhenius’s studies were not the first on the subject. Arrhenius [8]

cites several other authors, among whom Tyndall (1865, [10]) for pointing out the enormous

importance of atmospheric absorption of radiation and for having the opinion that water vapour has

the greatest influence on this. Interestingly, it appears that the first experiments on the ability of water

vapour and carbon dioxide to absorb heat were undertaken even earlier by Eunice Newton Foote

(1856, [11]), even though she did not recognize the underlying mechanisms or even the distinction of

short- and long-wave radiation [12–14]).

While the fact that the two variables are tightly connected is beyond doubt, the direction of the

simple causal relationship needs to be studied further. Today, additional knowledge has been

402

404

406

408

410

412

414

416

418

123456789101112

CO₂ concentration (ppm)

Month

2020

2019

2018

2017

Figure 2.

Atmospheric CO

2

concentration measured in Mauna Loa, Hawaii, USA, in the last four years.

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

Does the relationship of atmospheric carbon dioxide (CO

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.

Furthermore, following the Italian meteorologist De Marchi (1895, [

9

]), whom he cited, he rejected

what we call today Milankovi´c cycles as possible causes of the glacial periods. In addition, he substantially

overestimated the role of CO

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.

Arrhenius [

8

] also stated that “if the quantity of carbonic acid increases in geometric progression,

the augmentation of the temperature will increase nearly in arithmetic progression”. This Arrhenius’s

“rule” (which is still in use today) is mathematically expressed as

T−T0=αln [CO2]

[CO2]0!(1)

where Tand [CO

2

] denote temperature and CO

2

concentration, respectively, T

0

and [CO

2

]

0

represent

reference states, and αis a constant.

Here, it is useful to note that Arrhenius’s studies were not the ﬁrst on the subject. Arrhenius [

8

]

cites several other authors, among whom Tyndall (1865, [

10

]) for pointing out the enormous importance

of atmospheric absorption of radiation and for having the opinion that water vapour has the greatest

inﬂuence on this. Interestingly, it appears that the ﬁrst experiments on the ability of water vapour

and carbon dioxide to absorb heat were undertaken even earlier by Eunice Newton Foote (1856, [

11

]),

even though she did not recognize the underlying mechanisms or even the distinction of short- and

long-wave radiation [12–14]).

While the fact that the two variables are tightly connected is beyond doubt, the direction

of the simple causal relationship needs to be studied further. Today, additional knowledge has

Sci 2020,2, 83 4 of 33

been accumulated, particularly from palaeoclimatic studies, which allow us to examine Arrhenius’s

hypotheses on a sounder basis. In brief, we can state the following:

•

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

–

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 Milankovi´c 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.

Despite falsiﬁcation of some of Arrhenius’s hypotheses, his line of thought remained dominant.

Yet, there have been some important studies, based on palaeoclimatological reconstructions (mostly

the Vostok ice cores [

25

,

26

]), which have pointed to the opposite direction of causality, i.e., the change

in temperature as the cause and that of CO2concentration 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

–

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.

Since palaeoclimatic data suggest a direction opposite to that assumed by Arrhenius,

Koutsoyiannis [

30

], using palaeoclimatic data from the Vostok ice cores at a time resolution of 1000 years

and a stochastic framework similar to that of the present study (see Section 4.1), concluded that a

change in temperature precedes that of CO

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.

Sci 2020,2, 83 5 of 33

Sci 2020, 3, x FOR PEER REVIEW 4 of 33

accumulated, particularly from palaeoclimatic studies, which allow us to examine Arrhenius’s

hypotheses on a sounder basis. In brief, we can state the following:

• Indeed, CO₂ plays a substantial role as a greenhouse gas. However, modern estimates of the

contribution of CO₂ to the greenhouse effect differ largely from Arrhenius’s results, attributing

19% of the long-wave radiation absorption to CO₂ 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 CO₂ is beyond

doubt, this is not clear in Figure 3, where it is seen that the CO₂ 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 [14,18–23]. The T–CO₂ 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 CO₂

concentration, that explains the glaciation process. Specifically (quoting Roe [24]),

variations in atmospheric CO₂ appear to lag the rate of change of global ice volume. This implies only

a secondary role for CO₂—variations in which produce a weaker radiative forcing than the orbitally-

induced changes in summertime insolation—in driving changes in global ice volume.

Despite falsification of some of Arrhenius’s hypotheses, his line of thought remained dominant.

Yet, there have been some important studies, based on palaeoclimatological reconstructions (mostly

the Vostok ice cores [25,26]), which have pointed to the opposite direction of causality, i.e., the change

in temperature as the cause and that of CO₂ concentration as the effect. Such claims have been based

on the fact that temperature change leads and CO₂ 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 [27–32]. Claims that CO₂

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

CO₂ 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 CO₂ is an internal

feedback in Earth’s climate system rather than an initial trigger.

Figure 3. Proxy-based reconstructions of global mean temperature and CO₂ concentration during the

Phanerozoic Eon. The temperature reconstruction by Scotese [33] was mainly based on proxies from

[34–36], while the CO₂ 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].

10

15

20

25

30

35

40

05010015020025030035040045050055 0600

Temperature T(°C)

Scotese (2018)

10

100

1,000

10,000

05010015020025030035040045050055 0600

[CO₂] (ppmv)

Million years before present

Davis (2017) Berner (2008) Ekart et al. (1999)

Proterozoic

Ediacaran Cambrian Ordovic-

ian

Silur-

ian Devonian Carboni-

ferous Permian Triassi c Juras sic Cretaceous Paleo-

gene

Neo-

gene

Mesozoic Ceno zoic

Quaternary

Paleozoic

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

–

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].

Sci 2020, 3, x FOR PEER REVIEW 5 of 33

Figure 4. (upper) Time series of temperature and CO₂ 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]).

Since palaeoclimatic data suggest a direction opposite to that assumed by Arrhenius,

Koutsoyiannis [30], using palaeoclimatic data from the Vostok ice cores at a time resolution of 1000

years and a stochastic framework similar to that of the present study (see Section 4.1), concluded that

a change in temperature precedes that of CO₂ 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.

Studies exploring the rich body of modern datasets have also been published. Most of the studies

have been based on the so-called “Granger causality test” (see Section 4.2). To mention a few, Kodra

et al. [41], after testing several combinations and lags within the Granger framework, did not find

any statistically significant results at the usual 5% significance level (they only found 2 cases at the

10% significance level; see their Tables 2 and 3). Stern and Kaufmann [42] studied, again within the

Granger framework, the causality between radiative forcing and temperature, and found that both

natural and anthropogenic forcings cause temperature change, and also that the inverse is true, i.e.,

temperature causes greenhouse gas concentration changes. They concluded that their 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.

By contrast, Stips et al. [43] used a different method [44] to investigate the causal structure and

concluded that their

180

200

220

240

260

280

300

320

-10

-8

-6

-4

-2

0

2

4

0100200300400

Temperature difference from present (°C)

CO₂ concetration (ppmv)

Thousand years before present

TCO₂

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-40-30-20-10 0 10203040

Correlation coefficient

Lag (thousand years)

T

CO₂

T - CO₂

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]).

