Available via license: CC BY 4.0
Content may be subject to copyright.
Citation: Macedo, P.; Madaleno, M.
Global Temperature and Carbon
Dioxide Nexus: Evidence from a
Maximum Entropy Approach.
Energies 2023,16, 277. https://
doi.org/10.3390/en16010277
Academic Editor: Gabriele Di
Giacomo
Received: 10 November 2022
Revised: 22 December 2022
Accepted: 23 December 2022
Published: 27 December 2022
Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
energies
Perspective
Global Temperature and Carbon Dioxide Nexus: Evidence from
a Maximum Entropy Approach
Pedro Macedo 1and Mara Madaleno 2, 3, *
1CIDMA—Center for Research and Development in Mathematics and Applications,
Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
2Research Unit on Governance, Competitiveness and Public Policies (GOVCOPP), Universidade de Aveiro,
Campus Universitário de Santiago, 3810-193 Aveiro, Portugal
3Departamento de Economia, Gestão, Engenharia Industrial e Turismo (DEGEIT), Universidade de Aveiro,
Campus Universitário de Santiago, 3810-193 Aveiro, Portugal
*Correspondence: maramadaleno@ua.pt
Abstract:
The connection between Earth’s global temperature and carbon dioxide (CO
2
) emissions
is one of the highest challenges in climate change science since there is some controversy about
the real impact of CO
2
emissions on the increase of global temperature. This work contributes to
the existing literature by analyzing the relationship between CO
2
emissions and the Earth’s global
temperature for 61 years, providing a recent review of the emerging literature as well. Through a
statistical approach based on maximum entropy, this study supports the results of other techniques
that identify a positive impact of CO
2
in the increase of the Earth’s global temperature. Given the
well-known difficulties in the measurement of global temperature and CO
2
emissions with high
precision, this statistical approach is particularly appealing around climate change science, as it allows
the replication of the original time series with the subsequent construction of confidence intervals for
the model parameters. To prevent future risks, besides the present urgent decrease of greenhouse gas
emissions, it is necessary to stop using the planet and nature as if resources were infinite.
Keywords:
global temperature; carbon dioxide (CO
2
) emissions; maximum entropy; climate change
1. Introduction
Global warming is still widely debated due to divergent opinions. Some believe that it
results from human actions, others look at it as a natural cause, while some even see it as a
non-relevant problem, thinking it does not exist at all [
1
–
3
]. Many theories have emerged
around the inexistent consensus. One of the claims is that global warming exists due to
the sun’s effects. Another claim attributes global warming to human action, which rises
greenhouse gases (GHGs) like carbon dioxide [
1
,
4
,
5
]. Others ignore global warming and
pretend that the problem does not exist at all. As evidenced by Letcher [
6
], the root cause
of our present changing climate is the build-up of greenhouse gases, the most important of
these gases being carbon dioxide, mainly caused by the burning of fossil fuels [7].
The quick economic expansion of some economies has been done at the expense of
more environmental pollution, but greenhouse gas effects are only reflected in the long
run [
8
]. This is because a substantial amount of CO
2
emissions enters the atmosphere,
remaining there for centuries, and its effects on climate change are only reflected over
decades or even millennia [
9
–
13
]. If that point is reached, we cannot reverse it thereafter
by just stopping emissions, and these effects should be valued today by decision-makers.
Although other GHGs like CH
4
and N
2
O (methane, nitrous oxides) have a stronger ability
to absorb the radiation, their contribution to global warming is insignificant, since they have
a lower concentration in the atmosphere as compared with CO
2
. The scientific community
claims that, on the one hand, CO
2
doubling in the atmosphere will increase the average
surface temperature of the Earth by +3.8
◦
C, while on the other hand, its halving will
Energies 2023,16, 277. https://doi.org/10.3390/en16010277 https://www.mdpi.com/journal/energies
Energies 2023,16, 277 2 of 13
decrease the global temperature by
−
3.6
◦
C [
4
]. These pointed amounts depend, as well,
on the change in the humidity of the air, which in turn depends on the air’s temperature.
Thus, exploring the contribution of CO
2
emissions, maybe the strongest greenhouse
gas, to global warming and temperature is needed to develop cost-effective interventions.
Limiting this warming is essential. The latest carbon dioxide emissions reported affirm the
belief that the global warming compliance goal of “below 1.5
◦
C or 2
◦
C” will be achieved
shortly [
14
,
15
]. With the Paris Agreement in December 2015, the debate on whether limiting
warming to 1.5
◦
C is compatible with the current emissions level was opened. The Paris
Agreement participants agreed to restrict the global average temperature increase to less
than 2
◦
C and to limit global warming to 1.5
◦
C, and the latter was found by Millar et al. [
16
]
to not be a geophysical impossibility. The Intergovernmental Panel on Climate Change
(IPCC) scenarios (to produce negative emissions) highlight pathways to reduced climate
change impacts [
17
,
18
]. Indeed, “Holding the increase in the global average temperature to
well below 2
◦
C above pre-industrial levels and pursuing efforts to limit the temperature
increase to 1.5
◦
C above pre-industrial levels, recognizing that this would significantly
reduce the risks and impacts of climate change” (e.g., United Nations [19], p. Art 2.1 (a)).
Still, it should be noted that the global warming goal of 1.5
◦
C asserts that carbon
dioxide emissions due to human activity must reach “net zero” by 2050 to ensure the
average rise in global temperature at 1.5
◦
C above preindustrial levels. This has to be
achieved to reduce the catastrophic climate change risk over populations and the Earth.
The IPCC report highlights that “temperature extremes on land are projected to warm
more than the global mean surface temperature (high confidence); extreme hot days in
mid-latitudes warm by up to about 5
◦
C at global warming of 1.5
◦
C and about 4
◦
C at 2
◦
C,
and extremely cold nights in high latitudes warm by up to about 4.5
◦
C at 1.5
◦
C and about
6
◦
C at 2
◦
C (high confidence). The number of hot days is projected to increase in most
land regions, with the highest increases in the tropics (high confidence)” [
17
]. Thus, the
adoption of the mitigation and adaptation strategies is simultaneously the most effective
economic and technical solution for the global warming issue [4,20,21].
