Skip to main content

Advertisement

SpringerLink
Log in

A statistical analysis of sea temperature data

A statistical analysis of sea temperature data

  • Original Paper
  • Published:
Theoretical and Applied Climatology Aims and scope Submit manuscript

Abstract

The paper analyzes sea temperature series measured at two geographical locations along the coast of Norway. We address the question whether the series are stable over the sample period 1936–2012 and whether we can measure any signal of climate change in the regional data. We use nonstandard supF, OLS-based CUSUM, RE, and Chow tests in combination with the Bai-Perron’s structural break test to identify potential changes in the temperature. The augmented Dickey-Fuller, the KPSS, and the nonparametric Phillips-Perron tests are in addition applied in the evaluation of the stochastic properties of the series. The analysis indicates that both series undergo similar structural instabilities in the form of small shifts in the temperature level. The temperature at Lista (58° 06′ N, 06° 38′ E) shifts downward about 1962 while the Skrova series (68° 12′ N, 14° 10′ E) shifts to a lower level about 1977. Both series shift upward about 1987, and after a period of increasing temperature, both series start leveling off about the turn of the millennium. The series have no significant stochastic or deterministic trend. The analysis indicates that the mean temperature has moved upward in decadal, small steps since the 1980s. The result is in accordance with recent analyses of sea temperatures in the North Atlantic. The findings are also related to the so-called hiatus phenomenon where natural variation in climate can mask global warming processes. The paper contributes to the discussion of applying objective methods in measuring climate change.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  • Ådlandsvik B (2008) Marine downscaling of the future climate scenario for the North Sea. Tellus 60A:451–458

    Article  Google Scholar 

  • Årthun M, Eldevik T, Smedrun LH, Skagseth Ø, Ingvaldsen RB (2012) Quantifying the influence of atlantic heat on Barents Sea ice variability and retreat. J Clim 25:4736–4742

    Article  Google Scholar 

  • Ådlandsvik OE, Bentsen M (2007) Downscaling a 20th century global climate simulation to the North Sea. Ocean Dyn 57:453–466

    Article  Google Scholar 

  • ACIA (2005) Arctic climate impact assessment. Cambridge University Press. Overview Report, Cambridge University Press. http://www.acia.uaf.edu/pages/overview.html

  • Albretsen J, Aure J, Sætre R, Danielssen D (2011) Climate variability in the Skagerrak and coastal waters of Norway. ICES J Mar Sci. doi:10.1093/icesjms/fsr187

  • Anderson TW (1971) The statistical analysis of time series. Wiley, New York

    Google Scholar 

  • Andrews DWK (1993) Tests for parameter instability and statistical change with unknown change point. Econometrica 61:821–856

    Article  Google Scholar 

  • Andrews DWK (2003) Test for parameter instability and structural change with unknown change point: a corrigendum. Econometrica 71:395–397

    Article  Google Scholar 

  • Andrews DWK, Ploberger W (1994) Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62:1383–1414

    Article  Google Scholar 

  • Bai J (1997) Estimating multiple breaks one at a time. Econ Theory 13(3):315–352

    Article  Google Scholar 

  • Bai J, Perron P (1998) Estimating and testing linear models with multiple structural changes. Econometrica 66:47–78

    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 

  • Balmaseda M, Trenberth KE, Källén E (2013) Distinctive climate signals in reanalysis of global ocean heat content. Am Geophys Union. doi:10.1002/grl.50382

  • Bartlett MS (1966) An introduction to stochastic processes with special reference to methods and applications, 2nd edn. Cambridge University Press, Cambridge

    Google Scholar 

  • Bellman RE, Roth R (1969) Curve fitting by segmented stright lines. J Am Stat Assoc 64:1079–1084

    Article  Google Scholar 

  • Belkin IM (2009) Rapid warming of large marine ecosystems. Progr Oceanogr 81:207–213

    Article  Google Scholar 

  • Bengtsson L, Semenov VA, Johannessen OM (2004) The early twentiet century warming in the Arctic a possible mechanism. J Clim 17:4045–4057