Studies exploring the rich body of modern datasets have also been published. Most of the

studies have been based on the so-called “Granger causality test” (see Section 4.2). To mention a few,

Kodra et al. [

41

], after testing several combinations and lags within the Granger framework, did not

ﬁnd any statistically signiﬁcant results at the usual 5% signiﬁcance level (they only found 2 cases at the

Sci 2020,2, 83 6 of 33

10% signiﬁcance level; see their Tables 2 and 3). Stern and Kaufmann [

42

] studied, again within the

Granger framework, the causality between radiative forcing and temperature, and found that both

natural and anthropogenic forcings cause temperature change, and also that the inverse is true, i.e.,

temperature causes greenhouse gas concentration changes. They concluded that their 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.

By contrast, Stips et al. [

43

] used a diﬀerent method [

44

] to investigate the causal structure and

concluded that their

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 CO2, are the main causal drivers of the recent warming.

Here, we use a diﬀerent path to study the causal relation between temperature and CO

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.

Another diﬀerence of our study, from most of the earlier ones, is our focus on changes, rather than

current states, in the processes we investigate. This puts the technique of process diﬀerencing in

central place in our analyses. This technique is quite natural and also powerful for studying time

directionality [

30

]. We note that diﬀerencing has also been used in a study by Humlum et al. [

45

],

which has several similarities with our study, even though it is not posed in a formal causality context,

as well as in the study by Kodra et al. [

41

]. However, diﬀerencing has been criticized for potentially

eliminating long-run eﬀects and, hence, providing information only on short-run eﬀects [

42

,

46

]. Even if

this speculation were valid, it would not invalidate the diﬀerencing technique for the following reasons:

•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

Our investigation of the relationship of temperature with concentration of carbon dioxide in

the atmosphere is based on two time series of the former process and four of the latter. Speciﬁcally,

the temperature data are of two origins, satellite and ground-based. The satellite dataset, developed

at the University of Alabama in Huntsville (UAH), infers the temperature, T, of three broad levels

of the atmosphere from satellite measurements of the oxygen radiance in the microwave band using

advanced (passive) microwave sounding units on NOAA and NASA satellites [

47

,

48

]. The data are

publicly available on the monthly scale in the forms of time series of “anomalies” (deﬁned as diﬀerences

from long-term means) for several parts of earth as well as in maps. Here, we use only the global

average on monthly scale for the lowest level, referred to as the lower troposphere. The ground-based

data series we use is the CRUTEM.4.6.0.0 global T2m land temperature [

49

]. This originates from a

gridded dataset of historical near-surface air temperature anomalies over land. Data are available for

each month from January 1850 to the present. The dataset is a collaborative product of the Met Oﬃce

Sci 2020,2, 83 7 of 33

Hadley Centre and the Climatic Research Unit at the University of East Anglia. We note that both

sources of information, UAH and CRUTEM, provide time series over the globe, land, and oceans; here,

we deliberately use one source for the globe and one for the land.

The two temperature series used in the study are depicted in Figure 5. They are consistent with

each other (and correlated, r=0.8), though the CRUTEM4 series shows a larger increasing trend than

the UAH series. The diﬀerences can be explained by three reasons: (a) the UAH series includes both

land and sea, while the chosen CRUTEM4 series is for land only, in which the increasing trend is

substantially higher than in sea; (b) the UAH series refers to some high altitude in the troposphere

(see details in Koutsoyiannis [

50

]), while the CRUTEM4 series refers to the ground level; and (c) the

ground-based CRUTEM4 series might be aﬀected by urbanization (many ground stations are located

in urban areas). In any case, the diﬀerence in the increasing trends is irrelevant for the current study,

as the timing, rather than the magnitude, of changes is the determinant of causality. This will manifest

in our results.

Sci 2020, 3, x FOR PEER REVIEW 7 of 33

as the timing, rather than the magnitude, of changes is the determinant of causality. This will manifest

in our results.

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).

The most famous CO₂ 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 CO₂ 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 CO₂ 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]).

The period of data coverage varies, but all series cover the common 40-year period 1980–2019

which, hence, constitute the time reference of all our analyses. As a slight exception, the Barrow

(Alaska) and South Pole measurements have not yet been available in final form for 2019 and, thus,

this year was not included in our analyses of these two time series. The data of the latter two stations

are given in irregular-step time series, which was regularized (by interpolation) to monthly in this

study. All other data series have already been available on a monthly scale. While some of the earlier

studies refer to a longer time span (e.g., [42,43] which start from 1850s), here, we avoid using non-

systematic data earlier than 1980 due to their low reliability and bypass the raised controversies

explained in Appendix A.1.

All four CO₂ 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.

The annual cycle is better seen in Figure 7, where we have removed the trend with

standardization, namely by dividing each monthly value by the geometric average of the preceding

5-year period. The reason why we used division rather than subtraction and geometric rather than

arithmetic average (being thus equivalent to subtracting or averaging the logarithms of CO₂

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.

-1

-0.5

0

0.5

1

1.5

2

2.5

1980 1985 1990 1995 2000 2005 2010 2015 2020

T (°C)

𝑇(UAH) 𝑇(CRUTEM4)

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).

The most famous CO

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]).

The period of data coverage varies, but all series cover the common 40-year period 1980–2019

which, hence, constitute the time reference of all our analyses. As a slight exception, the Barrow

(Alaska) and South Pole measurements have not yet been available in ﬁnal form for 2019 and, thus,

this year was not included in our analyses of these two time series. The data of the latter two stations

are given in irregular-step time series, which was regularized (by interpolation) to monthly in this study.

All other data series have already been available on a monthly scale. While some of the earlier studies

Sci 2020,2, 83 8 of 33

refer to a longer time span (e.g., [42,43] which start from 1850s), here, we avoid using non-systematic

data earlier than 1980 due to their low reliability and bypass the raised controversies explained in

Appendix A.1.

All four CO

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.

Sci 2020, 3, x FOR PEER REVIEW 8 of 33

Figure 6. Plots of the data series of atmospheric CO₂ concentration measured in Mauna Loa (Hawaii,

USA), Barrow (Alaska, USA), and South Pole, and the global average.

Figure 7. Plots of atmospheric CO₂ 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.

4. Methods

4.1. Stochastic Framework

A recent study [30] investigated time irreversibility in hydrometeorological processes and

developed a theoretical framework in stochastic terms. It also studied necessary conditions for

causality, which is tightly linked to time irreversibility. A simple definition of time reversibility

within stochastics is the following, where underlined symbols denote stochastic (random) variables

and non-underlined ones denote values thereof or regular variables.