From the above, it may be argued that there is a strong, direct, and positive correspon-
dence between carbon dioxide concentration and temperature, turning climate change into
one of the strongest challenges faced by humankind [
22
,
23
]. To understand the true impact
of climate change on nature’s vital processes and how to mitigate these, we first need to
understand patterns and magnitudes of the relationship between global temperature and
emissions. The increased launching of anthropogenic greenhouse gases in the atmosphere,
in particular CO
2
emissions, moves average global temperatures upwards. Urgency is
now posed to the need to prevent global warming, which is leading to extreme weather
and related catastrophes [
5
,
24
]. The rising temperature worldwide is reflected in ocean
warming, ice melting, rising sea levels, and reduced snow cover [
3
]. Air pollution leads to
environmental degradation and is mainly responsible for Earth’s warming (powered by
the greenhouse gas effect) [17].
It has been highlighted by [
25
], while exploring the probability of achieving CO
2
emission targets set by the Paris Agreement of the top ten emitters during 1991–2015, that
the volume of CO
2
emissions is expected to raise in 2030 by 26.5–36.5% as compared to
the 2005 levels. Rau et al. [
26
] explored the global potential for converting renewable
electricity to negative CO
2
emissions hydrogen, pointing out that combinations can be done
by increasing energy generation and CO
2
removal by more than 50 times at equivalent or
lower costs. Kalra et al. [
1
] also modeled the relationship between global temperatures and
atmospheric concentrations of CO
2
, CH
4
, and N
2
O over a dataset of 65 years. However,
the authors used linear regression, decision tree regression, random forest regression, and
artificial neural networks, concluding that the latter performs better. They proved that the
contribution of carbon dioxide in the increase of global temperatures is the maximum of
the considered greenhouse gases. As such, the current analysis only concentrates on CO
2
greenhouse gas emissions by making use of a maximum entropy methodology to explore
its relationship with global temperature.
Energies 2023,16, 277 3 of 13
Recent studies also explored the relationship between CO
2
emissions and global tempera-
ture. The IPCC [
27
] report highlights the total net anthropogenic GHG emissions rise during
2010–2019, and the continuous cumulative growth of CO
2
emissions since 1850. However,
growth rates were stronger in 2000–2009 when compared to those of 2010–2019, evidencing en-
vironmental improvements. Additionally, evidenced are regional disparities reflecting different
development stages and income-dependent variations. Cost reductions and global adoptions of
low-emissions technologies are mainly attributed to innovation policies pursued [
27
]. Nationally
determined contributions announced before COP26 indicated that it was likely that warming
will exceed 1.5
◦
C during the 21st century, and recent contributions in the literature also dis-
cuss and put these problems at the forefront [
2
,
3
,
7
,
8
,
24
,
28
,
29
]. In [
2
] it was concluded that the
number of deaths increased significantly with the repetition of extreme weather events. Using a
methodology based on long memory and fractional integration, [28] concluded that emissions
present heterogeneous behavior in terms of persistence in pandemics, even if temperatures are
more homogeneous, evidencing mean reverting behaviors. In 2020, [
3
], also using fractional
integration and cointegration methodologies, concluded that CO
2
emissions and temperatures
are not cointegrated. However, assuming that emissions are weakly exogenous concerning the
temperatures, the results pointed to a positive relationship with a long memory pattern.
Accounting for the last 425 million years, [
8
] points to a pressing need for research on the
relationship between CO
2
, biodiversity extinction, and related carbon policies, concluding
that changes in emissions did not cause a temperature change in the ancient climate. Ad-
ditionally, [
24
] looks at both cause and causality relating to the “hen-or-egg” effect. Results
support the hypothesis that the dominant direction is from temperature to CO
2
emissions
(1980–2019). For China, [
29
] calculated the pathway in global emissions and its contribution
to global warming up to 2050 (since 2005). They called attention to the fact that the larger
differences in emission pathways of different atmospheric pollutant emissions. Proposing a
forecast error variance decomposition, [
7
] contradicts [
30
] findings of causality from various
forcings to global temperature. Therefore, and in the presence of multiple and contradicting
results, the present article tries to contribute to this stream of research.
The anthropogenic CO
2
emissions are mostly due to transportation, working machines,
and consumption [
31
], reduced recently with the COVID-19 confinement all over the
world [
32
]. Strict measures to limit the virus spreading were placed in the recent COVID-19
outbreak. These included grounded airlines, factories, businesses that were shut and closed
down, and people confined in their homes. Thus, a drastic reduction was registered in
anthropogenic carbon dioxide emissions, and [
32
] argues that this is the ideal time to
test if CO
2
emissions are the overwhelming contributor to CO
2
concentrations and global
warming, or even to check if these have a limited effect on CO
2
concentrations that are
driven by the temperature. Moreover, it would be possible to explore the effect of the
prolonged and unprecedented cut in carbon dioxide emissions of the CO2concentration.
As such, there is some evidence that CO
2
is the most relevant greenhouse gas in the
increase of global warming (e.g., [
30
,
33
]). Thus, this study aims to contribute to this discussion,
through a powerful methodology in the analysis of time series, namely, the maximum entropy
bootstrap, which, as far as we are aware, has not been used previously to explore this
relationship, probably due to its novelty. Furthermore, [
34
] argued that the response to
anthropogenic emission scenarios often requires a simple model linking emissions of carbon
dioxide to global temperature changes, given that future climate changes will largely be
determined by future cumulative CO
2
emissions (e.g., [
30
,
33
,
35
]), leaving room to the need to
explore the link existent between global temperature and CO2emissions.
Besides this introduction with the framework included, the rest of the article develops
as follows. Section 2exposes the data and the methodology employed in this work to
evidence its novelty and usefulness in the study of the relationship between CO
2
emissions
and global temperature. Afterward, Section 3presents all the results and discusses them,
while Section 4concludes by pointing out directions for decision-makers.
Energies 2023,16, 277 4 of 13
2. Data and Methodology
The data for global temperature and CO
2
were collected on 1 October 2020 from
(1) NASA Global Climate Change: Vital Signs of the Planet; (2) National Centers for
Environmental Information, National Oceanic and Atmospheric Administration; and
(3) Jet Propulsion Laboratory, California Institute of Technology, Education Section (These
were considered reliable sources to collect all the information needed for this work:
https://climate.nasa.gov/vital-signs/carbon-dioxide; https://www.ncdc.noaa.gov/cag/
global/time-series/globe/land_ocean/1/12/1880-2016; https://www.jpl.nasa.gov/edu/
teach/activity/graphing-global-temperature-trends). The monthly average data for CO
2
(in PPP; parts per million) were transformed to annual values and the information for global
temperature (annual global land and ocean temperature anomalies in
◦
C) was converted to
actual temperature (annual absolute values).
