Atmospheric Temperature and CO2: Hen-or-Egg Causality? (Preprints, ver. 1)

Abstract

It is common knowledge that increasing CO2 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 CO2 concentration due to human emissions causes an increase of temperature, by considering the reverse causality. Since increased temperature causes an increase in CO2 concentration, the relationship of atmospheric CO2 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 at the monthly time step, covering the time interval 1980–2019, in 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 → CO2. Changes in CO2 follow changes in T by about six months on a monthly scale, or about one year on an annual scale. We attempt to interpret this mechanism by involving biochemical reactions, as at higher temperatures soil respiration, and hence CO2 emission, are increasing.

Full text

Article
Atmospheric Temperature and CO2:
Hen-Or-Egg Causality? (Version 1)
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 or zkundze@man.poznan.pl
*Correspondence: dk@itia.ntua.gr
Received: 7 September 2020; Accepted: 11 September 2020; Published: 14 September 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 of 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 at the monthly time step, covering the time interval
1980–2019, in 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 CO2emission, are increasing.
Keywords: temperature; global warming; greenhouse gases; atmospheric CO2concentration
Π
ó
τερ
o
νἡ ὄρνις 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 Β
,
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., https://el.wikisource.org/wiki/
Συµπ
o
σιακά Β0
, “
ἡ ὄρνις
” appears as “
ἡ ἄρνις
”. After extended
search, we contend that this must be an error, either an old one in manuscript copying, e.g., by monks
in monasteries, or a modern one, e.g., in OCR. We are confident that the correct word is “ὄρνις”).
Sci 2020,2, 72; doi:10.3390/sci2030072 www.mdpi.com/journal/sci

Sci 2020,2, 72 1 of 26
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
proxy data-based palaeoclimatic studies, where change in temperature leads and change in CO
2
concentration follows. Next, we discuss the data bases 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., change in temperature leads
and 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, CO2emission.
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 unprecedented
decrease in carbon emission (Figure 1), there was increase in atmospheric CO
2
concentration,
which followed a pattern similar to previous years (Figure 2). Indeed, according to the IEA [
1
], 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 [
2
],
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 severest ever, including those in the world wars. It is also noteworthy in Figure 1that there were three
years in sequel without major increase in 2010s, where again there was increase in CO
2
concentration.
(At first glance, this does not sound reasonable and therefore we have cross-checked the data with other
sources, namely the Global Carbon Atlas, http://www.globalcarbonatlas.org/en/CO2-emissions and the
data base of Our World In Data, https://ourworldindata.org/grapher/annual-co-emissions-by-region;
we found only slight differences.) Interestingly, Figure 2also shows a rapid growth in emissions after
the 2008–2009 global financial crisis, which agrees with Peters et al. [3].
Sci 2020, 3, x FOR PEER REVIEW 3 of 27
Figure 1. Annual change in global energy‐related CO₂ emissions (adapted from IEA [1]).
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, [4]), the most renowned scientist who studied the causal
relationship between the two quantities, regarded the changes of the atmospheric carbon dioxide
concentration as the cause and the changes of the 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 [meant 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, [5]), 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, a
ratio of 1:0.6.
Arrhenius [4] 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:
402
404
406
408
410
412
414
416
418
1 2 3 4 5 6 7 8 9 10 11 12
CO₂ concentration (ppm)
Month
2020
2019
2018
2017
Figure 1. Annual change in global energy-related CO2emissions (adapted from IEA [1]).

Sci 2020,2, 72 2 of 26
Sci 2020, 3, x FOR PEER REVIEW 3 of 27
Figure 1. Annual change in global energy‐related CO₂ emissions (adapted from IEA [1]).
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, [4]), the most renowned scientist who studied the causal
relationship between the two quantities, regarded the changes of the atmospheric carbon dioxide
concentration as the cause and the changes of the 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 [meant 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, [5]), 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, a
ratio of 1:0.6.
Arrhenius [4] 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:
402
404
406
408
410
412
414
416
418
1 2 3 4 5 6 7 8 9 10 11 12
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, [
4
]), the most renowned scientist who studied the causal
relationship between the two quantities, regarded the changes of the atmospheric carbon dioxide
concentration as the cause and the changes of the 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 [meant 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, [
5
]), 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 CO2and water vapour as 1.5 and 0.88, respectively, a ratio of 1:0.6.
Arrhenius [
4
] 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
[CO2]
denote temperature and CO
2
concentration, respectively, T
0
and
[CO2]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 [
4
]
cites several other authors, among whom Tyndall (1865, [
6
]) for pointing out the enormous importance
of atmospheric absorption of radiation and for having the opinion that water vapour has the greatest
influence on it. However, it appears that the first experiments on the subject, for both water vapour and
carbon dioxide, were undertaken even earlier by the female scientist Eunice Newton Foote (1856, [
7
];
see also [8,9]).
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 accumulated, particularly from palaeoclimatic studies, which allow us to examine Arrhenius’s
hypotheses on a sounder basis. In brief, we can state the following:

