1 Introduction

Over the last 100 years, the temperature on the Earth's surface has been rising significantly (see Nicholls et al. 1996; Jones and Wigley 2010; and Folland et al. 2018; among others) caused by the effect of the burning and emissions of fossil fuels, industrialization and greenhouse gas concentration in the atmosphere (Anderegg et al. 2010; Beckage et al. 2018, etc.). Nevertheless, it is important to consider other factors such as solar irradiance, which are innate in the climate system, and which also affect this situation. According to Zickfeld et al. (2012), McMillan and Wohar (2013) and Zickfeld et al. (2016), the temperature and the concentration of carbon dioxide in the atmosphere exhibit a close correspondence. Also, National Oceanic and Atmospheric Administration (NOAA) and authors such as Laat and Maurellis (2004), Hansen et al. (2010), Cahill et al. (2015) and Sanz-Pérez et al. (2016) support the hypothesis that the carbon dioxide concentration and temperatures exhibit the same behavior and move in a very similar way.

In recent times, we have seen that an infectious disease named SARS-CoV-2, of the Coronaviridae family and which caused the COVID-19 disease, was identified in Wuhan City, China, in December 2019 (see Hui et al. 2020 and World Health OrganizationFootnote 1) causing an unprecedented cessation of human activities and affecting global energy use and CO2 emissions.

The confinement imposed on the population as a sanitary measure has brought about drastic changes in energy use with an impact on CO2 emissions. The OECD report (2020) indicates that the virus will cause a negative supply shock to the world economy, by forcing factories to shut down and disrupting global supply chains. This has resulted in a decrease of 5.8% in global fossil CO2 emissions during the first quarter of 2020 (see Liu et al. 2020). According to Le Quéré et al. (2020) and their sensitivity tests, the decrease in annual fossil CO2 emissions from the severe and forced confinement of world populations has been between –4.2% (if pandemic restrictions are lifted by mid-June) and –7.5% (if some restrictions remain worldwide until the end of 2020). According to some researchers, these rates of decrease are similar to those which are necessary year after year over the next few decades to limit climate change and prevent warming of 1.5 ºC.

Doing an extensive review of the bibliography, most of the literature tends to focus on studies based on temperatures and CO2, separately. On the one hand, researchers have focused their efforts to study global temperatures using stochastic processes and trends (see, e.g., Bloomfield 1992; Bloomfield and Nychka 1992; Galbraith and Green 1992; Woodward and Gray 1993, 1995; Koenker and Schorfheide 1994; Zhang and Basher 1999; Harvey and Mills 2001; Fomby and Vogelsang 2003; Gil-Alana 2003, 2005, 2008a,b; Vogelsang and Franses 2005; Mills 2006, 2010; Gay-Garcia et al. 2009; Hendry and Pretis 2013; Kaufmann et al. 2006, 2010, 2013; Estrada et al. 2013; Chang et al. 2015, etc.). On the other hand, emissions have also been studied by many authors: Sun and Wang (1996); Slottje et al. (2001); Alby (2006); Ezcurra (2007); Chang and Lee (2008); Romero-Avila (2008); Lee et al. (2008); Lee and Chang (2009); Nourry (2009); Panopoulou and Pantelidis (2009); Christidou et al. (2013); Yavuz and Yilanci (2013); Ahmed et al. (2016); Tiwari et al. (2016); Gil-Alana and Solarin (2018); Gil-Alana and Trani (2019); among others. Finally, other authors such as McMillan and Wohar (2013), Zhang et al. (2019), Ying et al. (2020) and Gil-Alana and Monge (2020) have taken into consideration the two variables together using various methodologies such as unit root tests and autorregression models (McMillan and Wohar 2013), multilayer and multivariable network methods (Zhang et al. 2019), multilayer climate network approach (Ying et al. 2020) and fractional integration (Gil-Alana and Monge 2020).