Monthly data for CO
2
only exists from 1958 (incomplete year and includes missing
values). In 1964 there is a lack of information for three months and in 1975 there is a lack of
information for one month, and in 1984. In these three years, the annual average of CO
2
is calculated with the existing information, and it was not considered necessary to apply
imputation techniques for missing values.
The maximum entropy bootstrap for time series proposed by H. D. Vinod ([
36
,
37
])
is a powerful technique that allows for statistical formulations free of restrictive and
unnecessary assumptions usually adopted in time series analysis. The technique creates a
large number of replicates for inference purposes that satisfy the ergodic theorem and the
central limit theorem. Those generated elements of the ensemble retain the shape of the
original time series, as well as the time-dependence structure of the autocorrelation and the
partial autocorrelation functions. As an illustration, Figure 1presents the original series of
the annual average of CO
2
, between 1959 and 2019, and five resamples provided by the
maximum entropy bootstrap.
Energies 2023, 16, x FOR PEER REVIEW 4 of 13
Besides this introduction with the framework included, the rest of the article devel-
ops as follows. Section 2 exposes the data and the methodology employed in this work to
evidence its novelty and usefulness in the study of the relationship between CO2 emis-
sions and global temperature. Afterward, Section 3 presents all the results and discusses
them, while Section 4 concludes by pointing out directions for decision-makers.
2. Data and Methodology
The data for global temperature and CO2 were collected on 1 October 2020 from (1)
NASA Global Climate Change: Vital Signs of the Planet; (2) National Centers for Environ-
mental Information, National Oceanic and Atmospheric Administration; and (3) Jet Pro-
pulsion Laboratory, California Institute of Technology, Education Section (These were
considered reliable sources to collect all the information needed for this work: https://cli-
mate.nasa.gov/vital-signs/carbon-dioxide; https://www.ncdc.noaa.gov/cag/global/time-
series/globe/land_ocean/1/12/1880-2016; https://www.jpl.nasa.gov/edu/teach/activ-
ity/graphing-global-temperature-trends). The monthly average data for CO2 (in PPP; parts
per million) were transformed to annual values and the information for global tempera-
ture (annual global land and ocean temperature anomalies in °C) was converted to actual
temperature (annual absolute values).
Monthly data for CO2 only exists from 1958 (incomplete year and includes missing
values). In 1964 there is a lack of information for three months and in 1975 there is a lack
of information for one month, and in 1984. In these three years, the annual average of CO2
is calculated with the existing information, and it was not considered necessary to apply
imputation techniques for missing values.
The maximum entropy bootstrap for time series proposed by H. D. Vinod ([36,37]) is
a powerful technique that allows for statistical formulations free of restrictive and unnec-
essary assumptions usually adopted in time series analysis. The technique creates a large
number of replicates for inference purposes that satisfy the ergodic theorem and the cen-
tral limit theorem. Those generated elements of the ensemble retain the shape of the orig-
inal time series, as well as the time-dependence structure of the autocorrelation and the
partial autocorrelation functions. As an illustration, Figure 1 presents the original series
of the annual average of CO2, between 1959 and 2019, and five resamples provided by the
maximum entropy bootstrap.
Figure 1. The annual average of CO2 between 1959 and 2019 (solid), and five resamples (dotted).
A general description of the maximum entropy bootstrap algorithm ([36,37]) for a
random replicate of a time series is provided next: (1) the original data are sorted to create
Figure 1. The annual average of CO2between 1959 and 2019 (solid), and five resamples (dotted).
A general description of the maximum entropy bootstrap algorithm ([
36
,
37
]) for a
random replicate of a time series is provided next: (1) the original data are sorted to
create order statistics, and the order index vector is stored; (2) the middle points from the
order statistics are computed; (3) the trimmed mean of deviations among all consecutive
observations, the lower limit for the left tail and the upper limit for the right tail are
computed; (4) the mean of the maximum entropy density [
38
] within each interval is
computed; (5) pseudorandom numbers from a [0, 1] uniform interval are generated and
the sample quantiles of the maximum entropy density at those points are computed and
sorted; (6) the sorted sample quantiles are reordered using the index vector stored in (1).
Then, steps (2) to (6) were repeated a large number of times (1000 replications in this study).
Energies 2023,16, 277 5 of 13
As noted by [
36
,
37
], the technique avoids all structural changes and unit root type
testing, and all the usual shape-destroying transformations like detrending or differencing
to achieve stationarity. See [
36
,
37
,
39
] for additional details of the algorithm and the advan-
tages of the technique, including the ones over the traditional bootstrap. For a review of
maximum entropy, see [38].
Furthermore, and in addition to the advantages mentioned above, the maximum
entropy bootstrap is particularly appealing in this area of climate change, given the diffi-
culties in the measurement of global temperature and CO
2
emissions with high precision,
widely discussed in the literature. Thus, the maximum entropy bootstrap, by not imposing
parametric restrictions, allows for greater freedom in statistical modeling and inference
through the replications of the original time series and the subsequent construction of
confidence intervals for the model parameters. A recent proposal to improve the estimation
of parameters is discussed in [
40
]. Moreover, since the inference is based on the analysis
of confidence intervals, the use and possible misinterpretations of p-values are avoided,
following recent recommendations from the statistical community (e.g., [41]).
Although other models were tested (These models and corresponding results are
available upon request to the authors. The Schwarz criterion was used to select the possible
best lag combinations among several experiments.), given the objective of this study, two
realistic models to evaluate the relationship between global temperature and CO
2
were
defined as
TE MPt=b1+b2CO2t−m+et, (1)
and
TE MPt=b1+b2TEMPt−1+b3CO2t−m+et, (2)
for
m=
0, 1, 2, 3, where
TE MP
represents global temperature (in
◦
C),
CO
2 represents car-
bon dioxide (CO
2
; in parts per million),
e
represents the noise component, and
t
represents
the period (year). For clarity and the reader’s convenience, the eight particular models are
described in Table 1.
Table 1. Models under study.
Model 1 TE MPt=b1+b2CO2t+et
Model 2 TE MPt=b1+b2CO2t−1+et
Model 3 TE MPt=b1+b2CO2t−2+et
Model 4 TE MPt=b1+b2CO2t−3+et
Model 5 TE MPt=b1+b2TEMPt−1+b3CO2t+et
Model 6 TE MPt=b1+b2TEMPt−1+b3CO2t−1+et
Model 7 TE MPt=b1+b2TEMPt−1+b3CO2t−2+et
Model 8 TE MPt=b1+b2TEMPt−1+b3CO2t−3+et
Three time periods were considered: 1959 to 2019 (the entire data available on the
sources; more complete data for CO
2
only exist from 1959), 1959 to 1989, and 1990 to 2019.