Sci 2020,2, 72 3 of 26
•
Indeed, CO
2
plays a substantial role as a greenhouse gas. However, modern estimates of the
CO
2
contribution to the greenhouse effect largely differ 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., [10]), a ratio of 1:4.
•
During the Phanerozoic Eon, Earth’s temperature has varied by even more than 5–10
◦
C, which was
postulated by Arrhenius—see Figure 3. The link of temperature and CO
2
is beyond doubt, even
though it 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.
The relationship becomes more legible in proxy data of the Quaternary (see Figure 4). It has
been demonstrated in a persuasive manner by Roe [
11
] 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 [11]),
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 [
12
,
13
]), which have pointed to the opposite direction of causality, i.e., the change of
temperature as the cause and that in the CO
2
concentration as the effect. Such claims have explained
the fact that temperature change leads and CO
2
concentration change follows. In agreement with
Roe [
11
], several papers have found the time lag positive, with estimates varying from 50 to 1000 years,
depending on the time period and the particular study [
14
–
18
]. Claims that CO
2
concentration leads
(i.e., a negative lag) have not been generally 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 [
18
]
while in one of them [
16
] it has been asserted that a “short lead of CO
2
over temperature cannot
be excluded”.
Sci 2020, 3, x FOR PEER REVIEW 5 of 27
Figure 3. Proxy‐based reconstructions of global mean temperature and CO₂ concentration during the
Phanerozoic Eon. The temperature reconstruction by Scotese [19] was mainly based on proxies from
[20–22], while the CO₂ concentration proxies have been taken from Davis [23], Berner [24] and Ekart
et al. [25]; 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
(https://stratigraphy.org/chart).
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 [17]).
Since palaeoclimatic data suggest a direction opposite to that assumed by Arrhenius,
Koutsoyiannis [17], 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
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
10
15
20
25
30
35
40
050100150200250300350400450500550600
Temperature T(°C)
Scotese (2018)
10
100
1000
10000
050100150200250300350400450500550600
[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 Triassic Jurassic Cretaceous Paleo-
gene
Neo-
gene
Mesozoic Cenozoic
Quaternary
Paleozoic
180
200
220
240
260
280
300
320
-10
-8
-6
-4
-2
0
2
4
0100000200000300000400000
Temperature difference from present
CO₂ concetration (ppmv)
Years before present
T CO₂
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-40 -30 -20 -10 0 10 20 30 40
Correlation coefficient
Lag (thousand years)
T
CO₂
T - CO₂
Figure 3.
Proxy-based reconstructions of global mean temperature and CO
2
concentration during
the Phanerozoic Eon. The temperature reconstruction by Scotese [
19
] was mainly based on proxies
from [
20
–
22
], while the CO
2
concentration proxies have been taken from Davis [
23
], Berner [
24
] and
Ekart et al. [
25
]; 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
(https://stratigraphy.org/chart).

Sci 2020,2, 72 4 of 26
Sci 2020, 3, x FOR PEER REVIEW 5 of 27
Figure 3. Proxy‐based reconstructions of global mean temperature and CO₂ concentration during the
Phanerozoic Eon. The temperature reconstruction by Scotese [19] was mainly based on proxies from
[20–22], while the CO₂ concentration proxies have been taken from Davis [23], Berner [24] and Ekart
et al. [25]; 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
(https://stratigraphy.org/chart).
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 [17]).
Since palaeoclimatic data suggest a direction opposite to that assumed by Arrhenius,
Koutsoyiannis [17], 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
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
10
15
20
25
30
35
40
050100150200250300350400450500550600
Temperature T(°C)
Scotese (2018)
10
100
1000
10000
050100150200250300350400450500550600
[CO₂] (ppmv)
Million years before present
Davis (2017) Berner (2008) Ekart et al. (1999)
Proterozoic
Ediacaran Cambrian O rdovic-
ian
Silur-
ian Devonian Carbon i-
ferous Permian Triassic Jurassic Cretaceous Paleo-
gene
Neo-
gene
Mesozoic Cenozoic
Quaternary
Paleozoic
180
200
220
240
260
280
300
320
-10
-8
-6
-4
-2
0
2
4
0100000200000300000400000
Temperature difference from present
CO₂ concetration (ppmv)
Years before present
T CO₂
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-40 -30 -20 -10 0 10 20 30 40
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 [17]).
Since palaeoclimatic data suggest a direction opposite to that assumed by Arrhenius,
Koutsoyiannis [
17
], 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 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 in the “hen-or-egg” causality
problems. Obviously, however, the proxy character of these data and the too-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. [26]
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 [
27
] 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 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.