A stochastic process 𝑥(𝑡) at continuous time t, with nth order distribution function:

𝐹(𝑥,𝑥,…,𝑥;𝑡,𝑡,…,𝑡) ≔𝑃𝑥(𝑡)≤𝑥

,𝑥(𝑡)≤𝑥

,…,𝑥(𝑡)≤𝑥

(2)

is time-symmetric or time-reversible if its joint distribution does not change after reflection of time

about the origin, i.e., if for any n, 𝑡,𝑡,…,𝑡,

𝐹(𝑥,𝑥,…,𝑥;𝑡,𝑡,…,𝑡)=𝐹(𝑥,𝑥,…,𝑥;−𝑡,−𝑡,…,−𝑡). (3)

If times 𝑡 are equidistant,i.e.,𝑡−𝑡 =𝐷, the definition can be also written by reflecting the

order of points in time, i.e.,

𝐹(𝑥,𝑥,…,𝑥,𝑥;𝑡,𝑡,…,𝑡,𝑡)=𝐹(𝑥,𝑥,…,𝑥,𝑥;𝑡,𝑡,…,𝑡,𝑡). (4)

330

340

350

360

370

380

390

400

410

420

198019851990199520002005201020152020

CO₂ concentration (ppm)

Mauna Loa, Hawaii

Barrow, Alaska

South Pole

Global

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1980 1985 1990 1995 2000 2005 2010 2015 2020

Standardized CO₂ concentration

Mauna Loa, Hawaii Barrow, Alaska South Pole Global

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

123456789101112

Standardized CO₂ concentration statistics

Month

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.

The annual cycle is better seen in Figure 7, where we have removed the trend with standardization,

namely by dividing each monthly value by the geometric average of the preceding 5-year period.

The reason why we used division rather than subtraction and geometric rather than arithmetic average

(being thus equivalent to subtracting or averaging the logarithms of CO

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.

Sci 2020, 3, x FOR PEER REVIEW 8 of 33

Figure 6. Plots of the data series of atmospheric CO₂ concentration measured in Mauna Loa (Hawaii,

USA), Barrow (Alaska, USA), and South Pole, and the global average.

Figure 7. Plots of atmospheric CO₂ 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.

4. Methods

4.1. Stochastic Framework

A recent study [30] investigated time irreversibility in hydrometeorological processes and

developed a theoretical framework in stochastic terms. It also studied necessary conditions for

causality, which is tightly linked to time irreversibility. A simple definition of time reversibility

within stochastics is the following, where underlined symbols denote stochastic (random) variables

and non-underlined ones denote values thereof or regular variables.

A stochastic process 𝑥(𝑡) at continuous time t, with nth order distribution function:

𝐹(𝑥,𝑥,…,𝑥;𝑡,𝑡,…,𝑡) ≔𝑃𝑥(𝑡)≤𝑥

,𝑥(𝑡)≤𝑥

,…,𝑥(𝑡)≤𝑥

(2)

is time-symmetric or time-reversible if its joint distribution does not change after reflection of time

about the origin, i.e., if for any n, 𝑡,𝑡,…,𝑡,

𝐹(𝑥,𝑥,…,𝑥;𝑡,𝑡,…,𝑡)=𝐹(𝑥,𝑥,…,𝑥;−𝑡,−𝑡,…,−𝑡). (3)

If times 𝑡 are equidistant,i.e.,𝑡−𝑡 =𝐷, the definition can be also written by reflecting the

order of points in time, i.e.,

𝐹(𝑥,𝑥,…,𝑥,𝑥;𝑡,𝑡,…,𝑡,𝑡)=𝐹(𝑥,𝑥,…,𝑥,𝑥;𝑡,𝑡,…,𝑡,𝑡). (4)

330

340

350

360

370

380

390

400

410

420

198019851990199520002005201020152020

CO₂ concentration (ppm)

Mauna Loa, Hawaii

Barrow, Alaska

South Pole

Global

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1980 1985 1990 1995 2000 2005 2010 2015 2020

Standardized CO₂ concentration

Mauna Loa, Hawaii Barrow, Alaska South Pole Global

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

123456789101112

Standardized CO₂ concentration statistics

Month

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.

Sci 2020,2, 83 9 of 33

4. Methods

4.1. Stochastic Framework

A recent study [

30

] investigated time irreversibility in hydrometeorological processes and

developed a theoretical framework in stochastic terms. It also studied necessary conditions for

causality, which is tightly linked to time irreversibility. A simple deﬁnition of time reversibility within

stochastics is the following, where underlined symbols denote stochastic (random) variables and

non-underlined ones denote values thereof or regular variables.

A stochastic process x(t)at continuous time t, with nth order distribution function:

F(x1,x2,. . . ,xn;t1,t2,. . . ,tn):=Pnx(t1)≤x1,x(t2)≤x2,. . . ,x(tn)≤xno(2)

is time-symmetric or time-reversible if its joint distribution does not change after reﬂection of time

about the origin, i.e., if for any n,t1,t2,. . . ,tn,

F(x1,x2,. . . ,xn;t1,t2,. . . ,tn)=F(x1,x2,. . . ,xn;−t1,−t2,. . . ,−tn). (3)

If times

tiare equidistant

,

i

.

e

.,

ti−ti−1=D

, the deﬁnition can be also written by reﬂecting the

order of points in time, i.e.,

F(x1,x2,. . . ,xn−1,xn;t1,t2,. . . ,tn−1,tn)=F(x1,x2,. . . ,xn−1,xn;tn,tn−1,. . . ,t2,t1). (4)

A process that is not time-reversible is called time-asymmetric, time-irreversible, or time-directional.

Important results related to time (ir)reversibility are the following:

•A time reversible process is also stationary (Lawrance [53]).

•

If a scalar process

x(t)

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]).

Time asymmetry of a process can be studied more conveniently (or even exclusively in a scalar

process) through the diﬀerenced process, i.e.,

˜

xτ,ν:=xτ+ν−xτ(5)

for an appropriate time-step

ν

of diﬀerencing. The diﬀerenced process represents change of the original

process within a time period of length

ν

. We further deﬁne the cumulative process of

xτ

for discrete

time κas

Xκ:=x1+x2+. . . +xκ. (6)

Through this, we ﬁnd that the time average of the original process xτfor discrete time scale κis

x(κ)

τ:=x(τ−1)κ+1+x(τ−1)κ+2+. . . +xτκ

κ=Xτκ −X(τ−1)κ

κ. (7)

Similar equations for the cumulative and averaged processes for the diﬀerenced process

˜

xτ,ν

are

given in Appendix A.2.

The variance of the process

x(κ)

τ

is a function of the time scale

κ

, which is termed the climacogram

of the process:

γκ:=var[x(κ)

τ]. (8)

Sci 2020,2, 83 10 of 33

The autocovariance function for time lag

η

is derived from the climacogram through the

relationship [56]

cη=(η+1)2γ|η+1|+(η−1)2γ|η−1|

2−η2γ|η|. (9)

For suﬃciently large κ(theoretically as κ→ ∞), we may approximate the climacogram as

γκ∝κ2H−2(10)

where His termed the Hurst parameter. The theoretical validity of such (power-type) behaviour of

a process was implied by Kolmogorov (1940 [

57

]). The quantity 2H—2 is visualized as the slope of

the double logarithmic plot of the climacogram for large time scales. In a random process, H=1/2,

while in most natural processes, 1/2

≤

H

≤

1, as ﬁrst observed by Hurst (1951 [

58

]). This natural

behaviour is known as (long-term) persistence or Hurst–Kolmogorov (HK) dynamics. A high value of H

(approaching 1) indicates enhanced presence of patterns, enhanced change, and enhanced uncertainty

(e.g., in future predictions). A low value of H(approaching 0) indicates enhanced ﬂuctuation or

antipersistence (sometimes misnamed as quasi-periodicity as the period is not constant).