3. Results and Discussion
Tables 2–7present the results provided by the maximum entropy bootstrap, consider-
ing 1000 replications of the original series. The highest density regions (HDR) were adopted
here to compute the confidence intervals [
42
]. (The R packages meboot ([
36
]) and hdrcde
([
43
]) were used in this work.) For all tables, the column Estimate represents the median of
the estimates obtained from the 1000 models. Additionally, at the bottom of all the tables,
the adjusted R
2
values are presented. The hypothesis test for the parameters of interest in
(1) and (2) is defined by
H0:bi=0 vs. H1:bi6=0, (3)
for
i=
2 (for Model 1 to Model 4) and
i=
3 (for Model 5 to Model 8). If the null
hypothesis,
H0
, is rejected (zero does not belong to a specified confidence interval), then the
Energies 2023,16, 277 6 of 13
corresponding variable (
CO
2
t−m
) is considered relevant to explain the response variable
(
TE MPt
) at the corresponding significance level, assuming the statistical model and the
sample used in the study.
Table 2. Results for Model 1 to Model 4 (data from 1959 to 1989).
Estimate Highest Density Regions
CI 90% CI 95% CI 99%
Model 1 b1*** 10.8514
(10.3647, 11.3193) (10.2646, 11.3988)
(9.9705, 11.6188)
b2*** 0.0096 (0.0082, 0.0111) (0.0080, 0.0114) (0.0077, 0.0123)
Model 2 b1*** 10.5906
(10.0564, 11.0974)
(9.9559, 11.1907) (9.7232, 11.4025)
b2*** 0.0104 (0.0089, 0.0120) (0.0086, 0.0123) (0.0080, 0.0130)
Model 3 b1*** 10.3552 (9.7862, 10.9002) (9.6450, 11.0284) (9.3398, 11.2920)
b2*** 0.0111 (0.0095, 0.0129) (0.0091, 0.0133) (0.0086, 0.0143)
Model 4 b1*** 10.0218 (9.3833, 10.6493) (9.2229, 10.7935) (8.7793, 11.1610)
b2*** 0.0122 (0.0103, 0.0141) (0.0099, 0.0147) (0.0092, 0.0161)
Note 1: Adjusted R
2
values lie, approximately, between 0.55 and 0.58 for Model 1 to Model 4. Note 2: *** means
that the null hypothesis
H0
:
bi=
0
(i=1, 2)
is rejected at 1% significance level. This note is valid for Tables 2–4.
Note 3: (1) Estimate represents the median of the estimates from the 1000 replications; (2) CI represents Confidence
Intervals; (3) all the values are rounded to four decimals. This note is valid for Tables 2–7.
Table 3. Results for Model 1 to Model 4 (data from 1990 to 2019).
Estimate Highest Density Regions
CI 90% CI 95% CI 99%
Model 1 b1*** 10.7968
(10.2239, 11.2865) (10.0709, 11.3605)
(9.7997, 11.5144)
b2*** 0.0098 (0.0087, 0.0114) (0.0085, 0.0117) (0.0081, 0.0125)
Model 2 b1*** 10.5704 (9.9635, 11.0832) (9.7829, 11.1805) (9.4867, 11.3637)
b2*** 0.0104 (0.0092, 0.0121) (0.0088, 0.0126) (0.0085, 0.0134)
Model 3 b1*** 10.4452 (9.7737, 11.0476) (9.5662, 11.1694) (9.2934, 11.3482)
b2*** 0.0108 (0.0093, 0.0127) (0.0091, 0.0134) (0.0084, 0.0138)
Model 4 b1*** 10.5971 (9.8105, 11.3096) (9.6304, 11.4214) (9.3000, 11.6257)
b2*** 0.0105 (0.0087, 0.0126) (0.0084, 0.0135) (0.0078, 0.0140)
Note: Adjusted R2values lie, approximately, between 0.77 and 0.80 for Model 1 to Model 4.
Table 4. Results for Model 1 to Model 4 (data from 1959 to 2019).
Estimate Highest Density Regions
CI 90% CI 95% CI 99%
Model 1 b1*** 10.7702
(10.4785, 11.0369) (10.3841, 11.0907) (10.2669, 11.1820)
b2*** 0.0098 (0.0092, 0.0106) (0.0091, 0.0109) (0.0087, 0.0112)
Model 2 b1*** 10.6825
(10.3883, 10.9703) (10.2887, 11.0451) (10.1637, 11.1452)
b2*** 0.0101 (0.0094, 0.0111) (0.0090, 0.0113) (0.0087, 0.0117)
Model 3 b1*** 10.6056
(10.2914, 10.9179) (10.2122, 10.9922) (10.0784, 11.1155)
b2*** 0.0104 (0.0095, 0.0112) (0.0093, 0.0115) (0.0090, 0.0119)
Model 4 b1*** 10.5136
(10.1717, 10.8562) (10.0890, 10.9368)
(9.9576, 11.0637)
b2*** 0.0107 (0.0097, 0.0116) (0.0095, 0.0119) (0.0092, 0.0122)
Note: Adjusted R2values are, approximately, 0.90 for all the models.
Energies 2023,16, 277 7 of 13
Table 5. Results for Model 5 to Model 8 (data from 1959 to 1989).