Sci 2020,2, 72 5 of 26
In contrast, Stips et al. [
28
] used a different method [
29
] 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 Ganger statistical test, this is not central in our approach. Rather, we give
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 in central place in our analyses the
technique of process differencing. This technique is quite natural and also powerful in studying time
directionality [
17
]. We note that differencing (which has also been used in Reference [
26
]) has been
criticized for potentially eliminating long-run effects and hence providing information on short-run
effects only [
27
]. Even if this speculation were valid, it would not invalidate the differencing technique
for the following reasons:
•The short-run 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, which have already been
discussed above.
3. Data
Our investigation of the relationship of temperature and 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 [
30
,
31
]. The data are
publicly available on 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 [
32
]. 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 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), yet the CRUTEM4 series shows a larger increasing trend than
the UAH series. The differences are explainable 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 [
33
]), while the CRUTEM4 series refers to the ground level; and (c) the
ground-based CRUTEM4 series is affected by urbanization (a lot of ground stations are located in

Sci 2020,2, 72 6 of 26
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 be
manifest in our results.
Sci 2020, 3, x FOR PEER REVIEW 7 of 27
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 [32]. 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
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), yet the CRUTEM4 series shows a larger increasing trend than the
UAH series. The differences are explainable 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 [33]), while the CRUTEM4 series refers to the ground level; and (c) the
ground‐based CRUTEM4 series is affected by urbanization (a lot of 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 be
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 [34]. 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 has also 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 https://www.esrl.noaa.gov/gmd/ccgg/about/ global_means.html).
The period of data coverage varies, but all series cover the common 40‐year period 1980–2019,
which hence constituted 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
-1
-0.5
0
0.5
1
1.5
2
2.5
1980 1985 1990 1995 2000 2005 2010 2015 2020
T (°C)

(UAH)

(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 [
34
]. 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 has also 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 https:
//www.esrl.noaa.gov/gmd/ccgg/about/global_means.html).
The period of data coverage varies, but all series cover the common 40-year period 1980–2019,
which hence constituted 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.
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.
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 5-year period before it.
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 2020,2, 72 7 of 26
Sci 2020, 3, x FOR PEER REVIEW 8 of 27
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.
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 5‐year
period before it. 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.
Figure 6. Plots of the data series of atmospheric CO₂ concentration measured in Mauna Loa (Hawaii,
USA), Barrow (Alaska, USA) and South Pole, and 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 [17] has 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
330
340
350
360
370
380
390
400
410
420
1980 1985 1990 1995 2000 2005 2010 2015 2020
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
1 2 3 4 5 6 7 8 9 10 11 12
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 global average.
Sci 2020, 3, x FOR PEER REVIEW 8 of 27
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.
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 5‐year
period before it. 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.
Figure 6. Plots of the data series of atmospheric CO₂ concentration measured in Mauna Loa (Hawaii,
USA), Barrow (Alaska, USA) and South Pole, and 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 [17] has 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
330
340
350
360
370
380
390
400
410
420
1980 1985 1990 1995 2000 2005 2010 2015 2020
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
1 2 3 4 5 6 7 8 9 10 11 12
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.
4. Methods
4.1. Stochastic Framework
A recent study [
17
] has 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)

Sci 2020,2, 72 8 of 26
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 [35]).
•
If a scalar process
x(t)
is Gaussian (i.e., all its finite dimensional distributions are multivariate
normal) then it is reversible (Weiss [
36
]). 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 [37]).
Time asymmetry of a process can be studied more conveniently (or even exclusively in a scalar
process) through the differenced process, i.e.,:
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 −e
X(τ−1)κ
κ(7)
Similar equations for the cumulative and averaged processes for the differenced process
e
xτ,ν
are given
in Appendix A.1.
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)
The autocovariance function for time lag
η
is derived from the climacogram through the relationship [
38
]:
cη=(η+1)2γ|η+1|+(η−1)2γ|η−1|
2−η2γ|j|(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 [
39
]). 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 [
40
]). 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

