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Time series analysis of quarterly rainfall and temperature (1900–2012) in sub-Saharan African countries

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Abstract

In this paper, we examine the statistical properties of rainfall data and temperature in six sub-Saharan African countries in the western, eastern, and southern regions (Botswana, Ethiopia, Ghana, Nigeria, Uganda, and South Africa) using time series data spanning between 1900 and 2012. By using linear trends, seasonality, and long-range dependence models, in fractional or I(d) frameworks, the results first indicate that time trends are required in most cases to explain the time series properties of the climatic series. Evidence of anti-persistence (d < 0) or I(0) behavior is found for the rainfall data, while long memory (d > 0) is found for the temperature data. Evidence of structural breaks are only found in the cases of Ethiopia, Ghana, and Uganda for the temperature data. With both series displaying significant evidence of seasonality and by working with the seasonally differenced data, the results show evidence of I(0) behavior or anti-persistence (d < 0) for the rainfall data but long memory (d > 0) for the temperature data. Testing the causality between the two variables, the results indicate evidence of causality in the two directions in all cases except for the case of the temperature on the rainfall in South Africa. The implication of the results obtained here is that erratic or constant rainfall is expected in Africa in the future while temperature is likely to continue to increase, and these subsequently lead to future warming experiences.

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Fig. 1
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Notes

  1. For further information on the specification and estimation of Eq. (1), see Hamilton (1994) and Gil-Alana (2012).

  2. This approach is applied in Bloomfield and Nychka (1992), where the authors observed a 1% significant level in trend parameter of Hansen and Lebedeff’s (1987, 1988) temperature.

  3. Fractional integration was introduced by Granger and Joyeux (1980), Granger (1980, 1981), and Hosking (1981) as a follow-up work on the original seminal work of Adenstedt (1974) and Robinson (1978) in the case of heterogeneous aggregated data.

  4. Non-parametric methods reduce to the R/S statistic (Hurst 1951) and its variants (Lo 1991; etc.).

  5. Applications of this method can be found in Gil-Alana and Robinson (1997), Gil-Alana (2008a), Gil-Alana and Singh (2015), Cunado et al. (2016), etc.

  6. Updated versions of this method can be found in Velasco (1999), Phillips and Shimotsu (2004, 2005), and Abadir et al. (2007) among others.

  7. An extension of this to monthly frequency data is found in Beaulieu and Miron (1993).

  8. These factorizations were obtained by Hylleberg et al. (1990) as factors of the polynomial in Eq. (10).

  9. Applications using this approach can be found in Gil-Alana and Robinson (2001), Caporale et al. (2012), etc.

  10. At http://sdwebx.worldbank.org/climateportal/index.cfm?page=downscaled_data_download&menu=historical.

  11. The seasonal coefficients for the temperatures range between 0.812 (UGD) and 0.986 (BOT and ZAF) (see Table 2).

  12. Using the Dickey et al. (1986) tests, the results were supportive of the seasonal unit roots in practically all cases.

  13. The model of Bloomfield is a non-parametric approach of modeling I(0) processes that produces autocorrelations decaying exponentially as in the AR(MA) case.

  14. Note that with autocorrelated errors, the estimates of d for the temperature are negative but the intervals include the I(0) hypothesis in the majority of the cases.

References

  • Abadir KM, Distaso W, Giraitis L (2007) Nonstationarity-extended local Whittle estimation. J Econ 141:1353–1384

    Article  Google Scholar 

  • Adenstedt RK (1974) On large sample estimation for the mean of a stationary random sequence. Ann Stat 2:259–272

    Article  Google Scholar 

  • Bai J, Perron P (2003) Computation and analysis of multiple structural change models. J Appl Econ 18:1–22

    Article  Google Scholar 

  • Beaulieu JJ, Miron JA (1993) Seasonal unit roots in aggregate US data. J Econ 55:305–328

    Article  Google Scholar 

  • Bloomfield P (1973) An exponential model in the spectrum of a scalar time series. Biometrika 60:217–226

    Article  Google Scholar 

  • Bloomfield P, Nychka D (1992) Climate spectra and detecting climate change. Clim Chang 21:275–287

    Article  Google Scholar 

  • Boko M, Niang I, Nyong A, Vogel C, Githeko A, Medany M, Osman-Elasha B, Tabo R, Yanda P (2007) Africa. In: Parry ML, Canziani OF, Palutikof JP, van der Linden PJ, Hanson CE (eds) Climate change 2007: impacts, adaptation and vulnerability. Contribution of Working Group II to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, pp 433–467 https://www.ipcc.ch/pdf/assessment-report/ar4/wg2/ar4-wg2-chapter9.pdf. Accessed 20 June2016