Estimate Highest Density Regions
CI 90% CI 95% CI 99%
Model 5
b1*** 8.0595 (6.6394, 9.4517) (6.3287, 9.7472) (5.9504, 10.1514)
b2*** 0.2364 (0.1081, 0.3680) (0.0880, 0.3900) (0.0513, 0.4300)
b3*** 0.0078 (0.0061, 0.0096) (0.0059, 0.0100) (0.0053, 0.0105)
Model 6
b1*** 7.9913 (6.6003, 9.3576) (6.3089, 9.6422) (5.9299, 10.0183)
b2*** 0.2367 (0.1108, 0.3657) (0.0883, 0.3899) (0.0547, 0.4260)
b3*** 0.0081 (0.0062, 0.0100) (0.0060, 0.0104) (0.0055, 0.0108)
Model 7
b1*** 8.0505 (6.6420, 9.4314) (6.4061, 9.6673) (5.9973, 10.0762)
b2*** 0.2189 (0.0891, 0.3452) (0.0676, 0.3671) (0.0193, 0.4164)
b3*** 0.0088 (0.0069, 0.0109) (0.0064, 0.0114) (0.0058, 0.0122)
Model 8
b1*** 8.0624 (6.6410, 9.4922) (6.2816, 9.8576) (5.8244, 10.3240)
b2** 0.1880 (0.0555, 0.3198) (0.0277, 0.3474) (−0.0160, 0.3906)
b3*** 0.0100 (0.0080, 0.0124) (0.0075, 0.0129) (0.0068, 0.0143)
Note 1: Adjusted R
2
values lie, approximately, between 0.55 and 0.58 for Model 5 to Model 8. Note 2: *, **, and *** means
that the null hypothesis
H0
:
bi=
0
(i=1, 2, 3)
is rejected, respectively, at 10%, 5%, and 1% significance levels. This note
is valid for Tables 5–7.
Table 6. Results for Model 5 to Model 8 (data from 1990 to 2019).
Estimate Highest Density Regions
CI 90% CI 95% CI 99%
Model 5
b1*** 8.2304 (6.1863, 10.2813) (5.7369, 10.7287) (4.9466, 11.5143)
b2* 0.2146 (0.0216, 0.4079) (−0.0173, 0.4474) (−0.0865, 0.5183)
b3*** 0.0082 (0.0059, 0.0106) (0.0054, 0.0111) (0.0045, 0.0123)
Model 6
b1*** 8.1764 (6.1097, 10.1755) (5.6693, 10.5740) (4.8516, 11.3057)
b2* 0.2168 (0.0286, 0.4085) (−0.0051, 0.4447) (−0.0831, 0.5299)
b3*** 0.0084 (0.0060, 0.0107) (0.0056, 0.0112) (0.0049, 0.0119)
Model 7
b1*** 8.1411 (5.8628, 10.3898) (5.4782, 10.7756) (4.7647, 11.4932)
b2* 0.2156 (0.0051, 0.4227) (−0.0337, 0.4615) (−0.1170, 0.5458)
b3*** 0.0086 (0.0061, 0.0115) (0.0055, 0.0118) (0.0047, 0.0131)
Model 8
b1*** 7.9782 (5.7322, 10.2690) (5.3466, 10.6520) (4.6333, 11.3643)
b2* 0.2431 (0.0307, 0.4539) (−0.0133, 0.4978) (−0.0769, 0.5612)
b3*** 0.0078 (0.0052, 0.0108) (0.0047, 0.0116) (0.0039, 0.0129)
Note: Adjusted R2values lie, approximately, between 0.79 and 0.81 for Model 5 to Model 8.
The first and very important result is that the null hypothesis
H0
:
bi=
0
(i=2, 3)
is
rejected at a low significance level, whatever the model or the period considered, where
bi
is the parameter associated with CO
2
(i.e.,
b2
for Model 1 to Model 4; and
b3
for Model 5
to Model 8). Since both limits of the corresponding confidence intervals are positive, this
means that an increase in the annual average of CO
2
corresponds to an increase in global
temperature. Without loss of generality, considering, for example, Model 1 in Table 4, using
this sample with data from 1959 to 2019, and considering the statistical model described
by Model 1, it is estimated that, on average, a unit increase on the annual average of
CO
2
implies an increase between 0.0087
◦
C and 0.0112
◦
C on global temperature, with a
confidence level of 99%.
Energies 2023,16, 277 8 of 13
Figures 2–4present the highest density regions (HDR) for the
b2
estimates, considering
Model 1 to Model 4 under the three time periods in the study. (All the other HDR graphics
are available upon request to the authors. They are omitted here for the sake of simplicity.
Additionally, the percentile method was computed for comparison purposes, but the inter-
pretation was qualitatively the same.) The HDR reported is the graphical representation of
the corresponding results in Tables 2–4(CI 90% in blue; CI 95% in green; CI 99% in red).
The graphics reveal the shift of the HDR to the right of the zero value (revealing the positive
impact of CO2), with the zero value not being included in any of the HDR considered.
Table 7. Results for Model 5 to Model 8 (data from 1959 to 2019).
Estimate
Highest Density Regions
CI 90% CI 95% CI 99%
Model 5
b1*** 7.6958 (5.9925, 9.3377) (5.6427, 9.6643) (5.0849, 10.1839)
b2*** 0.2779 (0.1260, 0.4388) (0.0973, 0.4708) (0.0361, 0.5386)
b3*** 0.0073 (0.0056, 0.0088) (0.0052, 0.0091) (0.0044, 0.0100)
Model 6
b1*** 7.6392 (5.9650, 9.2738) (5.6486, 9.5633) (4.9699, 10.1846)
b2*** 0.2784 (0.1279, 0.4409) (0.1025, 0.4702) (0.0422, 0.5390)
b3*** 0.0074 (0.0057, 0.0090) (0.0053, 0.0092) (0.0044, 0.0103)
Model 7
b1*** 7.6838 (6.0082, 9.2874) (5.6970, 9.5721) (5.0672, 10.1429)
b2*** 0.2703 (0.1194, 0.4321) (0.0928, 0.4631) (0.0363, 0.5294)
b3*** 0.0076 (0.0058, 0.0093) (0.0054, 0.0097) (0.0045, 0.0104)
Model 8
b1*** 7.6759 (6.0352, 9.2988) (5.7103, 9.5465) (5.0206, 10.0625)
b2*** 0.2622 (0.1117, 0.4191) (0.0828, 0.4553) (0.0291, 0.5247)
b3*** 0.0079 (0.0061, 0.0097) (0.0057, 0.0101) (0.0048, 0.0107)
Note: Adjusted R2values lie, approximately, between 0.90 and 0.91 for Model 5 to Model 8.
Energies 2023, 16, x FOR PEER REVIEW 8 of 13
𝑏3***
0.0078
(0.0052, 0.0108)
(0.0047, 0.0116)
(0.0039, 0.0129)
Note: Adjusted R2 values lie, approximately, between 0.79 and 0.81 for Model 5 to Model 8.
Table 7. Results for Model 5 to Model 8 (data from 1959 to 2019).