Sci 2020,2, 72 9 of 26
(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 e
xτhas mean zero and variance:
e
γν,1:=varhe
xτ,νi=varhxκ+νi+var[xτ]−2covhxτ+ν,xτi=2(γ1−cν)(11)
where
γ1and cν
are the variance and lag
ν
autocovariance, respectively, of
xτ
. Furthermore, it has
been demonstrated [
17
] that the Hurst coefficient of the differenced process
e
xτ
precisely equals zero,
which means that e
xτis completely antipersistent, irrespective of γκ.
In vector processes, to study irreversibility 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, [
17
]), given two processes
xτand yτ
we could study the
cross-correlations:
re
xe
y[ν,η]=corre
xτ,ν,e
yτ+η,ν(12)
Time (ir)reversibility could then be characterized by studying the properties of symmetry or
asymmetry of
re
xe
y(ν,η)
as a function of the time lag
η
. In a symmetric bivariate process,
re
xe
y[ν,η]=re
xe
y[ν,−η]
and if the two components are positively correlated, the maximum of
re
xe
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 re
xe
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, [
41
]; see also [
42
]). In probabilistic definitions of causality, time asymmetry is determinant.
Thus, Suppes [
43
] 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(AtBt0)>P(At)
”.
Also, Granger’s [
44
] 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
re
xe
y[ν,η]≥
0 for any
η≥
0, while
re
xe
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
re
xe
y[ν,η],
0 for any
η
. Yet, we can define a dominant direction of
causality based on the time lag
η1
maximizing cross-correlation. Formally,
η1
is defined for a specified
νas:
η1:=argmax
ηre
xe
y(ν,η)(13)
We can thus distinguish the following three cases:
•If η1=0 then there is no dominant direction.
•If η1>0 then the dominant direction is xτ→yτ.
•If η1<0 then the dominant direction is yτ→xτ.
Justification and further explanations on these conditions are provided in Appendix A.2.
4.2. Complications in Seeking Causality
It must be stressed that the above conditions are put as necessary and not sufficient conditions for a
causative relationship between the processes
xτand yτ
. Following Koutsoyiannis [
17
] (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 [44] puts it,

Sci 2020,2, 72 10 of 26
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 [
43
] 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 [
45
] 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, then 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., [
46
]). 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 Reference [47], 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 Reference [
46
], 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
cloths. Thus, it is reasonably expected that in a prediction of sweat quantity temperature matters.
In absence of temperature measurements (e.g., when we have only visual information, like when
watching a video), algorithmically the weight of the cloths improves the prediction of the sweat
quantity. But we could not say that the decrease of cloth weight causes increase of sweat (the opposite
would be more reasonable and would most probably become evident in a three-variable regression,
temperature – cloth weight – sweat).
Cohen [
48
] suggested replacing the term “Granger causality” with “Granger prediction” after
correctly pointing out that:
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

Sci 2020,2, 72 11 of 26
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 [
29
] 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 [
28
].
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, [49]):
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 [
29
], 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 and the time 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. [50]
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 indeed happened, would be violation of logic and perhaps of
common sense.
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 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.

Sci 2020,2, 72 12 of 26
2.
While we occasionally use statistical tests (namely, the Granger test, Equations (14)–(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, it fully reflects the basic causality axiom of time precedence, and it is
more straightforward, transparent (free of algorithmic manipulations) and easily reproducible
(without a need for specialized software).
3.
For simplicity, we do not use here any statistic other than correlation. We stress that the system
we are examining indeed classifies as Gaussian and thus it is totally unnecessary to examine
any statistic additional to correlation. The evidence of Gaussianity is provided by Figure A1 in
Appendix A.3, 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 [
49
,
51
].
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 modern data (observations rather than proxies), available at the monthly time step, 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 been performed also in
previous studies, which consider the logarithm of CO
2
concentration as a proxy of total radiative
forcing (e.g., [
26
]). However, by calling this quantity “forcing” we indirectly give it a priori (i.e., before
investigating causation) the role of 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.
Somewhat more informative is Figure 9, based on the methodology in Section 4.1, by studying
lagged cross-correlations of the two processes 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 give a hint about the 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 2020,2, 72 13 of 26
Sci 2020, 3, x FOR PEER REVIEW 14 of 27
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
GRANGER_TEST [52,53]). 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 keeps being below 0.01 for subsequent η. In 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 see that they are not
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)

(UAH) ln [CO₂] (Mauna Loa)
-0.2
0
0.2
0.4
0.6
0.8
1
-60 -48 -36 -24 -12 0 1 2 24 36 48 60
Correlation coefficient, r
Lag, j (months)
T ln[CO₂]
T - ln[CO₂] Difference ⱼ – ₋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 8.
Synchronous plots of the time series of UAH temperature and logarithm of CO
2
concentration
at Mauna Loa at monthly scale.
Sci 2020, 3, x FOR PEER REVIEW 14 of 27
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
GRANGER_TEST [52,53]). 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 keeps being below 0.01 for subsequent η. In 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 see that they are not
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)

