1. Introduction
Temperature is a key indicator of climate change, and monitoring changes in temperature over time can help identify potential impacts on ecosystems, agriculture, and human health (FAO, 2021). In a report by the Intergovernmental Panel on Climate Change (IPCC, 2018), it is mentioned that global warming induced by human activity will produce temperatures around 1.0°C higher than in times of the preindustrial period, and that if current trends continue, global warming is expected to reach 1.5℃ between 2030 and 2052. The report also noted that some regions of the globe are experiencing more warming than others due to vegetation cover and other factors (Nicholls et al., 1996; Percival et al., 2001; Caballero et al., 2002; GilAlana, 2008; Franzke, 2012; Ludescher et al., 2016; Contractor et al., 2020; etc.). There are also clear evidences to show the dependency of global warming on the diurnal temperature range (DTR), that is, the difference between the maximum and minimum temperature (Karl et al., 1991). The National Oceanic and Atmospheric Administration (NOAA) and National Aeronautics and Space Administration (NASA) have measuredout 2010 to be the warmest year on record when compared to the average baseline of the 20^{th} century and of 1951–1980, respectively. Hansen et al. (2010) and Cahill et al. (2015) have proved that the temperature increase trend would likely continue due to the continuous increases in the greenhouse gas (GHG) concentration, while in the subSaharan African region, the warming is likely to be higher compared to global average, and the speed of temperature rise will be more than the rise in the global mean temperature (see, e.g., Shepard, 2019; Ray, 2021). Africa is particularly vulnerable to the effects of climate change due to its dependence on rainfed agriculture and limited infrastructure to cope with extreme weather events (IPCC AR6, 2021; Ngarukiyimana, Ruhinda and Mupenzi, 2020). Furthermore, the largescale climate patterns such as El NiñoSouthern Oscillation (ENSO), Indian Ocean Dipole (IOD), and Atlantic Multidecadal Oscillation (AMO) often influence African’s climate, causing heatwaves, droughts, and floodings in different regions of the continent (Ngarukiyimana, Ruhinda and Mupenzi, 2020).
In the very arid regions of Africa, the temperature is expected to rise faster compared to other parts of the world in the 21
^{st} century (CDKN, 2014). Thus, the African continent is very prone to climate variability changes due to its geographical location on the globe. The continent lies between 37
^{o}N and 35
^{o}S and in this position, the continent lies almost perfectly on the equator, covering a total land area of about 30.2 million square kilometres, which is about onefifth of the land area of the entire globe (
https://www.mapsofworld.com/lat_long/africa.html#:~:text=Africa's%20latitude%20and%20longitude%20lies,of%20the%20world's%20land%20area).
Figure 1 shows the map of African showing all countries, and how the equator divides the continent into two parts, that is, the continent lies within the intertropical zone between the Tropic of Cancer and the Tropic of Capricorn, where the tropics are regions of the globe surrounding the equator.
This latitudinal position of Africa around the tropics makes rainfall, temperature and humidity in Africa to be high compared to other continents (
https://en.wikipedia.org/wiki/Tropics). There are also different climate types experienced annually. These are the equatorial climate, tropical wet and dry climate, tropical monsoon climate, semiarid climate, desert climate and subtropical highland climate (
https://en.wikipedia.org/wiki/Climate_of_Africa). It is very rare to experience a temperate climate in any part of Africa except at very high elevations and along the fringes. African deserts are the hottest and driest worldwide due to subtropical ridges with hot, dry air masses. As predicted by IPCC AR6 (2021). temperatures in African countries are expected to have increased by 1.5°C  3°C in 2050. Putting temperature on one side, the total annual greenhouse gasses are also rising fast at the rate of about 1.6% per annum with carbon dioxide emissions alone increasing by about 2% per year. This together with the fastrising temperature could change the African climate in such a way that urgent intervention would be required coupled with energy issues and poor amenities that most countries are experiencing. Thanks to the IPCC for their efforts in monitoring the African climate (IPCC AR6, 2021).