Estimate
Highest Density Regions
CI 90%
CI 95%
CI 99%
Model 5
𝑏1***
7.6958
(5.9925, 9.3377)
(5.6427, 9.6643)
(5.0849, 10.1839)
𝑏2***
0.2779
(0.1260, 0.4388)
(0.0973, 0.4708)
(0.0361, 0.5386)
𝑏3***
0.0073
(0.0056, 0.0088)
(0.0052, 0.0091)
(0.0044, 0.0100)
Model 6
𝑏1***
7.6392
(5.9650, 9.2738)
(5.6486, 9.5633)
(4.9699, 10.1846)
𝑏2***
0.2784
(0.1279, 0.4409)
(0.1025, 0.4702)
(0.0422, 0.5390)
𝑏3***
0.0074
(0.0057, 0.0090)
(0.0053, 0.0092)
(0.0044, 0.0103)
Model 7
𝑏1***
7.6838
(6.0082, 9.2874)
(5.6970, 9.5721)
(5.0672, 10.1429)
𝑏2***
0.2703
(0.1194, 0.4321)
(0.0928, 0.4631)
(0.0363, 0.5294)
𝑏3***
0.0076
(0.0058, 0.0093)
(0.0054, 0.0097)
(0.0045, 0.0104)
Model 8
𝑏1***
7.6759
(6.0352, 9.2988)
(5.7103, 9.5465)
(5.0206, 10.0625)
𝑏2***
0.2622
(0.1117, 0.4191)
(0.0828, 0.4553)
(0.0291, 0.5247)
𝑏3***
0.0079
(0.0061, 0.0097)
(0.0057, 0.0101)
(0.0048, 0.0107)
Note: Adjusted R2 values lie, approximately, between 0.90 and 0.91 for Model 5 to Model 8.
Figures 2–4 present the highest density regions (HDR) for the 𝑏2 estimates, consid-
ering Model 1 to Model 4 under the three time periods in the study. (All the other HDR
graphics are available upon request to the authors. They are omitted here for the sake of
simplicity. Additionally, the percentile method was computed for comparison purposes,
but the interpretation was qualitatively the same.) The HDR reported is the graphical rep-
resentation of the corresponding results in Tables 2–4 (CI 90% in blue; CI 95% in green; CI
99% in red). The graphics reveal the shift of the HDR to the right of the zero value (reveal-
ing the positive impact of CO2), with the zero value not being included in any of the HDR
considered.
Figure 2. Cont.
Energies 2023,16, 277 9 of 13
Energies 2023, 16, x FOR PEER REVIEW 9 of 13
Figure 2. HDR of 𝑏2 estimates. Model 1 to Model 4 (data from 1959 to 1989).
Figure 3. HDR of 𝑏2 estimates. Model 1 to Model 4 (data from 1990 to 2019).
Figure 2. HDR of b2estimates. Model 1 to Model 4 (data from 1959 to 1989).
Energies 2023, 16, x FOR PEER REVIEW 9 of 13
Figure 2. HDR of 𝑏2 estimates. Model 1 to Model 4 (data from 1959 to 1989).
Figure 3. HDR of 𝑏2 estimates. Model 1 to Model 4 (data from 1990 to 2019).
Figure 3. HDR of b2estimates. Model 1 to Model 4 (data from 1990 to 2019).
Energies 2023,16, 277 10 of 13
Energies 2023, 16, x FOR PEER REVIEW 10 of 13
Figure 4. HDR of 𝑏2 estimates. Model 1 to Model 4 (data from 1959 to 2019).
From the analysis undertaken, we can infer that there is a positive impact of CO2 in
the increase of the Earth’s global temperature, supporting similar results obtained with
other statistical techniques available in the literature. This is an important finding because
it is obtained with a methodology (not usually used in climate change science, as far as
we know) that allows the replication of the original time series with the subsequent con-
struction of confidence intervals for the model parameters, which represents an important
advantage. The choice of the methodology for the construction of confidence intervals and
the estimation method used, although it does not compromise the results of this work,
could be seen as possible limitations of this approach in other theoretical scenarios.
As evidenced by [5], joint forces worldwide are needed to fight global warming and
climate change. Sometimes, governmental institutions’ unresponsiveness leads to ham-
pered effects, but the general population should be accountable. Moreover, poverty and
the lack of appropriate infrastructure pave the way to the negative worldwide dissemina-
tion of these effects ([5]). Thus, air pollution needs to be mitigated, and we advocate the
need to quickly reduce global warming through our results, such as to protect human life
and health, and to ensure a sustainable future for the entire Earth planet. For that to be
true, future human actions should be rethought as to the overuse of fossil fuels as energy
resources, and drivers of greenhouse gases, with the latter driving the increase in the av-
erage surface temperature of the Earth ([4]).
Figure 4. HDR of b2estimates. Model 1 to Model 4 (data from 1959 to 2019).
From the analysis undertaken, we can infer that there is a positive impact of CO
2
in
the increase of the Earth’s global temperature, supporting similar results obtained with
other statistical techniques available in the literature. This is an important finding because
it is obtained with a methodology (not usually used in climate change science, as far as
we know) that allows the replication of the original time series with the subsequent con-
struction of confidence intervals for the model parameters, which represents an important
advantage. The choice of the methodology for the construction of confidence intervals
and the estimation method used, although it does not compromise the results of this work,
could be seen as possible limitations of this approach in other theoretical scenarios.
As evidenced by [
5
], joint forces worldwide are needed to fight global warming and
climate change. Sometimes, governmental institutions’ unresponsiveness leads to ham-
pered effects, but the general population should be accountable. Moreover, poverty and the
lack of appropriate infrastructure pave the way to the negative worldwide dissemination
of these effects ([
5
]). Thus, air pollution needs to be mitigated, and we advocate the need
to quickly reduce global warming through our results, such as to protect human life and
health, and to ensure a sustainable future for the entire Earth planet. For that to be true,
future human actions should be rethought as to the overuse of fossil fuels as energy re-
sources, and drivers of greenhouse gases, with the latter driving the increase in the average
surface temperature of the Earth ([4]).
Energies 2023,16, 277 11 of 13
4. Conclusions
This work explores through a recent technique the relationship between carbon dioxide
emissions and global temperature on Earth. This methodology, in addition to its statistical
advantages, is particularly appealing in the area of climate change, given the difficulties in
the measurement of global temperature and CO
2
emissions with high precision (usually
reported in the literature and perhaps one of the reasons for some to discredit scientific
studies), as it allows for the replication of the original time series with the subsequent
construction of confidence intervals for the model parameters. However, under different
technical premises, it is estimated that an increase in the annual average of CO
2
will always
drive an increase in global temperature, regardless of the time series model considered.