For a stationary stochastic process xτ, the diﬀerenced process ˜

xτhas mean zero and variance:

˜

γν,1 :=var[˜

xτ,ν] = var[xκ+ν] + var[xτ]−2cov[xτ+ν,xτ] = 2(γ1−cν)(11)

where

γ1and cν

are the variance and lag

ν

autocovariance, respectively, of

xτ

. Furthermore, it has

been demonstrated [

30

] that the Hurst coeﬃcient of the diﬀerenced process

˜

xτ

precisely equals zero,

which means that ˜

xτis completely antipersistent, irrespective of γκ.

To study irreversibility in vector processes, we can use second-order moments and, in particular,

cross-covariances among the diﬀerent components of the vector. In particular (adapting and simplifying

the analyses and results in Koutsoyiannis, [

30

]), given two processes

xτand yτ

, we could study the

cross-correlations:

r˜

x˜

y[ν,η]:=corr˜

xτ,ν,˜

yτ+η,ν. (12)

Time (ir)reversibility could then be characterized by studying the properties of symmetry

or asymmetry of

r˜

x˜

y(ν,η)

as a function of the time lag

η

. In a symmetric bivariate process,

r˜

x˜

y[ν,η]=r˜

x˜

y[ν,−η]

, and if the two components are positively correlated, the maximum of

r˜

x˜

y[ν,η]will appear at lag η=

0. If the bivariate process is irreversible, this maximum will appear at a

lag η1,0 and its value will be r˜

x˜

y[ν,η1].

Time asymmetry is closely related to causality, which presupposes irreversibility. Thus, “no causal

process (i.e., such that of two consecutive phases, one is always the cause of the other) can be reversible”

(Heller, [

59

]; see also [

60

]). In probabilistic deﬁnitions of causality, time asymmetry is determinant.

Thus, Suppes [

61

] deﬁnes causation thus: “An event B

t0

[occurring at time t

0

] is a prima facie cause

of the event A

t

[occurring at time t] if and only if (i)

t0<t

,

(ii)P{Bt0}>

0, (iii)

P(AtBt0)>P(At)

”.

In addition, Granger’s [

62

] ﬁrst axiom in deﬁning causality reads, “The past and present may cause the

future, but the future cannot”.

Consequently, in simple causal systems, in which the process component

xτ

is the cause of

yτ

(like

in the clear case of rainfall and runoﬀ, respectively), it is reasonable to expect

r˜

x˜

y[ν,η]≥

0 for any

η≥

0,

while

r˜

x˜

y[ν,η]=

0 for any

η=

0. However, in “hen-or-egg” causal systems, this will not be the case,

and we reasonably expect

r˜

x˜

y[ν,η],

0 for any

η

. Yet, we can deﬁne a dominant direction of causality

based on the time lag η1maximizing cross-correlation. Formally, η1is deﬁned for a speciﬁed νas

η1:=argmax

ηr˜

x˜

y(ν,η). (13)

We can thus distinguish the following three cases:

•If η1=0, then there is no dominant direction.

Sci 2020,2, 83 11 of 33

•If η1>0, then the dominant direction is xτ→yτ.

•If η1<0, then the dominant direction is yτ

→xτ.

Justiﬁcation and further explanations of these conditions are provided in Appendix A.3.

4.2. Complications in Seeking Causality

It must be stressed that the above conditions are considered as necessary and not suﬃcient

conditions for a causative relationship between the processes

xτand yτ

. Following Koutsoyiannis [

30

]

(where additional necessary conditions are discussed), we avoid seeking suﬃcient conditions, a task

that would be too diﬃcult or impossible due to its deep philosophical complications as well as the

logical and technical ones.

Speciﬁcally, it is widely known that correlation is not causation. As Granger [62] puts it,

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.

Perhaps that is the reason why Suppes [

61

] uses the term “prima facie cause” in his deﬁnition

given above which, however, he does not explain, apart for attributing “prima facie” to Jaakko Hintikka.

Furthermore, Suppes discusses spurious causes and eventually deﬁnes the genuine cause as a “prima

facie cause that is not spurious”; he also discusses the very existence of genuine causes which under

certain conditions (e.g., in a Laplacean universe) seems doubtful.

Granger himself also uses the term “prima facie cause”, while Granger and Newbold [

63

] note

that a cause satisfying a causality test still remains prima facie because it is always possible that,

if a diﬀerent information set were used, it would fail the new test. Despite the caution issued by its

pioneers, including Granger, through the years, the term “Granger causality” has become popular

(particularly in the so-called “Granger causality test”, e.g., [

64

]). Probably because of that misleading

term, the technique is sometimes thought of as one that establishes causality, thus resolving or

overcoming the “correlation is not causation” problem. In general, it has rarely been understood that

identifying genuine causality is not a problem of choosing the best algorithm to establish a statistical

relationship (including its directionality) between two variables. As an example of misrepresentation

of the actual problems, see [65], which contains the statement:

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.

In essence, the “Granger causality test” studies the improvement of prediction of a process

yτ

by

considering the inﬂuence of a “causing” process xτthrough the Granger regression model:

yτ=

η

X

j=1

ajyτ−j+

η

X

j=1

bjxτ−j+ετ(14)

where

aj

and

bj

are the regression coeﬃcients and

ετ

is an error term. The test is based on the null

hypothesis that the process xτis not actually causing yτ, formally expressed as

H0:b1=b2=. . . =bη=0 (15)

Algorithmic details of the test are given in [

64

], among others. The rejection of the null hypothesis

is commonly interpreted in the literature with a statement that xτ“Granger-causes” yτ.

This is clearly a misstatement and, in fact, the entire test is based on correlation matrices. Thus,

it again reﬂects correlation rather than causation. The rejection of the null hypothesis signiﬁes

improvement of prediction and this does not mean causation. To make this clearer, let us consider the

following example: people sweat when the atmospheric temperature is high and also wear light clothes.

Thus, it is reasonably expected that in the prediction of sweat quantity, temperature matters. In the

Sci 2020,2, 83 12 of 33

absence of temperature measurements (e.g., when we have only visual information, like when watching

a video), the weight of the clothes algorithmically improves the prediction of sweat quantity. However,

we could not say that the decrease in clothes weight causes an increase in sweat (the opposite is more

reasonable and becomes evident in a three-variable regression, temperature–clothes weight–sweat,

as further detailed in Appendix A.4).