(UAH) ln [CO₂] (Mauna Loa)
-0.2
0
0.2
0.4
0.6
0.8
1
-60 -48 -36 -24 -12 0 1 2 24 36 48 60
Correlation coefficient, r
Lag, j (months)
T ln[CO₂]
T - ln[CO₂] Difference ⱼ – ₋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.
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 [
52
,
53
]). It appears that in the causality direction [CO
2
]
→
Tthe 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 keeps being below 0.01 for subsequent
η
. In 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

Sci 2020,2, 72 14 of 26
certainly not unambiguous and most probably they are spurious. To see that they are not unambiguous,
we have plotted 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
[CO2]→T
for 58% of the time. The attained p-value for direction T
→
[CO
2
] is lower than 1% for
1.4% of the time, much higher than in the opposite direction (0.3% of the time). All these favour the
T→[CO2] direction.
Sci 2020, 3, x FOR PEER REVIEW 15 of 27
unambiguous, we have plotted 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
these 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), the 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 a methodology which proves to be so sensitive to time windows used and data
processing assumptions is inappropriate to draw conclusions from. 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 not believe would 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 of 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) the 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
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
→[CO₂] – 10-year window
[CO₂]→– 10-year window
→[CO₂] – entire period
[CO₂]→– 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
) the 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 of
Figure 10. Here the results are stunning. For both lags
η
=1 and 2 and for the entire period (or almost),
the 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 a methodology
which proves to be so sensitive to time windows used and data processing assumptions is inappropriate
to draw conclusions from. 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 not believe would 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
of Sections 4.1 and 4.3
is more
appropriate. We apply this methodology in Section 5.2.

Sci 2020,2, 72 15 of 26
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, indeed the mass and energy fluxes are
represented by differences.
The time step of differencing was chosen 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 e
xτ,12 and e
yτ,12, respectively.
Sci 2020, 3, x FOR PEER REVIEW 16 of 27
also physically meaningful as both CO₂ concentration and temperature (equivalent to thermal
energy) represent “stocks”, i.e., stored quantities, and, thus, indeed the mass and energy fluxes are
represented by differences.
The time step of differencing was chosen 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
 = 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 Δ and Δln[CO
] are used interchangeably with ,
and ,
, respectively.
Comparing Figure 8 (undifferenced series) and 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 8 did not provide any relevant hint, 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 Δ 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.
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.
Comparing Figure 8(undifferenced series) and 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 hint, 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.

Sci 2020,2, 72 16 of 26
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.
Sci 2020, 3, x FOR PEER REVIEW 17 of 27
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 [54]. This
mechanism may be a complicating factor, in turn influencing the link between temperature and CO₂
concentration. However, this is not examined further here as, given the focus in examining just the
connection of the latter two processes, it lies out of our present scope.
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 data base 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, =
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 A3 (Figures A3‐A7). In all cases 
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 202 0
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.
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 [54]. This mechanism
may be a complicating factor, in turn influencing the link between temperature and CO
2
concentration.
However, this is not examined further here 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
process). Had the available data base 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.
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 2020,2, 72 17 of 26
Sci 2020, 3, x FOR PEER REVIEW 18 of 27
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 difference in the shape of the cross‐correlogram and 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).
Sci 2020, 3, x FOR PEER REVIEW 18 of 27
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 difference in the shape of the cross‐correlogram and 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 are the graphs of the other combinations of temperature and CO
2
datasets, which are shown in
Appendix A.3 (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

Sci 2020,2, 72 18 of 26
three sites, does not produce any noteworthy difference in the shape of the cross-correlogram and 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).
Monthly Time Series
Annual Time
Series—Sliding
Annual Window
Annual Time
Series—Fixed
Annual Window
Temperature—CO2Series 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 here its results:
•
For the monthly scale and the causality direction [CO
2
]
→
T, the null hypothesis is not rejected
at all usual significance levels for
η
=1 and is rejected for significance level 1% for
η
=2–8,
with minimum attained p-value 1.4 ×10−4f for η=6.
•
For the monthly scale and the causality direction T
→
[CO
2
], the null hypothesis is rejected at all
usual significance levels for all η, with minimum attained p-value 1.4 ×10−8for η=6.
•
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
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 rejected at significance level 1% for all
η
=1–6, with minimum attained p-value
0.05% for η=2, thus supporting this causality direction.
•
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
]
→
Tby about 1 to 3 orders of magnitude, again clearly
supporting T→[CO2] 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.
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 [
55
,
56
]. Observational data of
R
s
(e.g., [
57
]) 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 [
58
]) and is routinely used in engineering design.
The latest report of the IPCC [
56
] (Figure 6.1) gives quantification of the mass balance of the carbon
cycle in the atmosphere, representative of the recent years. The soil respiration, assumed to be the sum

