Time–frequency varying estimations: comparison of discrete and continuous wavelets in the market line framework

  • Roman MestreEmail author
  • Michel Terraza
Original Article


This paper focus on comparison between three wavelets methodologies to estimate a time–frequency varying parameter. In the discrete case, we oppose the intuitive application of the rolling regression on wavelets frequency bands to the time–frequency rolling window. We compare if we have to use the time rolling window directly on the wavelet’s frequency bands or apply the time–frequency rolling window on the series realizing the wavelet decomposition at each step of the process. A time–frequency varying estimator by continuous wavelets is also considerate in the comparison. Our objective is to show that the time–frequency rolling window and the Continuous estimates are more suitable than the intuitive way. We use in first time simulated data and also the daily returns of AXA and the CAC 40 index from 2005 to 2015 as empirical application. We show that the differences between discrete methods are more important at low-frequencies. Moreover, the continuous time–frequency Betas are closer to the time–frequency windows estimates.


Time–frequency rolling regression Wavelets Time–frequency Betas CWT MODWT 

JEL Classification

G00 G11 G12 C29 C18 C49 


  1. 1.
    Bekiros S, Marcellino M (2013) The multiscale causal dynamics of foreign exchange markets. J Int Money Finance 33:282–305Google Scholar
  2. 2.
    Bekiros S, Nguyen DK, Uddin GS (2016) On time scale behavior of equity commodity links: implications for portfolio management. J Int Financ Market Inst Money 41:79–121Google Scholar
  3. 3.
    Black F, Jensen M, Scholes M (1972) The capital asset pricing model: some empirical test; studies in the theory of capital markets. In: Jensen M (ed) Praeger Publishers, New York, pp 79–121Google Scholar
  4. 4.
    Brooks RD, Faff RW, Lee JHH (1992) The form of time variation of systematic risk: some Australian evidence. Appl Financ Econ 2:191–198Google Scholar
  5. 5.
    Brooks RD, Faff RW, McKenzie MD (1998) Time-varying Beta risk of Australian industry portfolios: a comparison of modelling techniques. Aust J Manag 23(1):1–22Google Scholar
  6. 6.
    Fabozzi F, Francis J (1978) Beta as random coefficient. J Financ Quant Anal 13(1):101–116Google Scholar
  7. 7.
    Faff RW, Lee JHH, Fry TRL (1992) Stationarity of systematic risk: some Australian evidence. J Finance Account 19(2):253–270Google Scholar
  8. 8.
    Faff RW, Brooks RD (1998) Time-varying risk for Australian industry portfolios, an exploratory analysis. J Bus Finance Account 25(5):721–774Google Scholar
  9. 9.
    Fama E, MacBeth J (1973) Risk, return, and equilibrium: empirical tests. J Political Econ 81:607–636Google Scholar
  10. 10.
    Gabor JG (1946) Theory of communication. J Chin Inst Electr Eng 93(3):429–457Google Scholar
  11. 11.
    Gençay R, Selçuk S, Whitcher B (2003) Systematic risk and timescales. Quant Finance 3(2):108–116MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gençay R, Selçuk S, Whitcher B (2005) Multiscale systematic risk. J Int Money Finance 24:55–70Google Scholar
  13. 13.
    Mallat S (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Mach Intell 11(7):674–693zbMATHGoogle Scholar
  14. 14.
    Mallat S (2009) Une exploration des signaux en ondelettes, Ecole polytechniqueGoogle Scholar
  15. 15.
    Mallat S (2009) Wavelet tour of signal processing: the sparse way. Academic Press, New YorkzbMATHGoogle Scholar
  16. 16.
    McNevin B, Nix J (2018) The Beta heuristic from a time/frequency perspective: a wavelets analysis of the market risk of sectors. Econ Modell 68:570–585Google Scholar
  17. 17.
    Mestre R, Terraza M (2018) Time-frequency analysis of CAPM-application to the CAC 40. Manag Glob Trans 16(2):141–157Google Scholar
  18. 18.
    Mestre R, Terraza M (2018) Time-frequency varying beta estimation-a continuous wavelets approach-. Econ Bull 38(4):1796–1810Google Scholar
  19. 19.
    Rua A, Nunes L (2009) International comovement of stock market returns: a wavelets analysis’. J Emp Finance 16(4):632–639Google Scholar
  20. 20.
    Rua A, Nunes L (2012) A wavelet based assessment of market risk: the emerging market case. Q Rev Econ Finance 52:84–92Google Scholar
  21. 21.
    Sharpe W (1964) Capital asset prices: a theory of market equilibrium under risk. J Finance 19(3):425–442Google Scholar
  22. 22.
    Vacha L, Barunik J (2012) Co-movement of energy commodities revisited: evidence from wavelets coherence analysis. Energy Econ 34:241–247Google Scholar
  23. 23.
    Torrence C, Compo GP (1989) A practical guide to wavelet analysis’. Bull Am Meteorol Soc 79(1):61–78Google Scholar
  24. 24.
    Torrence C, Webster PJ (1999) Interdecadal in the ENSO-monsoon system. J Clim 12:2679–2690Google Scholar

Copyright information

© Institute for Development and Research in Banking Technology 2019

Authors and Affiliations

  1. 1.MRE Université de Montpellier, UFR d’économie Avenue Raymond DugrandMontpellier cedex 2France

Personalised recommendations