Cohen [

66

] suggested replacing the term “Granger causality” with “Granger prediction” after

correctly pointing out this:

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.

To avoid such philosophical and logical complications, here, we replace the “prima facie” or

“Granger” characterization of a cause and, as we already explained, we abandon seeking for genuine

causes by using the notion of necessary conditions for causality. One could say that if two processes

satisfy the necessary conditions, then they deﬁne a prima facie causality, but we avoid stressing

that as we deem it unnecessary. Furthermore, we drop “causality” from “Granger causality test”,

thus hereinafter calling it “Granger test”.

Some have thought they can approach genuine causes and get rid of the caution “correlation is not

causation” by replacing the correlation with other statistics in the mathematical description of causality.

For example, Liang [

44

] uses the concept of information (or entropy) ﬂow (or transfer) between two

processes; this method has been called “Liang causality” in the already cited work he co-authors [

43

].

The usefulness of such endeavours is not questioned, yet their vanity to determine genuine causality is

easy to infer: It suﬃces to consider the case where the two processes, for which causality is studied,

are jointly Gaussian. It is well known that in any multivariate Gaussian process, the covariance matrix

(or the correlation matrix along with the variances) fully determines all properties of the multivariate

distribution of any order. For example, the mutual information in a bivariate Gaussian process is

(Papoulis, [67])

H[y|x] = ln σyq2πe(1−r2)(16)

where

σ

and rdenote standard deviation and correlation, respectively. Thus, using any quantity

related to entropy (equivalently, information), is virtually identical to using correlation. Furthermore,

in Gaussian processes, whatever statistic is used in describing causality, it is readily reduced to

correlation. This is evident even in Liang [

44

], where, e.g., in his Equation (102), the information ﬂow

turns out to be the correlation coeﬃcient multiplied by a constant. In other words, the big philosophical

problem of causality cannot be resolved by technical tricks.

From what was exposed above (Section 4.1), the time irreversibility (or directionality) is most

important in seeking causality. In this respect, we certainly embrace Suppes’s condition (i) and

Granger’s ﬁrst axiom, as stated above. Furthermore, we believe there is no meaning in refusing that

axiom and continuing to speak about causality. We note though that there have been recent attempts

to show that

coupled chaotic dynamical systems violate the ﬁrst principle of Granger causality that the cause

precedes the eﬀect. [68]

Apparently, however, the particular simulation experiment performed in the latter work which,

notably, is not even accompanied by any attempt for deduction based on stochastics, cannot show any

violation. In our view, such a violation, if it indeed happened, would be violation of logic and perhaps

of common sense.

Additional notes for other procedures detecting causality, which are not included in the focus of

our study, are given in Appendix A.4.

Sci 2020,2, 83 13 of 33

4.3. Additional Clariﬁcations of Our Approach

After the above theoretical and methodological discourse, we can clarify our methodological

approach by emphasizing the following points.

1.

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.

2.

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).

3.

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 Figures A1 and 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

].

4.

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

Here, we examine the relationship of atmospheric temperature and carbon dioxide concentration

using the available modern data (observations rather than proxies) in monthly time steps, as described in

Section 3. To apply our stochastic framework, we must ﬁrst make the two time series linearly compatible.

Speciﬁcally, based on Arrhenius’s rule (Equation (1)), we take the logarithms of CO

2

concentration

while we keep Tuntransformed. 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 ln[CO2].

A synchronous plot of the two processes (speciﬁcally, UAH temperature and

ln[CO2]

at Mauna

Loa) is depicted in Figure 8. Very little can be inferred from this ﬁgure alone. Both processes show

increasing trends and thus appear as positively correlated. On the other hand, the two processes

appear to have diﬀerent behaviours. Temperature shows an erratic behaviour while

ln[CO2]

has a

smooth evolution marked by the annual periodicity. It looks impossible to infer causality from that

graph alone.

Sci 2020,2, 83 14 of 33

Sci 2020, 3, x FOR PEER REVIEW 13 of 33

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

Here, we examine the relationship of atmospheric temperature and carbon dioxide

concentration using the available modern data (observations rather than proxies) in monthly time

steps, as described in Section 3. To apply our stochastic framework, we must first make the two time

series linearly compatible. Specifically, based on Arrhenius’s rule (Equation (1)), we take the

logarithms of CO₂ concentration while we keep T untransformed. Such a transformation has also

been performed in previous studies, which consider the logarithm of CO₂ 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 ln[CO].

A synchronous plot of the two processes (specifically, UAH temperature and ln[CO] at Mauna

Loa) is depicted in Figure 8. Very little can be inferred from this figure alone. Both processes show

increasing trends and thus appear as positively correlated. On the other hand, the two processes

appear to have different behaviours. Temperature shows an erratic behaviour while ln[CO] has a

smooth evolution marked by the annual periodicity. It looks impossible to infer causality from that

graph alone.

Somewhat more informative is Figure 9, which depicts lagged cross-correlations of the two

processes, based on the methodology in Section 4.1 but without differencing the processes.

Specifically, Figure 9 shows the cross-correlogram between UAH temperature and Mauna Loa

ln[CO] at monthly and annual scales; the autocorrelograms of the two processes are also plotted for

comparison. In both time scales, the cross-correlogram shows high correlations at all lags, with the

maximum attained at lag zero. This does not hint at a direction. However, the cross-correlations for

negative lags are slightly greater than those in the positive lags. Notice that to make this clearer, we

have also plotted the differences 𝑟−𝑟

in the graph. This behaviour could be interpreted as

supporting the causality direction [CO₂] → 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

ln[CO] are very close to 1 for all lags shown in the figure.

Figure 8. Synchronous plots of the time series of UAH temperature and logarithm of CO₂

concentration at Mauna Loa at monthly scale.

In our investigation, we also applied the Granger test on these two time series in both time

directions. To calculate the p-value of the Granger test, we used free software (namely the function

5.8

5.85

5.9

5.95

6

6.05

-1

-0.5

0

0.5

1

1.5

1980 1985 1990 1995 2000 2005 2010 2015 2020

ln [CO₂]

T(°C)

𝑇(UAH) ln [CO₂] (Mauna Loa)

Figure 8.

Synchronous plots of the time series of UAH temperature and logarithm of CO

2

concentration

at Mauna Loa at monthly scale.

Somewhat more informative is Figure 9, which depicts lagged cross-correlations of the two

processes, based on the methodology in Section 4.1 but without diﬀerencing the processes. Speciﬁcally,

Figure 9shows the cross-correlogram between UAH temperature and Mauna Loa

ln[CO2]

at monthly

and annual scales; the autocorrelograms of the two processes are also plotted for comparison. In both

time scales, the cross-correlogram shows high correlations at all lags, with the maximum attained

at lag zero. This does not hint at a direction. However, the cross-correlations for negative lags are

slightly greater than those in the positive lags. Notice that to make this clearer, we have also plotted

the diﬀerences

rj−r−j

in the graph. This behaviour could be interpreted as supporting the causality

direction [CO2]→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

ln[CO2]

are very close to 1

for all lags shown in the ﬁgure.