Our main objective in this research paper is to conduct a serious statistical analysis about the statistical properties of various time series dealing with global temperatures and global CO2 emissions. We use techniques based on long memory and fractional integration that allow the number of differences to be taken in the series to render them stationary be fractional numbers. More in particular, we use fractionally integrated autoregressive moving average ARMA (ARFIMA) models, thus allowing for a fractional degree of differentiation in the level of the series of global annual temperatures (land temperatures, land and ocean temperatures and Northern and Southern hemisphere temperatures) as well as annual global CO2 emissions from 1880 to 2014, taking into consideration the eight large pandemic events around the world, prior to the present one caused by COVID-19.

The motivation that is behind this work is that previous studies that have investigated the nonstationarity/stationarity of the series under investigation only have considered integer degrees of differentiation, i.e., 0 for stationary series and 1 for nonstationary ones, not considering cases where the degree of differentiation may be a fractional value between 0 and 1. In fact, many recent studies have shown that many climatological and CO2 emission-related time series display a long memory pattern, implying different results than those obtained based on classical analysis and that only used integer degrees of differentiation (see, e.g., Barassi et al. 2011; Belbute and Pereira 2017; Gil-Alana and Trani 2019 for papers dealing with CO2 emissions and Vera-Valdes 2020; Mangat and Reschenhofer 2020; Gil-Alana and Monge 2020; Awe and Gil-Alana 2021 and others for papers with temperature data).

The paper is organized as follows: Sect. 2 briefly describes the techniques used in the paper, while Sect. 3 presents the dataset and Sect. 4 contains the main empirical results. Finally, Sect. 5 concludes the paper.

2 Methodology

2.1 Unit roots methods

There exist many different ways of testing for unit roots. The most common ones are those of Fuller (1976) and Dickey and Fuller (1979), the ADF tests. They are asymptotically optimal when the data are stationary. Other more updated unit root methods are those proposed in Phillips and Perron (1988), Kwiatkowski et al. (1992) , Elliott et al. (1992) , Ng and Perron (2001), etc.

2.2 ARFIMA (p, d, q) model

To carry out this research, we employ long memory methods based on fractional integration where the number of differences required to render a series I(0) stationary is a fractional value.

Following a mathematical notation, given a time series \({x}_{t}, where t=1, 2, \dots\), we say it is integrated of order d (and denoted as \({x}_{t}\approx I\left(d\right)\)) if:

$${(1-L)}^{d}{x}_{t}={u}_{t}, t=1, 2, \dots ,$$
(1)

where \(d\) can be any real value, \(L\) is the lag operator \((L{x}_{t}={x}_{t-1})\) and \({u}_{t}\) is I(0), defined as a covariance stationary process with a spectral density function that is positive and finite at the zero frequency. Thus, \({u}_{t}\) may display some type of time dependence of the weak form, i.e., the type of an invertible and stationary Autoregressive Moving Average (ARMA) form, i.e.,

$${\phi \left(L\right) u}_{t} ={\theta \left(L\right) \varepsilon }_{t}, t=1, 2, \dots ,$$
(2)

where ϕ(L) refers to the AR polynomial, θ(L) refers to the MA one and εt is a white noise process. In such a case, if \({u}_{t}\) is ARMA (p, q), xt is said to be fractionally integrated ARMA, i.e., ARFIMA (p, d, q).

Depending on the value of the differencing parameter d, several specifications based on (1) can be observed: The process would be short memory or I(0) when \(d=0\) in (1). This occurs because \({x}_{t}={u}_{t}\). The high degree of association between observations which are far distant in time receives the name of long memory and occurs when \(d>0\). Within this last assumption, the process is still covariance stationary if \(d<0.5\) with the autocorrelations decaying hyperbolically slowly.

The reading that we can make of the results obtained from the fractional d is as follows: We consider a process of reversion which means that the shocks disappear in the long run when d is smaller than 1, and the lower the value of d is, the faster the reversion process is. In contrast to the above, the shocks are expected to be permanent when \(d\ge 1\).