Urgently, decision-makers should be aware that at the current rate of increase in CO
2
emissions, it would hardly be possible for countries to fulfill the Paris Agreement. For that,
measures against pollution increases, stricter CO
2
abatement policies, a strict reduction
of fossil fuel energy consumption and production, the promotion of renewable energy
sources, and others, should be promoted and mandatory. Provided that the release of CO
2
emissions is only reflected in the long-run global temperature effects, the recent COVID-19
pandemic will, fortunately, lead us to reflect on the need for changing life habits, given
that the reduction in CO
2
emissions will only be strongly noticed in the next decades and
centuries. Still, it remains to be seen if the stricter restrictions and confinements would be
enough to save the planet and allow us to enjoy a green and breathable planet Earth, with
a positive impact on human, plant, and animal health in the medium to long future.
Only CO
2
emissions are considered here as a factor contributing to global temperature
rises, and, therefore, to global warming. Other greenhouse gases should be considered,
along with other factors which able to explain the global temperature rise (excessive use of
fossil fuels, land exploration, forest harvesting, population growth, urbanization, demand-
ing and unsustainable harming lifestyles, technology misuses, quick industrialization, and
excessive use of resources, among many others). In other words, to prevent the future rise
of global temperature, we need to decrease carbon emissions, but mainly need to stop using
nature as a raw material for exploration.
Author Contributions:
Conceptualization, P.M. and M.M.; methodology, P.M.; software, P.M.; valida-
tion, P.M. and M.M.; formal analysis, P.M. and M.M.; investigation, P.M. and M.M.; resources, P.M.;
data curation, P.M.; writing—original draft preparation, P.M. and M.M.; writing—review and editing,
M.M. All authors have read and agreed to the published version of the manuscript.
Funding:
This work is supported by the Center for Research and Development in Mathematics and
Applications (CIDMA) and by the Research Unit on Governance, Competitiveness and Public Policies
(GOVCOPP) through the Portuguese Foundation for Science and Technology (FCT—Fundação para
a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDB/04058/2020.
Data Availability Statement: Not applicable.
Acknowledgments:
We would like to express our gratitude to the Editor and the three anonymous
Reviewers for all the helpful comments. The authors also thank NASA’s Jet Propulsion Laboratory
and National Centers for Environmental Information for providing all the information needed for
this work through the websites above mentioned.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
Kalra, S.; Lamba, R.; Sharma, M. Machine learning based analysis for relation between global temperature and concentrations of
greenhouse gases. J. Inf. Optim. Sci. 2020,41, 73–84. [CrossRef]
2.
Amirkhani, M.; Ghaemimood, S.; von Schreeb, J.; El-Khatib, Z.; Yaya, S. Extreme weather events and death based on temperature
and CO
2
emission—A global retrospective study in 77 low-, middle- and high-income countries from 1999 to 2018. Prev. Med.
Rep. 2022,28, 101846. [CrossRef] [PubMed]
3.
Gil-Alana, L.A.; Monge, M. Global CO
2
emissions and global temperatures: Are they related. Int. J. Climatol.
2020
,40, 6603–6611.
[CrossRef]
4.
Al-Ghussain, L. Global warming: Review on driving forces and mitigation. Environ. Prog. Sustain. Energy
2018
,38, 750. [CrossRef]
Energies 2023,16, 277 12 of 13
5. D’Amato, G.; Akdis, C.A. Global warming, climate change, air pollution and allergies. Allergy 2020,75, 2158–2160. [CrossRef]
6.
Letcher, T.M. (Ed.) Why do we have global warming? In Managing Global Warming; Academic Press: Cambridge, MA, USA, 2019;
Chapter 1; pp. 3–15. [CrossRef]
7. Goulet Coulombe, P.; Göbel, M. On Spurious Causality, CO2, and Global Temperature. Econometrics 2021,9, 33. [CrossRef]
8.
Davis, W.J. The Relationship between Atmospheric Carbon Dioxide Concentration and Global Temperature for the Last 425
Million Years. Climate 2017,5, 76. [CrossRef]
9.
Morgan, M.G.; Vaishnav, P.; Dowlatabadi, H.; Azevedo, I.L. Rethinking the Social Cost of Carbon Dioxide. Issues Sci. Technol.
2017,33, 43–50.
10.
Varotsos, C.; Mazei, Y.; Efstathiou, M. Paleoecological and recent data show a steady temporal evolution of carbon dioxide and
temperature. Atmos. Pollut. Res. 2020,11, 714–722. [CrossRef]
11. Varotsos, C.; Assimakopoulos, M.-N.; Efstathiou, M. Long-term memory effect in the atmospheric CO2concentration at Mauna
Loa. Atmos. Chem. Phys. 2007,7, 629–634. [CrossRef]
12.
Krapivin, V.F.; Varotsos, C.A.; Soldatov, V.Y. Simulation results from a coupled model of carbon dioxide and methane global
cycles. Ecol. Model. 2017,359, 69–79. [CrossRef]
13.
Varotsos, C.; Mazei, Y.; Novenko, E.; Tsyganov, A.N.; Olchev, A.; Pampura, T.; Mazei, N.; Fatynina, Y.; Saldaev, D.; Efstathiou, M.
A New Climate Nowcasting Tool Based on Paleoclimatic Data. Sustainability 2020,12, 5546. [CrossRef]
14.
Peters, G.P.; Andrew, R.M.; Boden, T.; Canadell, J.G.; Ciais, P.; Le Quéré, C.; Marland, G.; Raupach, M.R.; Wilson, C. The challenge
to keep global warming below 2 ◦C. Nat. Clim. Chang. 2013,3, 4–6. [CrossRef]
15.
Peters, G.P.; Andrew, R.M.; Canadell, J.G.; Friedlingstein, P.; Jackson, R.B.; Korsbakken, J.I.; Le Quéré, C.; Peregon, A. Carbon
dioxide emissions continue to grow amidst slowly emerging climate policies. Nat. Clim. Chang. 2020,10, 3–6. [CrossRef]
16.
Millar, R.J.; Fuglestvedt, J.S.; Friedlingstein, P.; Rogelj, J.; Grubb, M.J.; Matthews, H.D.; Skeie, R.B.; Forster, P.M.; Frame, D.J.; Allen,
M.R. Emission budgets and pathways consistent with limiting warming to 1.5 ◦C. Nat. Geosci. 2017,10, 741–747. [CrossRef]
17.