Sci 2020, 3, x FOR PEER REVIEW 14 of 33

GRANGER_TEST [70,71]). It appears that in the causality direction [CO₂] → 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 → [CO₂], 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.

Figure 9. Auto- and cross-correlograms of the time series of UAH temperature and logarithm of CO₂

concentration at Mauna Loa.

Therefore, one could directly interpret these results as unambiguously showing one-way

causality between the total greenhouse gases and temperature and, hence, validating the consensus

view that human activity is responsible for the observed rise in global temperature. However, these

results are certainly not unambiguous and, most probably, they are spurious. To demonstrate that

they are not unambiguous, we have plotted, as shown in the upper panels of Figure 10, the p-values

of the Granger test for moving windows with a size of 10 years for number of lags η = 1 and 2. The

values for the entire length of time series, as given above, are also shown as dashed lines. Now the

picture is quite different: each of the two directions appear dominating (meaning that the attained

significance level is lower in one over the other) in about equal portions of the time. For example, for

η = 2, the T → [CO₂] dominates over [CO₂] → T for 58% of the time. The attained p-value for direction

T → [CO₂] 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 → [CO₂] direction.

To show that the results are spurious and, in particular, affected by the very high

autocorrelations of ln [CO] and, more importantly, by its annual cyclicity, we have “removed” the

latter by averaging over the previous 12 months. We did that for both series and plotted the results

in the lower panels of Figure 10. Here, the results are stunning. For both lags η = 1 and 2 and for the

entire period (or almost), T → [CO₂] 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

-0.2

0

0.2

0.4

0.6

0.8

1

-60 -48 -36 -24 -12 0 12 24 36 48 60

Correlation coefficient, r

Lag, j (months)

Tln[CO₂]

T - ln[CO₂] Difference 𝑟ⱼ–𝑟₋ⱼ

-0.2

0

0.2

0.4

0.6

0.8

1

-60 -48 -36 -24 -12 0 12 24 36 48 60

Correlation coefficient, r

Lag, j (months)

Figure 9.

Auto- and cross-correlograms of the time series of UAH temperature and logarithm of CO

2

concentration at Mauna Loa.

Sci 2020,2, 83 15 of 33

In our investigation, we also applied the Granger test on these two time series in both time

directions. To calculate the p-value of the Granger test, we used free software (namely the function

GRANGER_TEST [

70

,

71

]). It appears that in the causality direction [CO

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.

Therefore, one could directly interpret these results as unambiguously showing one-way causality

between the total greenhouse gases and temperature and, hence, validating the consensus view that

human activity is responsible for the observed rise in global temperature. However, these results are

certainly not unambiguous and, most probably, they are spurious. To demonstrate that they are not

unambiguous, we have plotted, as shown in the upper panels of Figure 10, the p-values of the Granger

test for moving windows with a size of 10 years for number of lags

η

=1 and 2. The values for the

entire length of time series, as given above, are also shown as dashed lines. Now the picture is quite

diﬀerent: each of the two directions appear dominating (meaning that the attained signiﬁcance level is

lower in one over the other) in about equal portions of the time. For example, for

η

=2, the T

→

[CO

2

]

dominates over [CO

2

]

→

Tfor 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→[CO2] direction.

Sci 2020, 3, x FOR PEER REVIEW 15 of 33

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 Sections 4.1 and 4.3 is more appropriate. We apply this methodology in

Section 5.2.

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 CO₂ 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.

5.2. Differenced Time Series

We have already explained the advantages of investigating the differenced processes, which

quantify changes from a mathematical and logical point of view. In our case, taking differences is

also physically meaningful as both CO₂ 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.

The chosen time step of differencing is equal to one year (ν = 12 for the monthly time step of the

time series). For instance, from the value of January of a certain year, we subtract the value of January

of the previous year and so forth. A first reason for this choice is that it almost eliminates the effect of

the annual cycle (periodicity). A second reason is that the temperature data are given in terms of

“anomalies”, i.e., differences from an average which varies from month to month. By taking 𝜈=12,

the varying means are eliminated, and “anomalies” are effectively replaced by the actual processes

(as the differences in the actual values equal the differences of “anomalies”).

We perform all analyses on both monthly and annual time scales. Figure 11 shows the

differenced time series for the UAH temperature and Mauna Loa CO₂ concentration at monthly scale;

the symbols Δ𝑇 and Δln[CO] are used interchangeably with 𝑥, and 𝑦,, respectively.

Comparing Figure 8 (undifferenced series) with Figure 11 (differenced series), one can verify

that the latter is much more informative in terms of the directionality of the relationship of the two

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1980 1985 1990 1995 2000 2005 2010

p-value

Starting year of the 10-year window

𝑇→[CO₂] – 10-year window

[CO₂]→𝑇– 10-year window

𝑇→[CO₂] – entire period

[CO₂]→𝑇– entire period

1.E-33

1.E-30

1.E-27

1.E-24

1.E-21

1.E-18

1.E-15

1.E-12

1.E-09

1.E-06

1.E-03

1.E+00

1980 1985 1990 1995 2000 2005 2010

p-value

Starting year of the 10-year window

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1980 1985 1990 1995 2000 2005 2010

p-value

Starting year of the 10-year window

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1980 1985 1990 1995 2000 2005 2010

p-value

Starting year of the 10-year window

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.

To show that the results are spurious and, in particular, aﬀected by the very high autocorrelations

of

ln[CO2]

and, more importantly, by its annual cyclicity, we have “removed” the latter by averaging

over the previous 12 months. We did that for both series and plotted the results in the lower panels

Sci 2020,2, 83 16 of 33

of Figure 10. Here, the results are stunning. For both lags

η

=1 and 2 and for the entire period

(or almost), T

→

[CO

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 Sections 4.1 and 4.3 is more appropriate. We apply this methodology in Section 5.2.

5.2. Diﬀerenced Time Series

We have already explained the advantages of investigating the diﬀerenced processes,

which quantify changes from a mathematical and logical point of view. In our case, taking diﬀerences is

also physically meaningful as both CO

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.

The chosen time step of diﬀerencing is equal to one year (

ν

=12 for the monthly time step of the

time series). For instance, from the value of January of a certain year, we subtract the value of January

of the previous year and so forth. A ﬁrst reason for this choice is that it almost eliminates the eﬀect

of the annual cycle (periodicity). A second reason is that the temperature data are given in terms of

“anomalies”, i.e., diﬀerences from an average which varies from month to month. By taking

ν=

12,

the varying means are eliminated, and “anomalies” are eﬀectively replaced by the actual processes

(as the diﬀerences in the actual values equal the diﬀerences of “anomalies”).

We perform all analyses on both monthly and annual time scales. Figure 11 shows the diﬀerenced

time series for the UAH temperature and Mauna Loa CO

2

concentration at monthly scale; the symbols

∆Tand ∆ln[CO2]are used interchangeably with ˜

xτ,12 and ˜

yτ,12, respectively.