Although there are several procedures to estimate the degree of differentiation d (see Geweke and Porter-Hudak 1983; Phillips 1999, 2007; Sowell 1992; Robinson 1995; Beran 1995; etc.), we base our results on the maximum likelihood procedure (see Sowell 1992) and we use the Akaike information criterion (AIC, Akaike 1973) and the Bayesian information criterion (BIC; Akaike 1979) to select the right ARFIMA model.

3 Data

We use global annual temperature anomalies using data from meteorological stations; global annual temperature anomalies computed from land and ocean; and global annual temperature anomalies for the northern and southern hemispheres computed using land and ocean data,Footnote 2 and we also use annual data from the Carbon Dioxide Information Analysis Center (CDIAC) of the global CO2 emissions originating from fossil fuel burningFootnote 3 to analyze the behavior of these variables in the long term during the periods of pandemics for the time period from 1880 to 2009.

Following the research done by Jordà et al. (2020), the dates that we have used for our analysis are collected in the following tableFootnote 4:

Event

Start

End

Global Flu Pandemic

1889

1890

Sixth Cholera Pandemic

1899

1923

Encephalitis Lethargica Pandemic

1915

1926

Spanish Flu

1918

1920

Asian Flu

1957

1958

Hong Kong Flu

1968

1969

SARS

2002

2004

H1N1 Pandemic

2009

2009

Figure 1 plots the original data of the global fossil fuel CO2 emissions and the four annual anomalies in the temperature series mentioned above indicating the pandemic periods. Although there is a constant increase in the trend across the sample, in periods of a pandemic it is observed that the temperatures stabilize/decrease with respect to the trend.

Fig. 1
figure 1

Pandemic episodes, global fossil fuel C02 emissions and annual anomaly in temperatures

Full size image

4 Results

We start the analysis by performing the three standard unit root tests outlined in Sect. 2. We select the augmented Dickey–Fuller test (ADF) to examine the statistical properties of the original series and its differences to obtain robust results.

Table 1 displays the results, which suggest that the original data are nonstationary I(1) and the first differences are stationary I(0).

Table 1 Unit roots tests. (i) Model with no deterministic components; (ii) with an intercept and (iii) with a linear time trend. Inside the parenthesis the p-value is reflected, outside the t-statistic with test critical value at 0,1% (***); 1% (**); 5% (*)
Full size table

Identical results are obtained if other more updated unit root methods are used, such as those mentioned in the previous section. (These results are available from the authors upon request).

Our results so far indicate that all the original series are nonstationary I(1); however, due to the low power of the unit root methods under fractional alternatives,Footnote 5 we also perform different ARFIMA (p, d, q) models to study the persistence of the subsamples corresponding to the different periods of pandemic since 1880.

We select the most appropriate ARFIMA specification for each series for the Akaike information criterion (AIC; Akaike 1973) and Bayesian information criterion (BIC; Akaike 1979).Footnote 6 We allow for ARFIMA models of the form "(0, d, 0)," "(1, d, 0)," "(2, d, 0)," "(0, d, 1)," "(0, d, 2)," "(1, d, 1)," "(1, d, 2)," "(2, d, 1)," "(2, d, 2)," i.e., we choose any ARFIMA(p, d, q) with p and q being smaller than or equal to 2. Once the various configurations were calculated, and following the selection criteria mentioned above, the results are collected in Table 2.

Table 2 Results of long memory tests. The second column displays the selected model, indicating the orders for the AR and MA dynamics. Column 3 reports the estimates of d while Column 4 the associated standard error. The 95% confidence band is displayed in Column 5, and Column 6 indicates the nature of the process according to the estimated value of d
Full size table

Table 2 displays the estimates of the fractional parameter d and the AR and MA terms obtained using Sowell's (1992) maximum likelihood estimator of various ARFIMA (p, d, q) specifications with all combinations of (p, q) with p, q ≤ 2, for global annual temperatures (land temperatures, land and ocean temperatures and Northern and Southern hemispheres temperatures) and global annual CO2 emissions in each pandemic subperiod.