    Google Scholar 

  • Bounoua L, Collatz GJ, Los SO, Sellers PJ, Dazlich DA, Tucker CJ, Randall DA (2000) Sensitivity of climate to changes in NDVI. J Clim 13:2277–2292

    Article  Google Scholar 

  • Box GEP, Jenkins GM, Reinsel GC (2008) Time series analysis: forecasting and control, 4th edn. Wiley, Hoboken

    Book  Google Scholar 

  • Caporale GM, Cunado J, Gil-Alana LA (2012) Deterministic versus stochastic seasonal fractional integration and structural breaks. Stat Comput 22(2):349–358

    Article  Google Scholar 

  • Chappell A, Agnew CT (2004) Modelling climate change in West African Sahel rainfall (1931-90) as an artifact of changing station locations. Int J Climatol 24:547–554

    Article  Google Scholar 

  • Christensen JH, Hewitson B, Busuioc A, Chen A, Gao X, Held I, Jones R, Koli RK, Kwon W-T, Laprise R, Rueda VM, Mearns L, Menéndez CG, Räisänen J, Rinke A, Sarr A, Whetton P (2007) Regional climate projections. In: Qin D, Manning M, Chen Z, Marquis M, Averyt KB, Tignor M, Miller HL (eds) Climate change 2007: the physical science basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon. Cambridge University Press, Cambridge, pp 847–940

    Google Scholar 

  • Climate and Development Knowledge Network (CDKN) (2014) The IPCC’s Fifth Assessment Report What’s in it for Africa? http://cdkn.org/wp-content/uploads/2014/04/AR5_IPCC_Whats_in_it_for_Africa.pdf. Accessed 20 June 2016

  • Conway D, Mould C, Bewket W (2004) Over one century of rainfall and temperature observations in Addis Ababa, Ethiopia. Int J Climatol 24:77–91

    Article  Google Scholar 

  • Cunado J, Gil-Alana LA, Gupta R (2016) Persistence, mean reversion and nonlinearities in CO2 emissions. Evidence from the BRICS and G7 countries. Environ Resour Econ 67(4):869–883

  • Dahlhaus R (1989) Efficient parameter estimation for self-similar process. Ann Stat 17(4):1749–1766

    Article  Google Scholar 

  • Dai A, Lamb PJ, Trenberth KE, Hulme M, Jones PD, Xie P (2004) The recent Sahel drought is real. Int J Climatol 24:1323–1331

    Article  Google Scholar 

  • Dickey DA, Hasza DP, Fuller WA (1986) Testing for unit roots in seasonal time series. J Am Stat Assoc 79:355–367

    Article  Google Scholar 

  • Diebold FX, Inoue A (2001) Long memory and regime switching. J Econ 105:131–159

    Article  Google Scholar 

  • Epstein PR, Mills E (eds) (2005) Climate change futures: health, ecological and economic dimensions. Center for Health and the Global Environment, Harvard Medical School, Boston 142 pp

    Google Scholar 

  • Franses PH, Hobijn B (1997) Numbers from all the tables in critical values for unit root tests in seasonal time series. J Appl Stat 24:25–46

    Article  Google Scholar 

  • Gil-Alana LA (2003) An application of fractional integration to a long temperature time series. Int J Climatol 23:1699–1710

    Article  Google Scholar 

  • Gil-Alana LA (2005) Statistical model for the temperatures in the Northern hemisphere using fractional integration techniques. J Clim 18(24):5537–5369

    Article  Google Scholar 

  • Gil-Alana LA (2006) Testing seasonality in the context of fractionally integrated processes. Annales D’économie Et De Statistique 81:61–79

  • Gil-Alana LA (2008a) Time trends with breaks and fractional integration in temperature time series. Clim Chang 9:325–337

    Article  Google Scholar 

  • Gil-Alana LA (2008b) Fractional integration and structural breaks at unknown periods of time. J Time Ser Anal 29:163–185

    Article  Google Scholar 

  • Gil-Alana LA (2012) Long memory, seasonality and time trends in the average monthly temperatures in Alaska. Theor Appl Climatol 108:385–396

    Article  Google Scholar 

  • Gil-Alana LA, Robinson PM (1997) Testing of unit roots and other nonstationary hypotheses in macroeconomic time series. J Econ 80:241–268

    Article  Google Scholar 

  • Gil-Alana LA, Robinson PM (2001) Testing of seasonal fractional integration in UK and Japanese consumption. J Appl Econ 16:95–114

    Article  Google Scholar 

  • Gil-Alana LA, Singh P (2015) The impact of ethnic violence in Kenya on wheat and maize markets. J Afr Econ 24(4):502–529

    Article  Google Scholar 

  • Granger CWJ (1980) Long memory relationships and the aggregation of dynamic models. J Econ 14(2):227–238