Comparing Figure 8(undiﬀerenced series) with Figure 11 (diﬀerenced series), one can verify

that the latter is much more informative in terms of the directionality of the relationship of the two

processes. While Figure 8did not provide any relevant hints, Figure 11 clearly shows that, most often,

the temperature curve leads and that of CO

2

follows. However, there are cases where the changes in

the two processes synchronize in time or even become decoupled.

Figure 12 shows the same time series at the annual time scale, with the year being deﬁned as

July–June for

∆T

and February–January for

∆ln[CO2]

. The reason for this diﬀerentiation will be

explained below. Here, it is more evident that, most of the time, the temperature change leads and that

of CO2follows.

It is of interest here that the variability of global mean annual temperature is signiﬁcantly inﬂuenced

by the rhythm of ocean–atmosphere oscillations, such as ENSO, AMO, and IPO [

72

]. This mechanism

may be a complicating factor, in turn inﬂuencing the link between temperature and CO

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.

The climacograms of the diﬀerenced time series used (actually four of the six to avoid an

overcrowded graph) are shown in Figure 13. It appears that the diﬀerenced temperature time series are

consistent with the condition implied by stationarity, i.e., H=0 for the diﬀerenced process. The same

does not look to be the case for the CO

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

∆ln[CO2]

in Figures 11 and 12 are positive, while stationarity entails a zero mean of the diﬀerenced

Sci 2020,2, 83 17 of 33

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 (Figures 3and 4) would

justify a stationarity assumption.

Sci 2020, 3, x FOR PEER REVIEW 16 of 33

processes. While Figure 8 did not provide any relevant hints, Figure 11 clearly shows that, most often,

the temperature curve leads and that of CO₂ follows. However, there are cases where the changes in

the two processes synchronize in time or even become decoupled.

Figure 11. Differenced time series of UAH temperature and logarithm of CO₂ 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 shows the same time series at the annual time scale, with the year being defined as

July–June for Δ𝑇 and February–January for Δln[CO]. The reason for this differentiation will be

explained below. Here, it is more evident that, most of the time, the temperature change leads and

that of CO₂ follows.

It is of interest here that the variability of global mean annual temperature is significantly

influenced by the rhythm of ocean–atmosphere oscillations, such as ENSO, AMO, and IPO [72]. This

mechanism may be a complicating factor, in turn influencing the link between temperature and CO₂

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.

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.

Sci 2020, 3, x FOR PEER REVIEW 17 of 33

Figure 12. Annually averaged time series of differenced temperatures (UAH) and logarithms of CO₂

concentrations (Mauna Loa). Each dot represents the average of a one-year duration ending at the

time of its abscissa.

The climacograms of the differenced time series used (actually four of the six to avoid an

overcrowded graph) are shown in Figure 13. It appears that the differenced temperature time series

are consistent with the condition implied by stationarity, i.e., H = 0 for the differenced process. The

same does not look to be the case for the CO₂ 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 CO₂ 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 [CO₂] exhibits a monotonic increasing trend (as also verified by the fact

that all values of Δln[CO] in Figures 11 and 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 [CO₂] process based on palaeoclimatic data (Figures 3

and 4) would justify a stationarity assumption.

The preliminary qualitative observation from graphical inspection of Figures 11 and 12 suggests

that the temperature change very often precedes and the CO₂ change follows in the same direction.

We note, though, that temperature changes alternate in sign while CO₂ changes are always positive.

A quantitative analysis based on the methodology in Section 4.1 requires the study of lagged

cross-correlations of the two processes. Figure 14 shows the cross-correlogram between UAH

temperature and Mauna Loa CO₂ 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.

The maximum cross-correlation of the monthly series is 0.47 and appears at a positive lag, 𝜂=

5 months, thus suggesting T → [CO₂], rather than [CO₂] → T, as dominant causality direction. Similar

are the graphs of the other combinations of temperature and CO₂ datasets, which are shown in

Appendix A.5 (Figures A3–A7). In all cases, 𝜂 is positive, ranging from 5 to 11 months.

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

1980 1985 1990 1995 2000 2005 2010 2015 2020

Δln[CO₂]

ΔT

ΔΤ

Δln[CO₂]

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.

Sci 2020,2, 83 18 of 33

Sci 2020, 3, x FOR PEER REVIEW 18 of 33

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).

To perform similar analyses on the annual scale, we fixed the specification of a year for

temperature for the period July–June, as already mentioned, and then slid the initial month specifying

the beginning of a year for CO₂ 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 CO₂ 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 CO₂

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.

Figure 14. Auto- and cross-correlograms of the differenced time series of UAH temperature and

Mauna Loa CO₂ concentration.

H= 1

0.01

0.1

1

1 10 100

Variance relative to that of scale 1

Time scale (months)

ΔT, UAH

ΔT, CRUTEM4

Δln[CO₂], Mauna Loa

Δln[CO₂], Barrow

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-48 -36 -24 -12 0 12 24 36 48

Correlation coefficient

Lag (months)

ΔΤ

Δln[CO₂]

ΔΤ - Δln[CO₂], monthly

ΔΤ - Δln[CO₂], annual

ΔΤ - Δln[CO₂], fixed year

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).

The preliminary qualitative observation from graphical inspection of Figures 11 and 12 suggests

that the temperature change very often precedes and the CO

2

change follows in the same direction.

We note, though, that temperature changes alternate in sign while CO2changes are always positive.

A quantitative analysis based on the methodology in Section 4.1 requires the study of lagged

cross-correlations of the two processes. Figure 14 shows the cross-correlogram between UAH

temperature and Mauna Loa CO

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.

Sci 2020, 3, x FOR PEER REVIEW 18 of 33

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).

To perform similar analyses on the annual scale, we fixed the specification of a year for

temperature for the period July–June, as already mentioned, and then slid the initial month specifying

the beginning of a year for CO₂ 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 CO₂ 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 CO₂

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.

Figure 14. Auto- and cross-correlograms of the differenced time series of UAH temperature and

Mauna Loa CO₂ concentration.

H= 1

0.01

0.1

1

1 10 100

Variance relative to that of scale 1

Time scale (months)

ΔT, UAH

ΔT, CRUTEM4

Δln[CO₂], Mauna Loa

Δln[CO₂], Barrow

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-48 -36 -24 -12 0 12 24 36 48

Correlation coefficient

Lag (months)

ΔΤ

Δln[CO₂]

ΔΤ - Δln[CO₂], monthly

ΔΤ - Δln[CO₂], annual

ΔΤ - Δln[CO₂], fixed year

Figure 14.

Auto- and cross-correlograms of the diﬀerenced time series of UAH temperature and Mauna

Loa CO2concentration.

The maximum cross-correlation of the monthly series is 0.47 and appears at a positive lag,

η1=

5

months, thus suggesting T

→

[CO

2

], rather than [CO

2

]

→

T, as dominant causality direction. Similar

Sci 2020,2, 83 19 of 33

are the graphs of the other combinations of temperature and CO

2

datasets, which are shown in

Appendix A.5 (Figures A3–A7). In all cases, η1is positive, ranging from 5 to 11 months.