Sci 2020,2, 72 19 of 26
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 5 Gt C/year by Green and Byrne [59]).
We can expect that sea respiration would have increased, too. Also, photosynthesis must have
been increased as in the 21st century the Earth has been greening, mostly due to CO
2
fertilization
effects [
60
] and human land-use management [
61
]. Specifically, satellite data show a net increase in
leaf area of 2.3% per decade [
61
]. 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 Gt C/year 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.
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 the former 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 of 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 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 of 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.

Sci 2020,2, 72 20 of 26
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).
Appendix A
Appendix A.1. Some Notes on the Averaged Differenced Process
The cumulative process of the differenced process e
xτ,νwill be:
e
Xκ,ν:=e
x1,ν+e
x2,ν+. . . +e
xκ,ν=x1+ν−x1+x2+η−x2+. . . +xκ+ν−xκ=Xκ+ν−Xν−Xκ(A1)
Note that for η=1 this simplifies to:
e
Xκ,1 =Xκ+1−X1−Xκ=xκ+1−x1=e
xκ,1:=e
xκ(A2)
Following Equation (7), the average differenced process at discrete time scale κ=ηwill be:
e
x(κ)
τ=e
Xτκ,κ−e
X(τ−1)κ,κ
κ=
(Xτκ+κ−Xκ−Xτκ)−X(τ−1)κ+κ−Xκ−X(τ−1)κ
κ(A3)
which, noting that in the rightmost part the two terms
Xκ
cancel each other and by virtue of (7),
simplifies to:
e
x(κ)
τ=x(κ)
τ+1−x(κ)
τ=e
x(κ)
τ,1 (A4)
In other words, the average differenced process equals the differenced average process in case that
the differencing time step
η
has chosen equal to the averaging time scale
κ
. For
κ
=
η
=1 we have
e
x(1)
τ≡e
xτ,1 ≡e
xτ.
Appendix A.2. Some Notes on Time Directionality of Causal Systems
In a unidirectional causal system in continuous time t, in which the process
x(t)
is the cause of
y(t), an equation of the form:
y(t)=Z∞
0
α(s)x(t−s)ds(A5)
should hold [49], where α(t)is the impulse response function. The causality condition is thus:
α(t)=0 for t<0 (A6)
Here we consider systems with positive dependence, in which α(t)≥0 for t≥0, which possibly
are also excited by another process v(t), independent of x(t). Working in discrete time we write:
yτ=P∞
j=0αjxτ−j+vτ(A7)

Sci 2020,2, 72 21 of 26
Assuming (without loss of generality) zero means for all processes, multiplying by
xτ−η
, taking expected
values and denoting the cross-covariance function as
cxy [η]:=Exτ−ηyτ
and the autocovariance function
as cx[η]:=Ehxτ−ηxτiwe find:
cxy [η]=P∞
j=0αjcx[η−j](A8)
For η > 0, using the property that cx[η]is an even function (cx[η]=cx[−η]) we get:
cxy [η]=P∞
j=0αjcx[j−η]=Pη−1
j=0αjcx[η−j]+P∞
j=ηαjcx[j−η](A9)
and for the negative part:
cxy [−η]=P∞
j=0αjcx[j+η](A10)
With intuitive reasoning, assuming that the autocovariance function is decreasing (
cx[j0]<cx[j]
for
j0>j
), as usually happens in natural processes, we may see that the rightmost term of Equations
(A9) and (A10) should be decreasing functions of
η
(as for
j0>j
it will be
cx[j0−η]<cx[j−η]
and
cx[j0+η]<cx[j+η]
). However, the term
Pη−1
j=0αjcx[η−j]
of Equation (A9), is not decreasing. Therefore,
it should attain a maximum value at some positive lag
η=η1
. Thus, a positive maximizing lag,
η=η1>
0, is a necessary condition for causality direction from
xτto yτ
. Conversely, the condition that
the maximizing lag is negative is a sufficient condition to exclude the causality direction exclusively
from xτto yτ.
All above arguments remain valid if we standardize (divide) by the product of standard deviations
of the processes
xτ
and
yτ
, and thus we can replace cross-covariances
cxy [η]
with cross-correlations
rxy [η](or, in the case of differenced processes, re
xe
y[ν,η]).
Appendix A.3. Additional Graphical Depictions
Sci 2020, 3, x FOR PEER REVIEW 22 of 27
(A9) and (A10) should be decreasing functions of η (as for ′ >  it will be [− ]< [ − ] and
[+]< [+]). However, the term ∑