Focussing specifically on Africa, there are various studies investigating the temperature dynamics and its maximum and minimum ranges on the continent. The mean annual DTR for Ethiopia, Sudan, South Africa and Zimbabwe have been considered by Nicholls et al. (1996). where the authors observed a decrease of 0.5°C to 1.0°C in the mean annual DTR for Sudan and Ethiopia. For countries in the southern part of the continent such as Zimbabwe and South Africa, the DTR reduced during the period 1950 – 1960. Kruger and Sekele (2012) and Kruger and Nxumalo (2016) find evidence of increasing warming trends in South Africa. Extending the analysis to other countries, New et al. (2006) investigated trends in daily climate extremes over Southern and Western Africa and found a repeating pattern of temperature extremes associated with rising temperatures. Neumann et al. (2007) found that temperature in the Volta basin, West Africa exhibited highly significant positive trends. In a similar vein, Muthoni (2020) studied temperatures in West Africa and also found a strong warming trend.
Meanwhile, there have been controversies in relation to the speed of temperature increase, on the appropriate estimation method to model temperature across time. The most standard approach still recommends incorporating a linear trend in the modelling framework. Using the linear model with fractional integration methods, GilAlana et al. (2019) examined issues such as linear trends, seasonality and persistence in Western, Eastern, and Southern regions of Africa and found that time trends are required in most of the countries to explain the climate features in the areas. They also found evidence of structural breaks in some of the countries. Other papers that have applied fractional integration in a linear framework to analyse temperature and rainfall data include Ogunsola and Yaya (2019) and Yaya and Vo (2020).
Closelyrelated research to the present paper is the one conducted in three African countries, namely, South Africa, Kenya, and Côte d’Ivoire by Carcel and GilAlana (2015). They check for warming trends and using fractional integration, they showed that only Kenya has experienced a significant temperature increase in the last 30 years as at the time of the research.
In the current paper, we, therefore, investigated persistence and linear trend in the maximum and minimum annual average temperatures in 36 African countries chosen by data availability from Climate Change Knowledge Portal (CCKP). Due to the statistical distribution of maximum and minimum temperature, we also examine the difference between the two series which leads to the diurnal range as this informs climate differences in the study area. High maximum temperatures and low minimum temperatures are experienced in desert or arid climate with scanty vegetation cover, while moderate maximum temperatures and low minimum, temperatures are often recorded in temperature climate regions. The latter is unlikely to be experienced in the subSaharan region of Africa. Thus, changes in the distribution of maximum and minimum temperatures over time can indicates climatic changes as the frequency or intensity of extreme temperatures, and its events such as heat waves or cold snaps can be indicative of climate change.
Having obtained the fractional integration estimates based on a linear trend specification, we conducted homogeneity of paired Local Whittle (LW) estimates of fractional orders based on Hausmantype test of Robinson and Yajima (2002) since the statistical equality of the fractional orders is a prerequisite for cointegration, at least in the bivariate representation as is the case in this paper. We extended the analysis to testing for fractional cointegration using narrowband frequency domain least square (NBFDLS) estimates of Christensen and Nielsen (2006) in the fractional cointegration framework of Marinucci and Robinson (2001). The NBFDLS estimates for the cointegrating vector were obtained and used to compute the model residuals in models linking each country's maximum and minimum temperatures. Then, fractional integration estimates were obtained on the residuals based on the LW estimator. Note that cointegration is relevant in the present context since it will indicate us if there is a long run equilibrium relationship between maximum and minimum temperatures. On the other hand, lack of this property will suggest that the two series move apart supporting potential extreme changes in temperatures.
In conclusion, the hypothesis to be tested in this paper are the following: we first consider the possibility of long memory or long range dependence in the variables under examination since this is a property widely observed in climatological data. Then, based on this observation, we claim that the estimation of the linear trends in the data is clearly affected by the long memory property, and not to take into account this issue will clearly produce biased estimates of the time trend coefficient to explain CC. As a final issue, the possibility of cointegration is also examined looking at the difference between maximum and minimum temperatures, claiming that under normal circumstances both variables should be linked in a long run equilibrium relationship.
