**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30172

##### Study of a BVAR(p) Process Applied to U.S. Commodity Market Data

**Authors:**
Jan Sindelar

**Abstract:**

The paper presents an applied study of a multivariate AR(p) process fitted to daily data from U.S. commodity futures markets with the use of Bayesian statistics. In the first part a detailed description of the methods used is given. In the second part two BVAR models are chosen one with assumption of lognormal, the second with normal distribution of prices conditioned on the parameters. For a comparison two simple benchmark models are chosen that are commonly used in todays Financial Mathematics. The article compares the quality of predictions of all the models, tries to find an adequate rate of forgetting of information and questions the validity of Efficient Market Hypothesis in the semi-strong form.

**Keywords:**
Vector auto-regression,
forecasting,
financial,
Bayesian,
efficient markets.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1055256

**References:**

[1] E. Fama, "The behavior of stock market prices," Journal of Business, vol. 38, p. 34105, 1965.

[2] P. Samuelson, "Proof that properly anticipated prices fluctuate randomly," Industrial Management Review, vol. 6, pp. 44-49, 1965.

[3] E. Fama, "Efficient capital markets: A review of theory and empirical work," Journal of Finance, vol. 25, pp. 383-417, 1970.

[4] L. Bachelier, Th'eorie de la speculation. Paris: Gauthier-Villars, 1900, reprinted in P.H.Cootner: The Random Character of Stock Market Prices (1964).

[5] E. Fama, "Efficient capital markets: II," International Journal of Finance, vol. 46, pp. 1575-1617,1991.

[6] C. Granger, "Forecasting stock market prices: Lessons for forecasters," International Journal of Forecasting, vol. 8, pp. 3-13, 1992.

[7] A. Timmermann and C. Granger, "Efficient market hypothesis and forecasting," International Journal of Forecasting, vol. 20, pp. 15-27,2004.

[8] R. Litterman, "Forecasting with bayesian vector autoregressions - five years of experience," Journal of Business & Economic Statistics, vol. 4, pp. 25-38, 1986.

[9] S. Shreve and H. Soner, "Optimal investment and consumption with transaction costs," The Annals of Applied Probability, vol. 4, pp. 609-692, 1994.

[10] T. Yamamoto, "Asymptotic mean square prediction error for an autoregressive model with estimated coefficients," Applied Statistics, vol. 25, pp. 123-127, 1976.

[11] C. Ing, "Multistep prediction in autoregressive processes," Econometric theory, vol. 19, pp. 254-279, 2003.

[12] J. ╦ç Sindel'a╦çr, "Construction of multi-step ahead predictions in a normal bvar(p) model using monte carlo sampling," in Proceedings of the 10th International PhD workshop on Systems and Control, 2006.

[13] S. Shreve, Stochastic Calculus for Finance II., Continuous-Time Models. Springer, 2004.

[14] P. Billingsley, Probability and Measure, 3rd ed. Wiley, 1986.

[15] M. K'arn'y, J. B┬¿ohm, T. Guy, L. Jirsa, I. Nagy, P. Nedoma, and L. Tesa╦çr, Optimized Bayesian Dynamic Advising, Theory and Algorithms. Springer, 2005.

[16] T. Cipra, Finan╦çcn'─▒ Ekonometrie. EKOPRESS, 2008, in Czech.

[17] G. Box and G. Jenkins, Time Series Analysis. San Francisco, U.S.A.: Holden-Day, 1970.

[18] P. Brockwell and R. Davis, Introduction to Time Series and Forecasting. Springer, 1996.

[19] J. And╦çel, Z'aklady matematick'e statistiky. Matfyz-Press, 2005, in Czech.

[20] I. Nagy, L. Pavelkov'a, E. Suzdaleva, J. Homolov'a, and M. K'arn'y, Bayesian Decision Making. Czech Academy of Sciences, 2005.

[21] S. Shreve, Stochastic Calculus for Finance I., The Binomial Asset Pricing Model. Springer, 2004.

[22] I. Karatzas and S. Shreve, Methods of Mathematical Finance. Springer, 1998.

[23] H. F┬¿olmer and A. Schied, Stochastic Finance, An Introduction in Discrete Time. Walter de Gruyter, 2002.

[24] P. Billingsley, Convergence of Probability Measures, 2nd ed. Wiley, 1999.

[25] M. K'arn'y and R. Kulhav'y, "Structure determination of regression-type models for adaptive prediction and control," in Bayesian Analysis of Time Series and Dynamic Models, J. C. Spall, Ed. Marcel Dekker, 1988, pp. 313-345.

[26] C. Robert, The Bayesian Choice, From Decision Theoretic Foundations to Computational Implementation. Springer, 2007.

[27] J. Ghosh, M. Delampady, and T. Samanta, An Introduction to Bayesian Analysis, Theory and Methods. Springer, 2006.

[28] R. Horn and C. Johnson, Matrix Analysis. Cambridge University Press, 1985.

[29] A. Votava, "Estimation of forgetting factor in the frame of dynamic decision making Bachelor-s thesis, Faculty of Mathematics and Physics, Charles University," Prague, 2009.

[30] S. Kotz, B. N., and N. Johnson, Continuous Multivariate Distributions, 2nd ed. Wiley, 2000.

[31] D. P. Bertsekas, Dynamic Programming and Optimal Control, 2nd ed. Athena Scientific, 2000, vol.

[32] Dynamic Programming and Optimal Control, 2nd ed. Athena Scientific, 2001, vol. 2.

[33] G. Constantinides, "Multiperiod consumption and investment behavior with convex transaction costs," Management Science, vol. 25, pp. 1127- 1137, 1979.

[34] G. Genotte and A. Jung, "Investment strategies under transaction costs: The finite horizon case," Management Science, vol. 38, pp. 385-404, 1994.

[35] J. Zeman, "Futures trading:design of a strategy," accepted to ICORFE 2009 Conference, Venice, Italy, October 2009.