 [ − ] of Equation (A9), is not decreasing.
Therefore, it should attain a maximum value at some positive lag  = 
. Thus, a positive maximizing
lag,  = > 0, is a necessary condition for causality direction from  to . Conversely, the
condition that the maximizing lag is negative is a sufficient condition to exclude the causality
direction exclusively from  to .
All above arguments remain valid if we standardize (divide) by the product of standard
deviations of the processes  and , and thus we can replace cross‐covariances [] with cross‐
correlations [] (or, in the case of differenced processes, 

[,]).
Appendix A3. Additional Graphical Depictions
Figure A1. Normal probability plots of Δ and Δln[CO
] where T is the UAH temperature and [CO₂]
is the CO₂ concentration at Mauna Loa at monthly scale.
Figure A2. Scatter plot of Δ and Δ ln [CO
] where T is the UAH temperature and [CO₂] is the CO₂
concentration at Mauna Loa at monthly scale; the two quantities are lagged in time using the optimal
the lag of 5 months (Table 1). The two linear regression lines are also shown in the figure.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
ΔT
Standard normal variate, z
Empirical
Theoretical
95% confidence limits
-0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
-3 -2 -1 0 1 2 3
Δln[CO₂]
Standard normal variate, z
Empirical
Theoretical
95% confidence limits
0
0.002
0.004
0.006
0.008
0.01
0.012
-1 -0.5 0 0.5 1
y= Δln[CO₂]
x= ΔΤ
Figure A1.
Normal probability plots of
∆T
and
∆ln[CO2]
where Tis the UAH temperature and [CO
2
]
is the CO2concentration at Mauna Loa at monthly scale.

Sci 2020,2, 72 22 of 26
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(A9) and (A10) should be decreasing functions of η (as for ′ >  it will be [− ]< [ − ] and
[+]< [+]). However, the term ∑

 [ − ] of Equation (A9), is not decreasing.
Therefore, it should attain a maximum value at some positive lag  = 
. Thus, a positive maximizing
lag,  = > 0, is a necessary condition for causality direction from  to . Conversely, the
condition that the maximizing lag is negative is a sufficient condition to exclude the causality
direction exclusively from  to .
All above arguments remain valid if we standardize (divide) by the product of standard
deviations of the processes  and , and thus we can replace cross‐covariances [] with cross‐
correlations [] (or, in the case of differenced processes, 

[,]).
Appendix A3. Additional Graphical Depictions
Figure A1. Normal probability plots of Δ and Δln[CO
] where T is the UAH temperature and [CO₂]
is the CO₂ concentration at Mauna Loa at monthly scale.
Figure A2. Scatter plot of Δ and Δ ln [CO
] where T is the UAH temperature and [CO₂] is the CO₂
concentration at Mauna Loa at monthly scale; the two quantities are lagged in time using the optimal
the lag of 5 months (Table 1). The two linear regression lines are also shown in the figure.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
ΔT
Standard normal variate, z
Empirical
Theoretical
95% confidence limits
-0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
-3 -2 -1 0 1 2 3
Δln[CO₂]
Standard normal variate, z
Empirical
Theoretical
95% confidence limits
0
0.002
0.004
0.006
0.008
0.01
0.012
-1 -0.5 0 0.5 1
y= Δln[CO₂]
x= ΔΤ
Figure A2.
Scatter plot of
∆T
and
∆ln [CO2]
where Tis the UAH temperature and [CO
2
] is the CO
2
concentration at Mauna Loa at monthly scale; the two quantities are lagged in time using the optimal
the lag of 5 months (Table 1). The two linear regression lines are also shown in the figure.
Sci 2020, 3, x FOR PEER REVIEW 23 of 27
Figure A3. Auto‐ and cross‐correlograms of the differenced time series of UAH temperature and
Barrow CO₂ concentration.
Figure A4. Auto‐ and cross‐correlograms of the differenced time series of UAH temperature and
South Pole CO₂ concentration.
Figure A5. Auto‐ and cross‐correlograms of the differenced time series of UAH temperature and
global CO₂ concentration.
-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
-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
-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 A3.
Auto- and cross-correlograms of the differenced time series of UAH temperature and
Barrow CO2concentration.
Sci 2020, 3, x FOR PEER REVIEW 23 of 27
Figure A3. Auto‐ and cross‐correlograms of the differenced time series of UAH temperature and
Barrow CO₂ concentration.
Figure A4. Auto‐ and cross‐correlograms of the differenced time series of UAH temperature and
South Pole CO₂ concentration.
Figure A5. Auto‐ and cross‐correlograms of the differenced time series of UAH temperature and
global CO₂ concentration.
-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
-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
-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 A4.
Auto- and cross-correlograms of the differenced time series of UAH temperature and South
Pole CO2concentration.