2. Data
Maximum and minimum average annual temperature (
^{0}C) datasets for 36 African countries were analysed to provide insights into climate change on the continent. The countries are: Angola, Benin, Botswana, Burkina Faso, Cameroon, Central African Republic, Chad, Congo, Cote d’Ívoire, Egypt, Gabon, Ghana, Guinea, GuineaBissau, Kenya, Lesotho, Liberia, Libya, Madagascar, Malawi, Mali, Mauritania, Morocco, Namibia, Niger, Nigeria, Rwanda, Sierra Leone, Senegal, South Africa, Sudan, Tanzania, Tunisia, Uganda, Zambia and Zimbabwe. The datasets were retrieved from the World Bank Climate Change Knowledge Portal at
https://climateknowledgeportal.worldbank.org/. (see World Bank, 2021). This portal has been collecting historical monthly and annual climate data of countries in the world since 1901 till date, and the nature of the analysis to be carried out here required only annual data since monthly datasets could bias our results due to the interference of seasonality. In each of the five geographical zones of Africa, we displayed a graph showing the distribution of maximum and minimum temperature, whereas, in the case of West Africa zone, we have temperature plots for Guinea and Nigeria, others are Chad (Central Africa), Egypt (North Africa), Kenya (East Africa) and South Africa (Southern Africa). In each case, both the maximum and minimum temperatures are plotted on the left vertical axis as shown in
Figure 1.
In all the plots, it is obvious to notice shifts in temperature trend, signalling global warming over time. In the case of Chad, the temperature increased consistently from 1950 to reach astronomic thermometric readings in 2010, similarly to Kenya, Nigeria and South Africa. The temperature shift in temperature trends is noticed around 1970 in the case of Egyptian temperature plots, and between 19701980 in the case of Nigeria.
Table 1, therefore, displays a data summary, showing the starting maximum and starting minimum temperatures in 1901 and corresponding sample ending maximum and minimum temperatures in 2021 for the 36 countries. It is found that ending temperatures in 2021 are quite higher than starting temperatures in 1901, resulting in a positive shift in temperature trend due to global warming. The annual range, the difference between the annual maximum and annual minimum temperature, which is used to proxy the diurnal temperature range (DTR) is also presented in the table for 1901 and 2021 data.
3. Econometric Methods
3.1. Testing for a Linear Trend
The analysis of climatological time series datasets takes its root from the seminal work of Hamilton (1994) where a linear model was employed to fit climate datasets. Hamilton (1994) used a model of the form:
where
${y}_{t}$ is climatic series under investigation and t is the time trend t = 1, 2, … . Then,
${x}_{t}$ is the deviation term, derived from the original series
${y}_{t}$. The model's parameters are α and β, which represent the intercept and slope, respectively. The slope, β, measures the average change in
${y}_{t}\text{}$over time, when
$t={t}_{0}$. Thus, in the context of climatic research, there is longrun warming if the slope, β, is positive and statistically significant, implying that the temperature is rising. The estimation of the deviation term is critical in carrying out this assessment. This is determined by the distribution from which
${x}_{t}\text{}$is created, and hence by the overall estimation of the process.
In most early studies, the error term x_{t} in (1) was assumed to be well behaved, i.e., displaying a shortmemory (SM) structure, also denominated integrated of order 0 or I(0) process. In this context, the simplest structure was the white noise case where no time dependence is permitted. If that dependenc allowed, the ordinary AutoRegressive Moving Average (ARMA) models were employed. However, the time dependence between the observations may display a higher range. Thus, it may exhibit longterm dependence (LTD) or long memory (LM), as claimed in various climatological data by researchers such as Montanari et al. (1996). Stephenson et al. (2000). Percival et al. (2001). GilAlana (2003, 2005) and others. GilAlana (2012) then adapted the linear model of Hamilton (1994) to the fractional integration or I(d) framework where the dependency of historical temperature observations of Alaska over long past time periods is investigated. This phenomenon is known as the longrange dependency (LRD) or LTD. Climatic variables often exhibit this property due to the natural variability, from year to year, and to the next. Climatic variables can also vary on much longer time scales such as from decades to centuries, and due to human interference, there may be a shift in the mean state of a particular climatic variable, such as temperature or rainfall. In this study, we will LRD or shift or shift in mean state of climate variables based on fractional integration and cointegration. Note, LRD is an aspect of fractional integration, thus LRD is often explained based on this broader perspective.