    Article  Google Scholar 

  • Bentsen M, Drange H, Furevik T, Zhou T (2004) Simulated variability of Atlantic meridional overturning circulation

  • Box GEP, Jenkins GM (1970) Time series analysis: forecasting and control. Holden-Day, San Francisco

    Google Scholar 

  • Breusch TS (1978) Testing for autocorrelation in dynamic linear models. Aust Econ Pap 17:334–355

    Article  Google Scholar 

  • Breusch TS, Pagan AR (1979) A simple test for heteroscedasticity and random coefficient variation. Econometrica 47:1287–1294

    Article  Google Scholar 

  • California Department of Fish and Wildlife (2013) El Niño information. http://www.dfg.ca.gov/marine/elnino.asp

  • Chiu ST (1989) Detecting periodic components in a white gaussian time series. J R Stat Soc Ser B Methodol 51(2):249–259. http://www.jstor.org/page/info/about/policies/terms.jsp

    Google Scholar 

  • Chow GC (1960) Tests of equality between sets of coefficients in two linear regressions. Econometrica 28(3):591–605

    Article  Google Scholar 

  • CIMAS - Cooperative Institute for Marine and Atmospheric Studies (2010) http://oceancurrents.rsmas.miami.edu/index.html

  • Danielssen DS, Edler L, Fonselius S, Henroth L, Ostrowski M, Svendsen E, Talpsepp L (1997) Oceanographic variability in the Skagerrak and Northern Kattegat, May–June, 1990. ICES J Mar Sci 54:753–773

    Article  Google Scholar 

  • Dickey D, Fuller WA (1979) Distribution of the estimators for autoregressive time series with a unit root. J Am Stat Assoc 74:427–431

    Google Scholar 

  • Dickey D, Fuller DW (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49:1057–72

    Article  Google Scholar 

  • Dickson B, Østerhus S (2009) One hundred years in the Nordic Seas. Nor J Geogr 61:56–75. Oslo, ISSN, 0029-1954

    Article  Google Scholar 

  • Drange H, Dokken T, Furevik T, Gerdes R, Berger W, editors (2005) The Nordic Seas—an integrated perspective. Gepphys Monogr 158. Amaerican Geophysical Union Washington, DC. ISBN 978-0-87590-423-8

  • Easterling DR, Wehner MF (2009) Is the climate warming or cooling? Geophys Res Lett 36:L08706. doi:10.1029/2009GL037810

    Google Scholar 

  • Efron B (1979) The 1977 Rietz lecture. Nootstrap methods: another look at the Jackknife. Ann Stat 7(1):1–26

    Article  Google Scholar 

  • Eldevik T, Nilsen JEO, Iovino D, Andersen Olsson K, Sandø AB, Drange H (2009) Observed sources and variability of Nordic seas overflow. Nat Geosci 2:406–410

    Article  Google Scholar 

  • Ellingsen IH, Dalpadado P, Slagstad D, Loeng L (2008) Impact on climate change on the biological production in the Barents Sea. Clim Change 87:155–175

    Article  Google Scholar 

  • Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica 50:987–1008

    Article  Google Scholar 

  • Feng MM, McPhaden MJ, Lee T (2010) Decadal variability of the Pacific subtropical cells and their influence on the southeast Indian Ocean. Geophys Res Lett 37:L09606. doi:10.1029/2010GL042796

    Google Scholar 

  • Foster R, Rahmstorf S (2011) Global temperature evolution 1979–2010, Environmental Research Letters, 6. doi:10.1088/17489326/6/4/044022

  • Fuller WA (1976) Introduction to statistical time series. Wiley, New York

    Google Scholar 

  • Furevik T (2001) Annual and interannual variability of atlantic water temperatures in the Norwegian and Barents Seas: 1980–1996. Deep Sea Res 48(2):383–404

    Article  Google Scholar 

  • Furevik T, Bertsen M, Drange H, Kindem IKT, Kvamstø NG, Sorteberg A (2003) Description and validation of the Bergen Climate Modell (BCM): ARPEGE coupled with MICOM, vol 10, pp 249–266