Sci 2020,2, 72 23 of 26
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Figure A3. Auto‐ and cross‐correlograms of the differenced time series of UAH temperature and
Barrow CO₂ concentration.
Figure A4. Auto‐ and cross‐correlograms of the differenced time series of UAH temperature and
South Pole CO₂ concentration.
Figure A5. Auto‐ and cross‐correlograms of the differenced time series of UAH temperature and
global CO₂ concentration.
-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
-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
-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 A5.
Auto- and cross-correlograms of the differenced time series of UAH temperature and global
CO2concentration.
Sci 2020, 3, x FOR PEER REVIEW 24 of 27
Figure A6. Auto‐ and cross‐correlograms of the differenced time series of CRUTEM4 temperature and
Mauna Loa CO₂ concentration.
Figure A7. Auto‐ and cross‐correlograms of the differenced time series of CRUTEM4 temperature and
global CO₂ concentration.
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Shan, Y.; Canadell, J.G. et al. Temporary reduction in daily global CO2 emissions during the COVID‐19
forced confinement. Nat. Clim. Chang. 2020, 10, 647–653, doi:10.1038/s41558‐020‐0797‐x.
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https://archive.org/details/ heatamodemotion05tyndgoog/ (accessed on 1 September 2020).
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8. Jackson, R. Eunice foote, john tyndall and a question of priority. Notes Rec. 2020, 74, 105–118,
doi:10.1098/rsnr.2018.0066.
-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
-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 A6.
Auto- and cross-correlograms of the differenced time series of CRUTEM4 temperature and
Mauna Loa CO2concentration.
Sci 2020, 3, x FOR PEER REVIEW 24 of 27
Figure A6. Auto‐ and cross‐correlograms of the differenced time series of CRUTEM4 temperature and
Mauna Loa CO₂ concentration.
Figure A7. Auto‐ and cross‐correlograms of the differenced time series of CRUTEM4 temperature and
global CO₂ concentration.
References
1. IEA (International Energy Agency). Global Energy Review 2020; IEA: Paris, French, 2020. Available online:
https://www.iea.org/reports/global‐energy‐review‐2020 (accessed on 1 July 2020).
2. Le Quéré, C.; Jackson, R.B.; Jones, M.W.; Smith, A.J.; Abernethy, S.; Andrew, R.M.; De‐Gol, A.J.; Willis, D.R.;
Shan, Y.; Canadell, J.G. et al. Temporary reduction in daily global CO2 emissions during the COVID‐19
forced confinement. Nat. Clim. Chang. 2020, 10, 647–653, doi:10.1038/s41558‐020‐0797‐x.
3. Peters, G.P.; Marland, G.; Le Quéré, C.; Boden, T.; Canadell, J.G.; Raupach, M.R. Rapid growth in CO₂
emissions after the 2008–2009 global financial crisis. Nat. Clim. Chang. 2012, 2, 2–4,
doi:10.1038/nclimate1332.
4. Arrhenius, S. On the influence of carbonic acid in the air upon the temperature of the ground. Lond. Edinb.
Dublin Philos. Mag. J. Sci. 1896, 41, 237–276, doi:10.1080/14786449608620846.
5. De Marchi, L. Le Cause dell'Era Glaciale; Premiato dal R. Instituto Lombardo: Pavia, Italy, 1895.
6. Tyndall, J. Heat a Mode of Motion, 2nd ed.; Longmans: London, UK, 1865. Available online:
https://archive.org/details/ heatamodemotion05tyndgoog/ (accessed on 1 September 2020).
7. Foote, E. Circumstances affecting the heat of the sun’s rays. Am. J. Sci. Arts 1856, 22, 382–383.
8. Jackson, R. Eunice foote, john tyndall and a question of priority. Notes Rec. 2020, 74, 105–118,
doi:10.1098/rsnr.2018.0066.
-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
-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 A7.
Auto- and cross-correlograms of the differenced time series of CRUTEM4 temperature and
global CO2concentration.

Sci 2020,2, 72 24 of 26
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