3.2. Testing for Fractional Integration
The following is the standard nonseasonal fractional integration model i.e. an integrated of order d or I(d) model:
where d is any real number (such as integer or fractional values), L is the lagoperator
$L{x}_{t}={x}_{t1}$, and
${u}_{t}$ denotes the white noise process, which is an I(0) process in the context of fractional integration I(d) process (An I(0) process is defined as a covariance stationary process where the infinite sum of its autocovariances is finite. It includes the white noise model but also the stationary ARMAclass of models). The polynomial (1  L)
^{d} in (2) is a Binomial representation, which can be expanded as follows:
and therefore
depicting that equation (2) can be written as
Intuitively, the role of $d$ in equation (3) is clear as it is a slope coefficient between ${x}_{t}$ and its lagged values such as ${x}_{t1},\text{}{x}_{t2},\text{}...\text{}$. Thus, it determines the degree of correlation between ${x}_{t}\text{}$and its lagged values. So, regression, or correlation here in explaining fractional integration or LRD processes leads to time series dependency or persistency as they are often used interchangeably. The model in (2) is nonseasonal model type since the dataset at hand in annual temperature series, and testing seasonality in addition, is not of interest in this paper.
Many methods have been used to estimate the fractional dependence parameter d. Some of these methods are semiparametric, while others are parametric methods. Semiparametric techniques are typically implemented in the frequency domain. In this study, we use a parametric frequency domain Whittle estimation approach (see Dahlhaus, 1989) in conjunction with a testing procedure of Robinson (1994a) that depends on the Lagrange Multiplier (LM) principle. Robinson (1994a) uses the null hypothesis test:
for any real value d
_{0} in a model given by the equations (1) and (2), that is,
where
${u}_{t}$ is a white noise process. The fundamental advantage of this approach is that, because it depends on the LM principle, all of it is evaluated under the null, and because d
_{o} can be any actual number, it is valid even in nonstationary situations (i.e., d
_{o} ≥ 0.5), with a regular normal limit distribution. In addition, this standard normal distribution holds independently of the inclusion or not of deterministic terms in the model like an intercept and/or a linear time trend, being this method the most efficient one in the Pitman sense against local departures from the null. (See, Robinson, 1994).
In complement to the aforesaid parametric methodology, we also use a semiparametric method to conduct the analysis. It is named semiparametric because no functional form is imposed on the error term, making only the assumption that it is integrated of order zero, i.e., I(0). In other words, it might be a simple white noise process or an ARMA process with a weak dependency autocorrelation. We employ in the paper a "local" Whittle (LW) estimate in the frequency domain, which is based on a frequency band that degenerates to zero. This approach (Robinson, 1995) is defined implicitly by:
where m represents the bandwidth parameter, and I(λ
_{j}) represents the periodogram of the time series, x
_{t}, which is given by:
Robinson (1995) demonstrated under finiteness of the fourth moment and other very mild conditions:
where d
_{o} is the actual value of d and in addition to the requirement that m→∞ is slower than
T.
3.3. Homogeneity of Paired Fractional Integration Parameters
Following from the local Whittle estimate, it is important to do the test of homogeneity of paired integration orders. As earlier argue, this is a necessary condition for testing cointegration in a bivariate system. The following null hypothesis is used to assess the homogeneity of the orders of integration:
where the orders of integration of the individual series are represented by
${d}_{y1}$ and
${d}_{y2}$ (see Robinson and Yajima, 2002; Yaya, 2022). The test statistic is defined as,
where
$h\left(T\right)>0$ and
${\widehat{G}}_{y1.y2}$ denote the (y1.y2)
^{th} element of
$\widehat{\Lambda}{\left({\lambda}_{j}\right)}^{1}I\left({\lambda}_{j}\right)\widehat{\Lambda}\left({\lambda}_{j}\right)$ with
$\widehat{\Lambda}\left({\lambda}_{j}\right)=diag\left\{{e}^{i\pi {\widehat{d}}_{y1}/2}{\lambda}^{{\widehat{d}}_{y1}},\text{}{e}^{i\pi {\widehat{d}}_{y2}/2}{\lambda}^{{\widehat{d}}_{y2}}\right\}$.
3.4. NarrowBand Frequency Domain Least Square Approach
At this juncture, we considered the narrow band frequency domain least square (NBFDLS) estimates in obtaining the cointegrating vector linking the maximum and minimum temperatures. This approach is applied when the cointegrating pairs are deemed to possess long memory, having weakly dependent regressors. Since the regressors and the residuals are of long memory, thus, they are deemed to be correlated even at a very long time span. In that case, both the least squares and generalized least squares estimates will be inconsistent (see Robinson and Hidalgo, 1997). Robinson (1994b) earlier proposed a semiparametric NBFDLS estimator which uses OLS on a degenerated band of frequencies around the origin. An improved version of the test for the stationary time series is given in Christensen and Nielsen 2006.