    Article  Google Scholar 

  • Granger CWJ (1981) Some properties of time series data and their use in econometric model specification. J Econ 16:121–131

    Article  Google Scholar 

  • Granger CWJ, Hyung N (2004) Occasional structural breaks and long memory with an application to the S&P 500 absolute stock return. J Empir Financ 11:399–421

    Article  Google Scholar 

  • Granger CWJ, Joyeux R (1980) An introduction to long-memory time series models and fractional differencing. J Time Ser Anal 1:15–29

    Article  Google Scholar 

  • Grenander U, Rosenblatt M (1957) Statistical analysis of stationary time series. Chelsea Publishing Company, New York

    Book  Google Scholar 

  • Hamilton JD (1994) Time series analysis. Princeton University Press, Princeton 820 pages

    Google Scholar 

  • Hansen J, Lebedeff S (1987) Global trends of measured surface air temperature. J Geophys Res 92:345–372

    Article  Google Scholar 

  • Hansen J, Lebedeff S (1988) Global surface air temperatures. Update through 1987. Geophys Res Lett 15:323–326

    Article  Google Scholar 

  • Hosking JRM (1981) Fractional differencing. Biometrika 68:165–176

    Article  Google Scholar 

  • Hudson DA, Jones RG (2002) Regional climate model simulations of present day and future climates of Southern Africa. Technical Note 39. Hadley Centre, Bracknell 42 pp

    Google Scholar 

  • Hulme M, Doherty R, Ngara T, New M (2005) Global warming and African climate change. In: Low PS (ed) Climate Change and Africa. Cambridge University Press, Cambridge, pp 29–40

    Chapter  Google Scholar 

  • Hurst H (1951) The long-term storage capacity of reservoirs. Trans Am Soc Civ Eng 116:770–799

    Google Scholar 

  • Hylleberg S, Engle RF, Granger CWJ, Yoo BS (1990) Seasonal integration and cointegration. J Econ 44:215–238

    Article  Google Scholar 

  • IPCC (2001) Intergovernmental Panel on Climate change. Third assessment report: climate change 2001. WG1: The scientific basis, summary for policymakers, Q2:4-5, pp 4-6 Geneva, Switzerland

  • IPCC (2007a) Intergovernmental Panel on Climate change. Fourth Assessment Report: Climate change 2007. Summary for policy makers: understanding and attributing climate change. Cambridge University Press, Cambridge, pp 10–13

    Google Scholar 

  • IPCC (2007b) Intergovernmental Panel on Climate change. Fourth assessment report: climate change 2007: chapter 11—regional climate projections. Cambridge University Press, Cambridge, pp 899–904

    Google Scholar 

  • IPCC (2014). Fifth assessment report: climate change 2014: impacts, adaptation, and vulnerability. Chapter 22, Africa, pp 1201–1236, Geneva, Switzerland

  • Jenkins GS, Adamou G, Fongang S (2002) The challenges of modelling climate variability and change in West Africa. Clim Chang 52:263–286

    Article  Google Scholar 

  • Johansen S (2008) A representation theory for a class of vector autoregressive models for fractional models. Econom Theory 24:651–676

    Article  Google Scholar 

  • Kruger AC, Shongwe S (2004) Temperature trends in South Africa: 1960–2003. Int J Climatol 24:1929–1945

    Article  Google Scholar 

  • Lo AW (1991) Long-term memory in stock market prices. Econometrica 59(5):1279–1313

    Article  Google Scholar 

  • MacKinnon JG, Haug AA, Michelis L (1999) Numerical distribution functions of likelihood ratio tests for cointegration. J Appl Econ 14:563–577

    Article  Google Scholar 

  • Malhi Y, Wright J (2004) Spatial patterns and recent trends in the climate of tropical rainforest regions. Philos Trans R Soc Lond B Biol Sci 359:311–329

    Article  Google Scholar 

  • McMichael AJ, Woodruff RE, Hales S (2006) Climate change and human health: present and future risks. Lancet 367:859–869

    Article  Google Scholar 

  • Montanari A, Rosso R, Taqqu MS (1996) Some long-run properties of rainfall records in Italy. J Geophys Res 101:431–438

    Article  Google Scholar 

  • Moreno M (2000) Riding the temp. Weather Derivatives, FOW Special Supplement

  • New M, Hewitson B, Stephenson DB, Tsiga A, Kruger A, Manhique A, Gomez B, Coelho CAS et al (2006) Evidence of trends in daily climate extremes over southern and west Africa. J Geophys Res–Atmos 111:D14102. https://doi.org/10.1029/2005JD006289