IPCC. Intergovernmental Panel on Climate Change. Global Warming of 1.5
◦
C: Summary for policymakers. Geneva IPCC. 2018.
Available online: https://www.ipcc.ch/sr15/chapter/spm/ (accessed on 4 February 2021).
18.
Withey, P.; Johnston, C.; Guo, J. Quantifying the global warming potential of carbon dioxide emissions from bioenergy with
carbon capture and storage. Renew. Sustain. Energy Rev. 2019,115, 109408. [CrossRef]
19.
United Nations. Paris Agreement. 2015. Available online: https://unfccc.int/sites/default/files/english_paris_agreement.pdf
(accessed on 4 February 2021).
20.
Sun, J.; Chen, C.; Zhang, Y.; Li, W.; Song, Y. Competitive adsorption characteristics based on partial pressure and adsorption
mechanism of CO2/CH4 mixture in shale pores. Chem. Eng. J. 2022,430, 133172. [CrossRef]
21.
Hu, W.; Chen, C.; Sun, J.; Zhang, N.; Zhao, J.; Liu, Y.; Ling, Z.; Li, W.; Liu, W.; Song, Y. Three-body aggregation of guest molecules
as a key step in methane hydrate nucleation and growth. Commun. Chem. 2022,5, 33. [CrossRef]
22.
Demirhan, H. Impact of increasing temperature anomalies and carbon dioxide emissions on wheat production. Sci. Total Environ.
2020,741, 139616. [CrossRef]
23.
Krapivin, V.F.; Varotsos, C.A.; Soldatov, V.Y. The Earth’s Population Can Reach 14 Billion in the 23rd Century without Significant
Adverse Effects on Survivability. Int. J. Environ. Res. Public Health 2017,14, 885. [CrossRef]
24. Koutsoyiannis, D.; Kundzewicz, Z.W. Atmospheric Temperature and CO2: Hen-Or-Egg Causality? Sci 2020,2, 83. [CrossRef]
25.
Dong, C.; Dong, X.; Jiang, Q.; Dong, K.; Liu, G. What is the probability of achieving the carbon dioxide emission targets of the
Paris Agreement? Evidence from the top ten emitters. Sci. Total Environ. 2018,622–623, 1294–1303. [CrossRef]
26.
Rau, G.H.; Willauer, H.D.; Ren, Z.J. The global potential for converting renewable electricity to negative-CO
2
-emissions hydrogen.
Nat. Clim. Chang. 2018,8, 621–625. [CrossRef]
27.
IPCC. Summary for Policymakers. In Climate Change 2022: Mitigation of Climate Change. Contribution of Working Group III to
the Sixth Assessment Report of the Intergovernmental Panel on Climate Change; Shukla, P., Skea, J., Slade, R., Al Khourdajie, A.,
van Diemen, R., McCollum, D., Pathak, M., Some, S., Vyas, P., Fradera, R., et al., Eds.; Cambridge University Press: Cambridge,
UK; New York, NY, USA, 2022. [CrossRef]
28.
Monge, M.; Gil-Alana, L.A. Pandemic episodes, CO
2
emissions and global temperatures. Theor. Appl. Climatol.
2022
,148, 481–489.
[CrossRef] [PubMed]
29.
Xiong, W.; Tanaka, K.; Ciais, P.; Yan, L. Evaluating China’s Role in Achieving the 1.5
◦
C Target of the Paris Agreement. Energies
2022,15, 6002. [CrossRef]
30.
Stips, A.; Macias, D.; Coughlan, C.; Garcia-Gorriz, E.; Liang, X.S. On the causal structure between CO
2
and global temperature.
Sci. Rep. 2016,6, 21691. [CrossRef] [PubMed]
31. Skytt, T.; Nielsen, S.N.; Jonsson, B.-G. Global warming potential and absolute global temperature change potential from carbon
dioxide and methane fluxes as indicators of regional sustainability—A case study of Jämtland, Sweden. Ecol. Indic.
2020
,110,
105831. [CrossRef]
32. Boretti, A. COVID 19 impact on atmospheric CO2concentration. Int. J. Glob. Warm. 2020,21, 317–323. [CrossRef]
33.
Lacis, A.A.; Schmidt, G.A.; Rind, D.; Ruedy, R.A. Atmospheric CO
2
: Principal Control Knob Governing Earth’s Temperature.
Science 2010,330, 356–359. [CrossRef]
Energies 2023,16, 277 13 of 13
34.
Millar, R.J.; Nicholls, Z.R.; Friedlingstein, P.; Allen, M.R. A modified impulse-response representation of the global near-surface
air temperature and atmospheric concentration response to carbon dioxide emissions. Atmos. Chem. Phys.
2017
,17, 7213–7228.
[CrossRef]
35.
Allen, M.R.; Frame, D.J.; Huntingford, C.; Jones, C.D.; Lowe, J.A.; Meinshausen, M.; Meinshausen, N. Warming caused by
cumulative carbon emissions towards the trillionth tonne. Nature 2009,458, 1163–1166. [CrossRef] [PubMed]
36.
Vinod, H.D.; López-de-Lacalle, J. Maximum Entropy Bootstrap for Time Series: The meboot R Package. J. Stat. Softw.
2009
,29,
1–19. [CrossRef]
37. Vinod, H.D. Maximum entropy ensembles for time series inference in economics. J. Asian Econ. 2006,17, 955–978. [CrossRef]
38. Jaynes, E.T. Probability Theory—The Logic of Science; Cambridge University Press: Cambridge, UK, 2003. [CrossRef]
39.
Vinod, H.D. Hands-On Intermediate Econometrics Using R: Templates for Extending Dozens of Practical Examples; World Scientific:
Hackensack, NJ, USA, 2008. [CrossRef]
40.
Macedo, P. A two-stage maximum entropy approach for time series regression. Commun. Stat. B Simul. Comput.
2022
. [CrossRef]
41.
Wasserstein, R.L.; Lazar, N.A. The ASA’s Statement on p-Values: Context, Process, and Purpose. Am. Stat.
2016
,70, 129–133.
[CrossRef]
42. Hyndman, R.J. Computing and Graphing Highest Density Regions. Am. Stat. 1996,50, 120–126.
43.
Hyndman, R.; Einbeck, J.; Wand, M. hdrcde: Highest Density Regions and Conditional Density Estimation. 2018. Available
online: https://CRAN.R-project.org/package=hdrcde (accessed on 6 November 2020).
Disclaimer/Publisher’s Note:
The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.