In the twovariable case, where
${y}_{1t}$ is for maximum temperature and
${y}_{2t}$ is for minimum temperature series, both of fractional integration order
$d<0.5$ and residuals of order
${d}_{e}<d$, the NBFDLS estimator is given by,
which is asymptotically distributed as,
where
${g}_{y1}$ and
${g}_{e}$ are the elements of a
$G$ diagonal
$2\times 2$ matrix. From (10), the normality is ensured as long as
$d+{d}_{e}<0.5$ (Christensen and Nielsen, 2006).
4. Main Results
Having presented the dataset used in
Section 2, we therefore present the main results obtained based on the econometric methods presented in
Section 3 above. In
Table 2, the results of temperature data stationarity are presented by employing Augmented DickeyFuller (ADF, Dickey and Fuller, 1979) unit root test for no deterministic term, intercept only, and intercept with the trend. These are the three options available for the test in the case of a linearly assumed time series process. The choice of the unit root test is motivated by its design in a linear framework which mimics the assumed linear specification of climatological time series as in Hamilton (1994) (see equation (1)). The results, in
Table 2 showed nonrejection of the unit root null hypothesis when no deterministic term (none) is assumed for maximum, minimum and range temperature series. The stern acceptance of nonstationarity of the three time series could be due to the high power of the test when the degree of freedom, as a result of no deterministic term in the test framework leaving only the augmented component, is compromised. By testing with only intercept, very few countries out of the 36 countries showed rejection of unit root null hypotheses in the case of both maximum and minimum temperature, while in the case of temperature range series, null hypotheses of unit root were rejected in almost all the 36 cases. For intercept with trend specification, the results for the range temperature are similar to those from the intercept only. For intercept with trend, as in maximum and minimum temperatures, more rejections of unit root null were observed compared to that of intercept only specification of the test. This mixed decision of the results of the unit root test for temperature series is due to the fact that the unit root test lacks power against trendstationarity and fractional alternatives (Diebold and Rudebusch, 1991; Hassler and Wolter, 1994; Lee and Schmidt, 1996). Also, when the persistence is likely to fall in long range dependence range (i.e.
$0<d<1$) or long memory range (
$0<d<0.5$), ADFlike unit root tests might find it difficult to detect correctly the stationarity/nonstationarity of the series.
This weakness of the unit root test makes fractional integration (fractional unit root) attractive since the test allows for persistence level (series differencing) in fractional value other than unit differencing as in the unit root test to be determined. Robinson (1994b) can be seen as an ADF test in a fractional sense since it allows testing fractional integration based on no deterministic term, intercept only and intercept with the trend. This test is earlier described above in equations (4)(6). Here in
Table 3, the results are presented for only the case of intercept with trend. Evidences of long memory and longrange dependence, i.e.,
$0<d<1$ are found in maximum and minimum temperatures, with
$d$ less than 0.5 in a number of countries and fairly above 0.5 in others. The highest bound values for
$d$ are 0.72 (Rwanda) and 0.76 (Uganda) in the case of maximum temperature, and these values correspond to
$d$ values 0.43 (0.27, 0.60) and 0.48 (0.32, 0.65), respectively for minimum temperatures. In the maximum, temperature, evidence of
$d$ fairly above the long memory stationary range (i.e
$d>0.5$) are found in Kenya, Madagascar, Rwanda and Uganda, while in the case of minimum temperature, we have Benin, Burkina Faso, Ghana and Madagascar. These results imply that long memory exists generally in maximum and minimum temperature distributions even though the upper bound of the confidence limit shows that some estimates might be in the longrange dependence range which is still close to 0.5. Differing persistence estimates explain the dynamics of temperature predictions, not in terms of trend often expected in climatological study but in terms of inherent correlations of current observations to past lagged historical values, large enough to compare the present climate with climate of the same regions over many decades. In that case, using only the linear trend approach such as that employed in Hamilton (1994) could lead to bias due to ignorance of exploring the (long memory) time series properties of the climatological observations.