  • Furevik T, Drange H, Sorteberg A (2002) Anticipated changes in the Nordic Seas marine climate Scenarios for 2020, 2050, and 2080. Fisken og Havet, No. 4, 2002. Havforskningsinstituttet. The Fish and the Sea, no. 4, 2002. Institute of Marine Research Bergen, Norway

  • Godfrey LG (1978) Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables. Econometrica 46:1293–1302

    Article  Google Scholar 

  • Goldfeld S, Quandt R (1972) Nonlinear methods in econometrics. North-Holland

  • Gulev SK, Latif M, Keelyside N, Park W, Koltermann KP (2013) North Atlantic Ocean control on surface heat flux on multidecadal timescale. Nature 499:464–467

    Article  Google Scholar 

  • Guthery SB (1974) Partition regression. J Am Stat Assoc 69:945–947

    Article  Google Scholar 

  • Hansen B, Østerhus S (2000) North Atlantic—Nordic Seas exchanges. J Oceanogr 45:109–208

    Google Scholar 

  • Hansen BE (1992) Testing the parameter instability in linear models. J Policy Model 14:517–533

    Article  Google Scholar 

  • Hansen BE (2000) Testing for structural change in the conditional models. J Econ 97:93–115

    Article  Google Scholar 

  • Hansen JE (2005) A slippery slope: how much global warming constitutes dangerous anthropogenic interference? An editorial essay. Clim Chang 68:269–279. doi:10.1007/s10584-005-4135-0

    Article  Google Scholar 

  • Hansen J (2008) Tipping point: perspective of a climatologist. In: Ward woods. State of the wild 2008–2009: a global portrait of wildlife, Wildlands, and Oceans (State of the Wild). Island Press, Washington, DC, pp 6–15. ISBN 1-59726-135-1. OCLC 429495689

  • Hansen J (2012) Perception of climate change. PNAS early edn. www.pnas.org/cgi/doi/10.1073/pnas.1205276109, Ruedy R

  • Hansen J, Ruedy R, Sato M, Lo K (2010) Global surface temperature change. Rev Geophys 48:RG4004. doi:10.1029/2010RG000345

  • Hátun H, Sandø AB, Drange H, Hansen B, Valdemarsen H (2005) Influence of the Atlantic Subpolar Gyre on the thermohaline circulation. Science 309:1841–1844

    Article  Google Scholar 

  • Haugan PM, Evensen G, Johannessen JA, Johannessen OM, Pettersson LH (1991) Modeled and observed mesoscale circulation and wave current refraction during the 1988 Norwegian continental shelf experiment. J Geophys Res 96:10487–10506

    Article  Google Scholar 

  • Hayashi F (2000) Econometrics. Princeton University Press, Princeton

    Google Scholar 

  • Hobijn B, Franses PH, Ooms M (1998) Generalizations of the KPSS-test for Stationarity. Econometric Institute, Erasmus University Rotterdam, The Netherlands. http://www.eur.nl/few/ei/papers

  • Hoerling MP, Hurrell J, Xu T (2001) Tropical origins for North Atlantic climate change. Science 292:90–92

    Article  Google Scholar 

  • Holliday NP, et al (2008) Reversal of the 1960s to 1990s freshening trend in the northeast North and Nordic Seas. Geophys Res Lett 35:L03614

    Google Scholar 

  • Huges SL, Holliday PN, Gaillard F (2012) Variability in the ICES/NAFO region between 1950 and 2009: observations from the ICES Report on Ocean Climate. ICES J Mar Sci. doi:10.1093/icesjms/fss044

  • Hurrell JW (1995) Decadal trends in the North Atlantic Oscillation: regional temperatures and precipitation. Science 269:676–679

    Article  Google Scholar 

  • Ingvaldsen R, Loeng H, Ottersen H, Ådlandsvik B (2003) Climate variability in the Barents Sea during the 20th century with focus on the 1990s. ICES Mar Sci Symp 219:160–168

    Google Scholar 

  • IPCC (2001) Intergovernmental panel on climate change: Climate Change 2001: synthesis report, intergovernmental panel on climate change. Cambridge University Press

