Competitive and Collaborative Mixtures of Experts for Financial Risk Analysis

  • José Miguel Hernández-Lobato
  • Alberto Suárez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4132)


We compare the performance of competitive and collaborative strategies for mixtures of autoregressive experts with normal innovations for conditional risk analysis in financial time series. The prediction of the mixture of collaborating experts is an average of the outputs of the experts. If a competitive strategy is used the prediction is generated by a single expert. The expert that becomes activated is selected either deterministically (hard competition) or at random, with a certain probability (soft competition). The different strategies are compared in a sliding window experiment for the time series of log-returns of the Spanish stock index IBEX 35, which is preprocessed to account for the heteroskedasticity of the series. Experiments indicate that the best performance for risk analysis is obtained by mixtures with soft competition, where the experts have a probability of activation given by the output of a gating network of softmax units.


Portfolio Return Conditional Likelihood Expect Shortfall Slide Window Analysis Competition Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Jorion, P.: Value at Risk. McGraw-Hill Professional, New York (2000)Google Scholar
  2. 2.
    Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Mathematical Finance 9(3), 203–228 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cont, R.: Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1(2), 223–236 (2001)CrossRefGoogle Scholar
  4. 4.
    Kon, S.J.: Models of stock returns–a comparison. Journal of Finance 39(1), 147–165 (1984)CrossRefGoogle Scholar
  5. 5.
    Mandelbrot, B.: The variation of certain speculative prices. Journal of Business 36(4), 394–419 (1963)CrossRefGoogle Scholar
  6. 6.
    Fama, E.F., French, K.R.: Permanent and temporary components of stock prices. The Journal of Political Economy 96(2), 243–276 (1988)CrossRefGoogle Scholar
  7. 7.
    Akgiray, V.: Conditional heteroscedasticity in time series of stock returns: Evidence and forecasts. The Journal of Business 62(1), 55–80 (1989)CrossRefGoogle Scholar
  8. 8.
    Engle, R.: Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 987–1008 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bollerslev, T.: Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307–327 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Suárez, A.: Mixtures of autoregressive models for financial risk analysis. In: Dorronsoro, J.R. (ed.) ICANN 2002. LNCS, vol. 2415, p. 1186. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Vidal, C., Suárez, A.: Hierarchical mixtures of autoregressive models for time-series modeling. In: Kaynak, O., Alpaydın, E., Oja, E., Xu, L. (eds.) ICANN 2003 and ICONIP 2003. LNCS, vol. 2714, pp. 597–606. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Jacobs, R.A., Jordan, M.I., Nowlan, S., Hinton, G.E.: Adaptive mixtures of local experts. Neural Computation 3, 1–12 (1991)CrossRefGoogle Scholar
  13. 13.
    Jordan, M.I., Jacobs, R.A.: Hierarchical mixtures of experts and the EM algorithm. Neural Computation 6, 181–214 (1994)CrossRefGoogle Scholar
  14. 14.
    Sociedad de Bolsas: Histórico Cierres Índices Ibex (2006),
  15. 15.
    Hamilton, J.D.: Time Series Analysis. Princeton University Press, Princeton (1994)zbMATHGoogle Scholar
  16. 16.
    Bishop, C.: Neural Networks for Pattern Recognition. Oxford University Press, Oxford (1996)Google Scholar
  17. 17.
    Jacobs, R.A., Jordan, M.I., Barto, A.G.: Task decompostiion through competition in a modular connectionist architecture: The what and where vision tasks. Machine Learning: From Theory to Applications, 175–202 (1993)Google Scholar
  18. 18.
    Mathworks: Matlab Optimization toolbox 2.2. Mathworks, Inc., Natick (2002)Google Scholar
  19. 19.
    Hamilton, J.D.: A quasi-bayesian approach to estimating parameters for mixtures of normal distributions. Journal of Business & Economic Statistics 9(1), 27–39 (1991)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Rosenblatt, M.: Remarks on a multivariate transformation. The Annals of Mathematical Statistics 23(3), 470–472 (1952)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kerkhof, J., Melenberg, B.: Backtesting for risk-based regulatory capital. Journal of Banking & Finance 28(8), 1845–1865 (2004)CrossRefGoogle Scholar
  22. 22.
    Kupiec, H.: Techniques for verifying the accuracy of risk management models. Journal of Derivatives 3 (1995)Google Scholar
  23. 23.
    van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (2000)Google Scholar
  24. 24.
    Wilcoxon, F.: Individual comparisons by ranking methods. Biometrics Bulletin 1(6), 80–83 (1945)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • José Miguel Hernández-Lobato
    • 1
  • Alberto Suárez
    • 1
  1. 1.Computer Science Department, Escuela Politécnica SuperiorUniversidad Autónoma de MadridMadridSpain

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