Also, the temperature range series, which is the difference between the maximum and minimum temperature display long memory as well in a number of countries but with an upper bound limit above 0.5 in many cases.
Table 3 also indicates 29 cases of significant trends in the case of maximum temperatures, and 33 cases for the minimum temperatures, all them with significant positive coefficients, while only 9 countries show significant trends in the range, three with a positive value (Lesotho, Morocco and Botswana) and five with a negative trend (Sudan, Burkina Faso, Nigeria, Egypt, Kenya, and Central Africa).
For clarity,
Table 4, therefore, summarized the results in
Table 3 for cases where
$d$ value for range series is less than either or both
$d$ values for maximum and minimum temperatures. These countries include: Angola, Botswana, Central African Republic, Chad, Congo, Egypt, Gabon, Kenya, Libya, Madagascar, Malawi, Namibia, Nigeria, Sudan, Uganda, and Zimbabwe. These are the countries in which maximum and minimum temperatures are likely to have longrun relationships but the high confidence band for the range series poses suspicion which requires further robust analysis before one concludes the long run relationship existing between the pair.
The framework of fractional integration employed here also allows us to simultaneously test for positive trends in the series under investigation, as we know that significant positive trends in temperature series suggest evidence of global warming induced by temperature increases. In
Table 3, positive and significant trend coefficients are found in 29 countries for the case of maximum temperature, and in 33 countries in the case of minimum temperature. Thus, in
Table 4, it is put together that evidence of temperature increase is observed in 34 countries, that is, all except Liberia and Sierra Leone.
In order to further establish the long run relationship between maximum and minimum temperature series, we carried out homogeneity of fractional integration orders, where we ignore the confidence band since the test is defined in the frequency domain using periodogram bands. The estimation of
$d$ parameter is based on Local Whittle (LW) estimators and the significant difference in paired values of
$d$ are tested based on the approach described earlier in equations (7)(8). Estimates of
${\widehat{T}}_{xy}$ statistics, as given in Robinson and Yajima (2002) are presented in
Table 5 where none of these estimates are significant at 5% level. Recall that the test is asymptotically normally distributed with 1.96 as its twosided rejection value, and all these estimates are below the value. Evidence of homogeneity in all the pairs (maximum and minimum temperature pairs) further supports the possibility of longrun relationship, based on fractional cointegration in the comovement of maximum and minimum temperatures in the 36 African countries under investigation.
We further the analysis by using fractional cointegration, and following now, the twostep approach developed in Engle and Granger (1987). Thus, we first conduct the regression of one of the variables (maximum) against the other (minimum). However, instead of using OLS regressions either in the time or in the frequency domain, we follow Nielsen (2015) and perform the narrow band frequency domain least square (NBFDLS) estimation approach, using as bandwidth numbers m = T
^{0.5} and T
^{0.6}. The estimated regression coefficients are displayed in
Table 6. In the second step, we estimated the order of integration in the residuals of the estimated relationships by using Whittle estimates in the frequency domain. Results are reported across
Table 7.
in the cointegrating residuals are 19 less 11 countries.
The results are similar in the two cases in relation with the bandwidth numbers. Evidence of I(0) residuals, supporting thus short memory in the long run equilibrium relationship is found in the cases of Angola, Botswana, Cameroon, Congo, Gabon, Kenya, Madagascar, Mauritania, Namibia and Zimbabwe. If m = T
^{0.5}, this hypothesis is also supported by Malawi and Morocco. Thus, for these groups, our results support the hypothesis of cointegration with a rapid reversion of the series to a long run equilibrium relationship. On the other extreme, evidence of nonstationary I(1) is found for Liberia and Sierra Leone for the two bandwidth numbers, and also for Ghana if m = T
^{0.5}. In all the other countries, the estimate of the differencing parameter is in the interval (0, 1), and evidence of cointegration (in the sense that the order of integration of the residuals is lower than the minimum of the two individual series) is found in the cases of Angola, Botswana, Cameroon, Central African Republic, Congo, Gabon, Kenya, Madagascar, Malawi, Morocco, Namibia, Nigeria, Tanzania, Uganda and Zimbabwe for the two bandwidth numbers, but also for Benin, Burkina Faso, Mauritania and Tunisia if m = T
^{0.6}. Thus, the countries with no evidence of cointegration are Chad, Cote d’Ivoire, Egypt, Ghana, Guinea, Guinea Bissau, Lesotho, Liberia, Libya, Mali, Niger, Rwanda, Sierra Leone, Senegal, South Africa, Sudan and Zambia. Among the 19 countries listed in
Table 7 with evidence of cointegration, 11 group of countries display short memory equilibrium relationships between their maximum and minimum temperatures, i.e.