  • IPCC (2003) Intergovernmental panel on climate change: international panel for climate change. Website, www.ipcc.ch

  • IPCC (2007a) Intergovernmental panel on climate change: climate change 2007: physical science basis. Summary for Policymakers. IPCC Secretariat, c/o WMO, Geneva, Switzerland

  • IPCC (2007b) A report of working group I. Summary for policy makers. http://www.ipcc.ch/pdf/assessment-report/ar4/wg1/ar4-wg1-spm.pdf

  • IPCC (2013) Summary for policymakers. In: Stocker TF, Qin D, Plattner G-K, Tignor M, Allen SK, Boschung J, Nauels A, Xia Y, Bex V, Midgley PM (eds) Climate Change 2013: the physical science basis. Contribution of working group I to the fifth assessment report of the intergovernmental panel on climate change. Cambridge University Press, Cambridge, United Kingdom and New York, NY

    Google Scholar 

  • Iversen T et al (2005) RegClim-Norges klima om 100 år. Usikkerheter og risiko. RegClim-Norway’s climate in 100 years. Unscertainty and risk, Oslo, Norway. http://regclim.met.no

  • Jones PD, Lister DH, Osborn TJ, Harpham C, Salmon M, Morice CP (2012) Hemispheric and large-scale land-surface air temperature variations: an extensive revision and an update to 2010. Journal of Geophysical Research 117:2012. doi:10.1029/2011JD017139

    Google Scholar 

  • Katsman CA, von Oldenborgh GJ (2011) Tracing the upper ocean’s “missing heat”. Geophys Res Lett 38:L14610. doi:10.1029/2011GL048417

    Google Scholar 

  • Keeling CD (1960) The concentration and isotopic abundances of carbon dioxide in the atmosphere. Tellus 12:200–2003

    Article  Google Scholar 

  • Keenlyside NS, Latif M, Jungclaus J, Kornblueh L, Roeckner E (2008) Advancing decadal-scale climate prediction in the North Atlantic sector. Nature 453:84–88

    Article  Google Scholar 

  • Kosaka Y, Ping-Xie S (2013) Recent global-warming hiatus tied to equitorial pacific surfcae cooling. Lett, Nat 501:403–407. doi:10.1038/nature12534

    Article  Google Scholar 

  • Krämer W, Ploberger W, Alt R (1988) Testing for structural change in dynamic models. Econometrica 56:1355–1369

    Article  Google Scholar 

  • Kuan CM, Hornik K (1995) The generalized fluctuation test: a unified view. Econ Rev 14:135–161

    Article  Google Scholar 

  • Kushnir Y (1994) Interdecadal variations in North Atlantic sea surface temperature and associated atmospheric conditions. J Clim 7:141–157

    Article  Google Scholar 

  • Kwiatowski D, Phillips PCB, Schmidt P, Shin Y (1992) Testing the null hypothesis of stationarity against the alternative of unit root. J Econ 54:159–178

    Article  Google Scholar 

  • Lenton TM, Held H, Kriegler E, Hall JW, Lucht W, Rahmstorf S, Schellnhuber HJ (2008) Tipping elements in the Earth’s climate system. Proc Natl Acad Sci 105(6):1786–1793. doi:10.1073/pnas.0705414105

    Article  Google Scholar 

  • Levitus S, Antonov JI, Boyer TP, Baranova OK, Garcia HE, Locarnini RA, Mishonov AV, Reagan JR, Seidov D, Yarosh ES, Zweng MM (2012) World ocean heat content and thermosteric sea level change (0-2,000 m). pp 1955–2010

  • Lockwood M, Harrison RG, Woollings T, Solanki SK (2010) Are cold winters in Europe associated with low solar activity? Environ Res Lett. doi:10.1088/1748-9326/5/2/024001

  • Lorentzen T (2013) Statistical analysis of temperature data sampled at station-M in the Norwegian Sea. J Mar Syst. doi:10.1016/j.marsys.2013.09.009