    Article  Google Scholar 

  • Niang I, Ruppel OC, Abdrabo MA, Essel A, Lennard C, Padgham J, Urquhart P (2014) Africa. In: Barros VR, Field CB, Dokken DJ, Mastrandrea MD, Mach KJ, Bilir TE, Chatterjee M, Ebi KL, Estrada YO, Genova RC, Girma B, Kissel ES, Levy AN, MacCracken S, Mastrandrea PR, White LL (eds) Climate change 2014: impacts, adaptation, and vulnerability. Part B: regional aspects. Contribution of Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, pp 1199–1265 http://ipcc-wg2.gov/AR5/images/uploads/WGIIAR5-Chap22_FINAL.pdf. Accessd 20 June 2016

    Google Scholar 

  • Nicholls N, Gruza GV, Jouzel J, Karl TR, Ogallo LA, Parker DE (1996) Observed climate variability and change. In: Houghton JT, Meiro Filho LG, Callendar BA, Kattenburg A, Maskell K (eds) Climate change 1995: the science of climate change. Cambridge University Press, Cambridge, pp 133–192

    Google Scholar 

  • Nicholson SE, Selato JC (2000) The influence of La Nina on African rainfall. Int J Climatol 20:1761–1776

    Article  Google Scholar 

  • Ohanissian A, Russell JR, Tsay RS (2008) True or spurious long memory? A new test. J Bus Econ Stat 26(2):161–175

    Article  Google Scholar 

  • Park RE, Mitchell BM (1980) Estimating the autocorrelated error model with trended data. J Econ 13:185–201

    Article  Google Scholar 

  • Pascual M, Ahumada JA, Chaves LF, Rodó X, Bouma M (2006) Malaria resurgence in the East African highlands: temperature trends revisited. Proc Natl Acad Sci U S A 103:5829–5834

    Article  Google Scholar 

  • Patz JA, Olson SH (2006) Climate change and health: global to local influences on disease risk. Ann Trop Med Parasitol 100:535–549

    Article  Google Scholar 

  • Percival D, Overland J, Mofjeld H (2001) Interpretation of North Pacific variability as a short and long memory process. Tech Rep Series NRCSE 065:35

    Google Scholar 

  • Phillips PCB, Shimotsu K (2004) Local Whittle estimation in nonstationary and unit root cases. Ann Stat 32:656–692

    Article  Google Scholar 

  • Phillips PCB, Shimotsu K (2005) Exact local Whittle estimation of fractional integration. Ann Stat 33:1890–1933

    Article  Google Scholar 

  • Prais SJ, Winsten CB (1954) Trend estimators and serial correlation, Cowles Commission Monograph, No. 23. Yale University Press, New Haven

    Google Scholar 

  • Robinson PM (1978) Statistical inference for a random coefficient autoregressive model. Scand J Stat 5:163–168

    Google Scholar 

  • Robinson PM (1994) Efficient tests of nonstationary hypotheses. J Am Stat Assoc 89:1420–1437

    Article  Google Scholar 

  • Robinson PM (1995) Gaussian semiparametric estimation of long range dependence. Ann Stat 23:1630–1661

    Article  Google Scholar 

  • Stephenson DB, Pavan V, Bojariu R (2000) Is the North Atlantic oscillation a random walk? Int J Climatol 20:1–18

    Article  Google Scholar 

  • Swart R, Mitchell J, Morita T, Raper S (2002) Stabilisation scenarios for climate impacts assessment. Glob Environ Chang 12:155–165

    Article  Google Scholar 

  • van Lieshout M, Kovats RS, Livermore MTJ, Martens P (2004) Climate change and malaria: analysis of the SRES climate and socio-economic scenarios. Glob Environ Chang 14:87–99

    Article  Google Scholar 

  • Velasco C (1999) Gaussian semiparametric estimation of nonstationary time series. J Time Ser Anal 20:87–127

    Article  Google Scholar 

  • Wakaura M, Ogata Y (2007) A time series analysis on the seasonality of air temperature anomalies. Meteorol Appl 14:425–434

    Article  Google Scholar 

  • Woodward WA, Gray HL (1993) Global warming and the problem of testing for trend in time series data. J Clim 6:953–962

    Article  Google Scholar 

  • Yaya OS, Fashae OA (2015) Seasonal fractional integrated time series models for rainfall data in Nigeria. Theor Appl Climatol 120:99–108

    Article  Google Scholar 

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The authors gratefully acknowledge the comments of the Editor and two anonymous reviewers.

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Correspondence to OlaOluwa S. Yaya.

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Gil-Alana, L.A., Yaya, O.S. & Fagbamigbe, A.F. Time series analysis of quarterly rainfall and temperature (1900–2012) in sub-Saharan African countries. Theor Appl Climatol 137, 61–76 (2019). https://doi.org/10.1007/s00704-018-2583-5

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