$I\left(0\right)$ evidences. These countries include Angola, Botswana, Cameroon, Congo, Gabon, Kenya, Madagascar, Mauritania, Morocco, Namibia, and Zimbabwe, while the remaining eight countries indicate long memory cointegration, i.e.
$I(d>0)$.
5. Conclusion
This paper investigates persistence and linear trends in the maximum and minimum annual average temperatures in 36 African countries. The statistical distribution of maximum and minimum temperature series, with the diurnal range in the climatological study as they act as an important factors driving global warming, have gingered our interest. Historical datasets from 1901 to 2021 are analysed with the annual range series. The results first establish long term memory in maximum and minimum temperatures over the historic years implying that temperature series is often strongly correlated with its past lag values. This is a persistence property, often measured as fractional integration. It assists in uncover the correlation property of this series, classified as long memory, meanreverting and nonmeanreverting, and these have implications for erratic and predicted climatic series. Thus, long term memory observed in this paper further indicates the possibility of temperatures in Africa being predicted for future values. The trend analysis shows evidence of trend shifts in the plots and these are revealed further in the positive significant trend coefficients with insignificant positive trend in range series. Note that trend analysis allows one in making future predictions, but the approach used in this paper checking future predictions based on fractional integration is relatively novel and hardly adopted in climatological studies. Few papers along this line are those of GilAlana, Yaya and Fagbamigbe, (2019). and GilAlana (2003, 2008, 2012), among others.
Our results support the hypothesis of fractional cointegration. This implies that both maximum and minimum temperatures follow a long run equilibrium relationship with shocks in the range displaying transitory effects and not producing permanent discrepancies between the maximum and minimum temperatures. Also, 17 countries among the 36 African countries are at the very high risk of climate change due to the absence of longterm comovement in the maximum and minimum temperatures in those countries, compared to other countries with mild effect. Though, most African countries are prone to the effect of climatic change as the econometric findings reveal, but some regions of the continents are fairer than others.
Findings in this paper are of relevance for climatological studies in a number of ways. It allows for proper understanding of historical temperature patterns in Africa which is crucial for predicting and mitigating the impacts of climate change. Thus, it helps in studying the largescale climate patterns that influence African’s climate in its different regions (see Guo and Wang, 2015; World Bank, 2021). Understanding these patterns and their potential impacts on different parts of Africa is essential for developing effective adaptation and mitigation strategies. Additionally, research on largescale climate patterns in Africa can contribute to global climate models, improving the accuracy of climate projections and enhancing our understanding of the global climate system.
As a final issue, we should note that the last century was a particularly convulse era of humankind; due to this, issues such as heterogeneity (changes in the variance) and stability (structural breaks) in the data should also be taken into account, noting that some authors have found that long memory and nonlinearities and breaks are issues which are intimately related (Diebold and Inoue, 2001; Granger and Hyung, 2004; etc.). Thus, nonlinear deterministic trend structures, based, for instance, on Chebyshev polynomial in time (Cuestas and GilAlana, 2016), Fourier functions (GilAlana and Yaya, 2021) or neural networks (Yaya et al., 2021) could be examined in these data still under the assumption of long range dependence. Work in this line is now in progress.
Declarations
Not applicable
Ethics approval/declarations (include appropriate approvals or waivers)
Not applicable.
Consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare no competing interests.
Author Contributions
OSY: conceptualization, methodology, formal analysis, validation, writing original draft, writing review and editing, visualization, and supervision. OAA: conceptualization, methodology, formal analysis, validation, writing original draft, writing review and editing, visualization, and supervision. HAO & OEO: writingreview and editing, visualization, and validation. LAG: methodology, formal analysis, writing original draft, writingreview and editing, and visualization.
Funding
No funding was received
Data availability statement
The data and the codes that support the findings of this study is available from the corresponding author upon request.
Acknowledgments
Comments from the Editor and four anonymous reviewers are gratefully acknowledged.
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