Computational Economics

, Volume 52, Issue 1, pp 145–166 | Cite as

A Hybrid Metaheuristic for the Efficient Solution of GARCH with Trend Models

  • Lourdes Uribe
  • Benjamin Perea
  • Gerardo Hernández-del-Valle
  • Oliver Schütze


GARCH with trend models represent an efficient tool for the analysis of different commodities via testing for a linear trend in the volatilities. However, to obtain the volatility of a given time series an instance from a particular class of scalar optimization problems (SOPs) has to be solved which still represents a challenge for existing solvers. We propose here a novel algorithm for the efficient numerical solution of such global optimization problems. The algorithm, DE–N, is a hybrid of Differential Evolution and the Newton method. The latter is widely used for the treatment of GARCH related models, but cannot be used as standalone algorithm in this case as the SOPs contain many local minima. The algorithm is tested and compared to some state-of-the-art methods on a benchmark suite consisting of 42 monthtly agricultural commodities series of the Mexican Consumer Price Index basket as well as on two series related to international prices. The results indicate that DE–N is highly competitive and that it is able to reliably solve SOPs derived from GARCH with trend models.


Time series GARCH Trend Differential Evolution Newton method Hybrid algorithm 



Benjamin Perea acknowledges support from the Conacyt to pursue his M.Sc. studies at the Cinvestav-IPN. Lourdes Uribe acknowledges support from Project SIP20162103 and from a CONACyT scholarship to pursue graduate studies at ESFM-IPN.


  1. Andrzej, O., & Stanislaw, K. (2006). Evolutionary algorithms for global optimization. In Global optimization (pp. 267–300). Springer.Google Scholar
  2. Bäck, T., & Schwefel, H.-P. (1993). An overview of evolutionary algorithms for parameter optimization. Evolutionary Computation, 1(1), 1–23.CrossRefGoogle Scholar
  3. Baillie, R. T., Bollerslev, T., & Mikkelsen, H. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 74(1), 3–30.CrossRefGoogle Scholar
  4. Bauer, C. (2007). A better asymmetric model of changing volatility in stock and exchange rate returns: Trend-garch. The Europen Journal of Finance, 13(1), 65–87.CrossRefGoogle Scholar
  5. Beck, S. (1993). A rational expectations model of time varying risk premia in commodities futures markets: Theory and evidence. International Economic Review, 34(1), 149–168.CrossRefGoogle Scholar
  6. Beck, S. (2001). Autoregressive conditional heteroscedasticity in commodity spot prices. Journal of Applied Econometrics, 16(2), 115–132.CrossRefGoogle Scholar
  7. Beyer, H.-G., & Schwefel, H.-P. (2002). Evolution strategies: A comprehensive introduction. Natural Computing, 1(1), 3–52.CrossRefGoogle Scholar
  8. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–327.CrossRefGoogle Scholar
  9. Bollerslev, T. (1987a). A conditionally heteroskedastic time series model for speculative prices and rates of return. The Review of Economics and Statistics, 69(3), 542–547.Google Scholar
  10. Bollerslev, T. (1987b). A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics, 69(1), 542–547.Google Scholar
  11. Domowitz, I., & Hakkio, C. S. (1985). Conditional variance and the risk premium in the foreign exchange market. Journal of International Economics, 19(1–2), 47–66.CrossRefGoogle Scholar
  12. Eiben, A. E., & Smith, J. E. (2003). Introduction to Evolutionary Computing. Berlin: Springer.CrossRefGoogle Scholar
  13. Engle, R . F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica Journal of the Econometric Society, 987–1007Google Scholar
  14. Engle, R. F., & Bollerslev, T. (1986). Modeling the persistence of conditional variances. Econometric Reviews, 5(1), 1–50.CrossRefGoogle Scholar
  15. Engle, R. F., & Victor, K. N. (1993). Measuring and testing the impact of news on volatility. The Journal of Finance, 48(5), 1747–1778.CrossRefGoogle Scholar
  16. Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. The Journal of Finance, 48(5), 1779–1801.CrossRefGoogle Scholar
  17. Guerrero, S., Hernandez-del-Valle, G., & Miriam, J.-T. (2016a). A functional approach to test trending volatility. Documentos de Investigación Banco de México 2016–03.Google Scholar
  18. Guerrero, S., Hernandez-del-Valle, G., & Miriam, J.-T. (2016b). A functional approach to test trending volatility: Evidence of trending volatility in the price of Mexican and international agricultural products. Agricultural Economics, 48, 1–11.Google Scholar
  19. Hart, W. E., Krasnogor, N., & Smith, J. E. (2005). Memetic evolutionary algorithms. In: Recent advances in memetic algorithms, (pp. 3–27). Springer.Google Scholar
  20. Hung, J.-C. (2009). A fuzzy GARCH model applied to stock market scenario using a genetic algorithm. Expert Systems with Applications, 36(9), 11710–11717.CrossRefGoogle Scholar
  21. Kukkonen, S., & Lampinen, J. (2006). Constrained real-parameter optimization with generalized differential evolution. In: IEEE Congress on evolutionary computation, 2006. CEC 2006, (pp. 207–214).Google Scholar
  22. Muth, J. F. (1961). Rational expectations and the theory of price movements. Econometrica, 29(3), 315–335.CrossRefGoogle Scholar
  23. Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59(2), 347–370.CrossRefGoogle Scholar
  24. Nelson, D. B. (1994). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 52(2), 347–370.Google Scholar
  25. Neri, F., & Cotta, C. (2012). Memetic algorithms and memetic computing optimization: A literature review. Swarm and Evolutionary Computation, 2, 1–14.CrossRefGoogle Scholar
  26. Neri, F., Cotta, C., & Moscato, P. (2011). Handbook of memetic algorithms (Vol. 379). Berlin: Springer.Google Scholar
  27. Nocedal, J., & Wright, S. (2006). Numerical optimization. Springer series in operations research and financial engineering. Berlin: Springer.Google Scholar
  28. Price, K., Storn, R. M., & Lampinen, J. (2005). Differential evolution: A practical approach to global optimization. Natural computing series. Berlin: Springer.Google Scholar
  29. Schwefel, H.-P. (1993). Evolution and optimum seeking. New York: Wiley.Google Scholar
  30. Setiawan, K., & Maekawa, K. (2014). Estimation of vector error correction model with garch errors: Monte Carlo simulation and applications. EcoMod: Technical report.Google Scholar
  31. Storn, R., & Price, K. (1995). Differential evolution-a simple and efficient adaptive scheme for global optimization over continuous spaces (Vol. 3). Berkeley: ICSI.Google Scholar
  32. Wright, S. J. (2001). On the convergence of the Newton/log-barrier method. Mathematical Programming, 90(1), 71–100.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Lourdes Uribe
    • 1
  • Benjamin Perea
    • 3
  • Gerardo Hernández-del-Valle
    • 2
  • Oliver Schütze
    • 3
  1. 1.MexicoMexico
  2. 2.Dirección General de Investigación EconómicaBanco de MéxicoMexicoMexico
  3. 3.MexicoMexico

Personalised recommendations