  • Lyman JM, Good CA, Gouretski VV, Ishii M, Johanson GC, Palmer MD, Smith DM, Willis JK (2010) Robust warming of the global upper ocean

  • Matulla C et al (2007) European storminess: late nineteenth century to present. Clim Dyn doi:10.1007/s00382-007-0333-y, published online, 6 November 2007

  • McLeod A, Li W (1983) Diagnostic checking ARMA time series models using squared residual correlations. J Time Ser Anal 4:269–273

    Article  Google Scholar 

  • Meehl GA, Arblaster JM, Fasullo J, Hu A, Trenberth KE (2011) Model-based evidence of deep ocean heat uptake during surface temperature hiatus periods. Nat Clim Chang 1:360–364. doi:10.1038/nclimate1229

    Article  Google Scholar 

  • Meehl GA, Hu A, Arblaster JM, Fasullo J, Trenberth KE (2013) Externally forced and internally generated decadal climate variability associated with the interdecadal pacific oscillation. J Clim 26:7298–7310

    Article  Google Scholar 

  • Melsom A, Lien V, Budgell WP (2009) Using the regional Ocean Modeling System (ROMS) to improve the ocean circulation from a GCM 20th century simulation. Ocean Dyn 59:969–981

    Article  Google Scholar 

  • Newey W, West K (1987) A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix. Econometrica 55:703–708

    Article  Google Scholar 

  • NOU (2010) Noregs offentlege utgreiingar 2010:10. Tilpassing til eit klima i endring. Samfunnet si sårbarhet og behov for tilpassing til konsekvensar av klimaendringane. Oslo, 2010

  • Nyquist H (1924) Certain factors affecting telegraph speed. Bell Syst Tech J 3:324–346

    Article  Google Scholar 

  • Nyquist H (1928) Certain topics in telegraph transmission theory. AIEE Trans 47:617–644

    Google Scholar 

  • Orvik KA, Skagseth O (2005) Heat flux variations in the eastern Norwegian Atlantic Current toward the Artic from moored instruments, 1995–2005. Geophys Res Lett 32:L14610. doi:10.1029/2005GL023487

    Google Scholar 

  • Overland JE, Spillane MC, Perceival DB, Wang M, Mofjeld HO (2004) Seasonal and regional variation of pan-Arctic surface air temperature over the instrumental record. J Clim 17:3263–3282

    Article  Google Scholar 

  • Perron P (1988) Trends and random walks in macroeconomic time series: further evidence from a new approach. J Econ Dyn Control 12:297–332

    Article  Google Scholar 

  • Perron P (1989) The great crash, the oil price shock and the unit root hypothesis. Econometrica 57:1361–1401

    Article  Google Scholar 

  • Perron P (1990) Testing for a unit root in a time series with a changing mean. J Bus Econ Stat 8:153–162

    Google Scholar 

  • Perron P, Vogelsang TJ (1992) Testing for a unit root in the time series with a changing mean: corrections and extensions. J Bus Econ Stud 41:153–171

    Google Scholar 

  • Phillips PCB (1987) Time series regression with a unit root. Econometrica 55:277–301

    Article  Google Scholar 

  • Phillips PCB, Perron P (1988) Testing for a unit root in time series regression. Biometrica 75:335–446

    Article  Google Scholar 

  • Ploberger W, Krämer W (1992) The CUSUM test with OLS residuals. Econometrica 60(2):271–285

    Article  Google Scholar 

  • Ploberger W, Krämer W, Kontrus K (1989) A new test for structural stability in the linear regression model. J Econ 40:307–318

    Article  Google Scholar 

  • Quenouille MH (1952) Associated measurements. Butterworths, London

    Google Scholar 

  • Ramsey JB (1969) Test for specification error in classical linear least squares regression analysis. J R Stat Soc B 31:350–371

    Google Scholar 

  • Rohde R, Muller RA, Perlmutter S, Rosenfeld A, Wurtele J, Curry J, Wickham C, Mosher S (2013) Berkeley earth temperature averaging process. Geoinfor Geostat: An Overview 1:2 doi:10.4172/gigs.1000103