To perform similar analyses on the annual scale, we ﬁxed the speciﬁcation of a year for temperature

for the period July–June, as already mentioned, and then slid the initial month specifying the beginning

of a year for CO

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 1summarizes 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.

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—CO2Series

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

While, as explained in Sections 4.2 and 5.1, the Granger test has weaknesses that may not help in

drawing conclusions, for completeness and as a conﬁrmation, we list its results here:

•

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 ×10−4for 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 ×10−8for η=7.

•

For the monthly scale, the attained p-values in the direction T

→

[CO

2

] are always smaller than in

direction [CO

2

]

→

Tby 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.

We note that the test cannot be applied for the sliding time window case and, hence, we cannot

provide results for this case.

Sci 2020,2, 83 20 of 33

We add a ﬁnal remark, in view of a comment by Masters and Benestad [

73

] on the already

mentioned study by Humlum et al. [

45

], in which they claim that “the inter-annual ﬂuctuations in

atmospheric CO

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.

∆ln[CO2]

has been linearly regressed with

∆T

and the running average of SOI for the

previous 12 months. At synchrony (without applying any time lag), the correlation of SOI with

∆T

is

0.40, higher than that of

∆T

and

∆ln[CO2]

(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

∆ln[CO2]

and

∆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

r2=

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.

In brief, all above conﬁrm the results of our methodology that the dominant direction of causality

is T→[CO2].

6. Physical Interpretation

The omnipresence of positive lags on both monthly and annual time scales and the conﬁrmation

by Granger tests reduce the likelihood that our results are statistical artefacts. Still, our results require

physical interpretation which we seek in the natural process of soil respiration.

Soil respiration, R

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.

The Figure 6.1 of the latest report of the IPCC [

75

] provides a quantiﬁcation of the mass balance

of the carbon cycle in the atmosphere that is representative of recent years. The soil respiration,

assumed to be the sum of respiration (plants) and decay (microbes), is 113.7 Gt C/year (IPCC gives

a value of 118.7 including ﬁre, which along with biomass burning, is estimated to be 5 Gt C/year by

Green and Byrne [80]).

We can expect that sea respiration would also have increased. Moreover, outgassing from

the oceans must also have increased as the solubility of CO

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

The relationship between atmospheric concentration of carbon dioxide and the global temperature

is widely recognized, and it is common knowledge that increasing CO

2

concentration plays a major

role in enhancement of the greenhouse eﬀect and contributes to global warming.

Sci 2020,2, 83 21 of 33

While the fact that these two variables are tightly connected is beyond doubt, the direction of

the causal relationship needs to be studied further. The purpose of this study is to complement the

conventional and established theory, that increased CO

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”

(“

ὄρνιςἢ 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.

We examine the relationship of global temperature and atmospheric carbon dioxide concentration

using the most reliable global data that are available—the data gathered from several sources,

covering the common time interval 1980–2019, available at the monthly time step.

The results of the study support the hypothesis that both causality directions exist, with T

→

CO

2

being the dominant, despite the fact that CO

2→

Tprevails in public, as well as in scientiﬁc, perception.

Indeed, our results show that changes in CO

2

follow changes in Tby about six months on a monthly

scale, or about one year on an annual scale.

The opposite causality direction opens a nurturing interpretation question. We attempted to

interpret this mechanism by noting that the increase in soil respiration, reﬂecting the fact that the

intensity of biochemical process increases with temperature, leads to increasing natural CO

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:

Conceptualization, D.K.; methodology, D.K.; software: D.K.; validation, Z.W.K.; formal

analysis, D.K.; investigation, D.K. and Z.W.K.; data curation, D.K.; writing—original draft preparation, D.K. and

Z.W.K.; writing—review and editing, D.K. and Z.W.K.; visualization, D.K. and Z.W.K. All authors have read and

agreed to the published version of the manuscript.

Funding:

This research received no external funding but was motivated by the scientiﬁc curiosity of the authors.

Acknowledgments:

Some negative comments of two anonymous reviewers for an earlier submission of this

manuscript in another journal (which we have posted online: manuscript at http://dx.doi.org/10.13140/RG.2.

2.29154.15045/1, reviews at http://dx.doi.org/10.13140/RG.2.2.14524.87681) helped us improve the presentation

and strengthen our arguments against their comments. We appreciate these reviewers’ suggestions of relevant

published works, which we were unaware of. We are grateful to the reviewers of the submission of our study to

Sci, Yog Aryal, Ronan Connolly, and Stavros Alexandris, whose constructive comments helped us to improve the

paper, mostly by adding a lot of additional information and clariﬁcation in appendices. The discussion of DK with

Antonis Christoﬁdes helped to substantially improve an initial version of Appendix A.4, which he informally

reviewed. We thank the journal Sci for the interesting experience it oﬀered us.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

Data Availability:

The two temperature time series and the Mauna Loa CO

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).

Sci 2020,2, 83 22 of 33

Appendix A

Appendix A.1. On Early Non-Systematic Measurements of CO2

This Appendix (not contained in Version 1 of our paper) addresses comments by all three reviewers

of Version 1, Yog Aryal [

85

], Ronan Connolly [

14

], and Stavros Alexandris [

86

], about the reasons why

we delimit our analysis to the period 1980–2019. The two latter reviewers suggested using earlier data

compiled by Beck (2007), who referred to old chemical analyses of atmospheric concentration of CO

2

.

We are sympathetic to the passion of the late Ernst-Georg Beck who, being a biology teacher,

sacriﬁced a lot of time and eﬀort to the exciting exercise of digging out old CO

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.

Historically, it was not the ﬁrst review paper of this sort. For instance, in his Table 1, Beck [

87

]

refers to old works by Letts and Blake (~1900; [88]), who considered 252 papers with data (all in 19th

century), and to Stepanova [

89

], who considered 229 papers with data (130 in 19th century and 99 in

20th century). Beck himself [87] considered 156 papers with data (82 in 19th and 74 in 20th century).

As usual, it is instructive to consider the paper by Beck [

87

] jointly with critical commentaries

published later in the journal where the original paper appeared [

90

,

91

]. In particular, R.F. Keeling [

90

]

opined that the old chemical measurements examined by Beck [87] “exhibit far too much geographic

and short-term temporal variability to plausibly be representative of the background. The variability of

these early measurements must therefore be attributed to ‘local or regional’ factors or poor measurement

practice”. Keeling [

90

] also noted “basic accounting problems”. “Beck’s 11-year averages show large

swings, including an increase from 310 to 420 ppm between 1920 and 1945 (Beck’s Figure 11)”. “To drive

an increase of this magnitude globally requires the release of 233 billion metric tons of carbon to the

atmosphere. The amount is equivalent to more than a third of all the carbon contained in land plants

globally. [

. . .

] To make a credible case, Beck needed to oﬀer evidence for losses or gains of carbon of

this magnitude from somewhere. He oﬀered none.”

Meijer [

91

] expressed the opinion that