Starting with the CO2 emissions, we see that the values of d range widely between 0.0007 (ELP) and 1.9997 (GFP), and though the confidence intervals are, in some cases, very wide (clearly due to the small sample sizes in some of the periods examined), we observe that the I(0) hypothesis cannot be rejected in the cases of the Sixth Cholera Pandemia (SCP), the Encephalitis Lethargica Pandemia (ELP) and the Asian Flu (AF), while the I(1) null cannot be rejected for the Global Flu Pandemia (GFP), the Sixth Cholera Pandemia (SCP), the Asian Flu (AF) and the SARS; finally, these two hypotheses are rejected in favor of I(d, 0 < d < 1) behavior in the cases of Spanish Flu (SF), Hong Kong Flu (HKF) and the HINI Pandemia. Thus, the results here are very heterogeneous across the different periods of pandemics.

Focusing next on the temperatures, all values of d are now in the range (0, 1) implying fractional integration, and the highest values correspond to the Sixth Cholera Pandemia ( SCP), with the values of d ranging between 0.4091 (Land Temp.) and 0.5473 (Land Oc. Temp.). In many cases, the I(0) hypothesis cannot be rejected in any single case (GFP, AF, HKF or SARS) but neither for SF in three out of the four temperature series. In general, we observe that the orders of integration are smaller than 1 in all cases for the temperature series (the only exception is Land Oc. Temp. for the Spanish Flu (SF)), implying mean reversion, with shocks having temporary effects and disappearing by themselves in the long run.

5 Concluding remarks

In this paper we have examined forty time series corresponding to the eight pandemic subsamples (Global Flu Pandemic, Sixth Cholera Pandemic, Encephalitis Lethargica Pandemic, Spanish Flu, Asian Flu, Hong Kong Flu, SARS Pandemic and H1N1 Pandemic) that have been taken place during the last 120 years to understand if these pandemic episodes follow a similar pattern.

Our first focus has been to analyze the statistical properties of these time series using unit roots methods. We started by performing ADF unit root tests and the results of these and other similar methods suggest that the series are nonstationary I(1) while the first differences are stationary I(0).

On the other hand, and in order to be more general, we also estimated the differencing parameter d in terms of a fractional model using an ARFIMA (p, d, q) approach. To select the right model, we combined all the possible (p, d, q) cases, with p and q smaller than or equal to 2 to find the best specification throughout AIC and BIC methods.

Our results indicate that for the CO2 emissions the results are quite heterogeneous across the different pandemic periods and the intervals are in some cases very wide such that for example, for the Global Flu Pandemic, the I(1) and the I(2) hypotheses cannot be rejected, and for the SCP and AF, the same happens for the I(0) and I(1) hypotheses; for SARS only the I(1) cannot be rejected and for ELP, the I(0) one; finally for SF, HKF and HINI, the estimated values of d are constrained between 0 and 1. Thus, only for the last three subsamples (SF, HKF and HINI) there is some evidence of mean reversion and transitory shocks contrary to what happens in the rest of the cases. For the temperature series, mean reversion occurs in all cases, since all the estimated values of d are strictly smaller than 1, and the highest levels of persistence occur in the case of SCP and SP. For the remaining periods, the I(0) hypothesis is rarely rejected. and thus the recovery of a shock will take place in a shorter period of time. These results are consistent with those presented in Gil-Alana and Monge (2020) where the emissions are found to be I(1) or I(d) with d close to 1 (as in the cases of the Global Flu Pandemic, the Sixth Cholera Pandemic, the Asian Flu and SARS Pandemics), while the temperatures display orders of integration strictly smaller than 1, and thus show mean reverting behavior. These results suggest that in the event of exogenous shocks, temperatures will recover by themselves unlike what happens with the emissions in the majority of the cases where there is no reversion to the mean and strong actions should be adopted to recover the original long term projections.