    Google Scholar 

  • Sætre R(ed) (2007) The Norwegian coastal current—oceanography and climate. Institute of Marine Research, Tapir Academic Press, Trondheim, Norway

  • Sandø AB, Nilsen JEØ, Gao DY, Lohmann K (2010) Importance of heat transport and local air-sea heat fluxes for Barents Sea climate variability. J Geophys Res 115:CO1713. doi:10.1029/2009JC005884

    Google Scholar 

  • Scripps Institute of Oceanography (2013) http://scrippsco2.ucsd.edu/home/index.php. See also the reference to Charles David Keeling article from 1960

  • Shannon CE (1949) Communication in the present of noise. Proc Inst Radio Eng 37(1):10–21, January 1949. Reprint as classic paper In: Proc. IEEE, vol 86, No 2, February 1996

  • Skagseth Ø, Furevik T, Ingvaldsen R, Loeng H, Mork KA, Orvik KA, Ozhigin V (2008) Volume and heat transports to the arctic ocean via the Norwegian and Barents Seas. In: Dickson, RR, Meincke, J, Rhines, P (eds) Arctic-subarctic ocean fluxes: 45 defining the role of the northern seas in climate. Springer Science, Business Media B.V. ISBN 978-1-4020-6773-0

  • Solomon S, et al (2010) Contributions of stratospheric water vapor to decadal changes in the rate of global warming. Science 327:1219–1223

    Article  Google Scholar 

  • Solomon S, et al (2011) The persistently variable “background” stratospheric aerosol layer and global climate change. Science 333:866–870

    Article  Google Scholar 

  • Sorteberg A, Andersen MS (2008) Regional precipitation an temperature changes for Norway 2010 and 2025. Report No. 28, Bjerknes Centre for Climate Research, Bergen, Norway

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

    Article  Google Scholar 

  • Stroeve JC, Holland MM, Kay JE, Malanik J, Barrett APB (2012) The Arctic’s rapidly sea ice cover: a research synthesis. Clim Change 110:1005–1027. doi:10.1007/s10584-011-0101-1

    Article  Google Scholar 

  • Subba Rao E, Priestly MB, Lessi O (1997) Application of time series analysis in astronomy and meterology. Chapman and Hall, London

    Google Scholar 

  • Thomson DJ (1982) Spectrum estimation and harmonic analysis. Proc IEEE 70(9):1055–1096

    Article  Google Scholar 

  • White H (1980) A heteroscedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity. Econometrica 48:827–838

    Google Scholar 

  • Wichert S, Fokianos K, Strimmer K (2004) Identifying periodically expressed transcripts in microarray time series data. Bioinformatics 20(1):5–20. doi:10.1093/bioinformatics/btg364

    Article  Google Scholar 

  • UN’s World Meteorological Organization (2010) Homepage, http://www.wmo.int/pages/index_en.html

  • Whistler D, White KJ, Donna Wong S, Bates D (2006) Shazam, version 10. Northwest Econometrics, Ltd, Vancouver

  • Zeileis A (2005) A unified approach to structural change tests based on ML scores, F-statistics and OLS residuals. Econ Rev 24(4):445–466

    Article  Google Scholar 

Download references

Acknowledgments

I thank an anonymous reviewer, Nils Gunnar Kvamstøand, Iselin Medhaug, both Geophysical Institute, University of Bergen, Norway, for valuable comments. I also thank oceanographer Jan Aure at the Institute of Marine Research (IMR), Bergen, Norway for information about the hydrographic stations located at Skrova and Lista.

Author information

Authors and Affiliations

  1. Uni Research Climate, Allegaten 55, NO-5007, Bergen, Norway

    Torbjørn Lorentzen

Authors
  1. Torbjørn Lorentzen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Torbjørn Lorentzen.

Additional information

This work is funded by the Norwegian Research Council and Uni Research Climate.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lorentzen, T. A statistical analysis of sea temperature data. Theor Appl Climatol 119, 585–610 (2015). https://doi.org/10.1007/s00704-014-1119-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00704-014-1119-x

JEL Classifications

Mathematics Subject Classification 2010

Search

Navigation