Recurrent Support and Relevance Vector Machines Based Model with Application to Forecasting Volatility of Financial Returns

ABSTRACT

In the recent years, the use of GARCH type (especially, ARMA-GARCH) models and computational-intelligence-based techniques—Support Vector Machine (SVM) and Relevance Vector Machine (RVM) have been successfully used for financial forecasting. This paper deals with the application of ARMA-GARCH, recurrent SVM (RSVM) and recurrent RVM (RRVM) in volatility forecasting. Based on RSVM and RRVM, two GARCH methods are used and are compared with parametric GARCHs (Pure and ARMA-GARCH) in terms of their ability to forecast multi-periodically. These models are evaluated on four performance metrics: MSE, MAE, DS, and linear regression R squared. The real data in this study uses two Asian stock market composite indices of BSE SENSEX and NIKKEI225. This paper also examines the effects of outliers on modeling and forecasting volatility. Our experiment shows that both the RSVM and RRVM perform almost equally, but better than the GARCH type models in forecasting. The ARMA-GARCH model is superior to the pure GARCH and only the RRVM with RSVM hold the robustness properties in forecasting.

In the recent years, the use of GARCH type (especially, ARMA-GARCH) models and computational-intelligence-based techniques—Support Vector Machine (SVM) and Relevance Vector Machine (RVM) have been successfully used for financial forecasting. This paper deals with the application of ARMA-GARCH, recurrent SVM (RSVM) and recurrent RVM (RRVM) in volatility forecasting. Based on RSVM and RRVM, two GARCH methods are used and are compared with parametric GARCHs (Pure and ARMA-GARCH) in terms of their ability to forecast multi-periodically. These models are evaluated on four performance metrics: MSE, MAE, DS, and linear regression R squared. The real data in this study uses two Asian stock market composite indices of BSE SENSEX and NIKKEI225. This paper also examines the effects of outliers on modeling and forecasting volatility. Our experiment shows that both the RSVM and RRVM perform almost equally, but better than the GARCH type models in forecasting. The ARMA-GARCH model is superior to the pure GARCH and only the RRVM with RSVM hold the robustness properties in forecasting.

Cite this paper

nullA. Hossain and M. Nasser, "Recurrent Support and Relevance Vector Machines Based Model with Application to Forecasting Volatility of Financial Returns,"*Journal of Intelligent Learning Systems and Applications*, Vol. 3 No. 4, 2011, pp. 230-241. doi: 10.4236/jilsa.2011.34026.

nullA. Hossain and M. Nasser, "Recurrent Support and Relevance Vector Machines Based Model with Application to Forecasting Volatility of Financial Returns,"

References

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[3] R. F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, Vol. 50, No. 2, 1982, pp. 987-1007. doi:10.2307/1912773

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[5] S. H. Poon and C. Granger, “Forecasting Volatility in Financial Markets: A Review,” Journal of Economic Literature, Vol. 41, No. 2, 2003, pp. 478-539. doi:10.1257/002205103765762743

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[7] H. Tang, K. C. Chun and L. Xu, “Finite Mixture of ARMA-GARCH Model for Stock Price Prediction,” Proceedings of 3rd International Workshop on Computational Intelligence in Economics and Finance (CIEF 2003), North Carolina, 26-30 September 2003, pp. 1112-1119.

[8] A. Hossain and M. Nasser, “Comparison of Finite Mixture of ARMA-GARCH, Back Propagation Neural Net-works and Support-Vector Machines in Forecasting Financial Returns,” Journal of Applied Statistics, Vol. 38, No. 3, 2011, pp. 533-551.

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[19] L. B. Tang, H. Y. Sheng and L. X. Tang, “Forecasting Volatility Based on Wavelet Support Vector Machine,” Ex- pert Systems with Applications, Vol. 36, No. 2, 2009, pp. 2901-2909.

[20] L. B. Tang, H. Y. Sheng and L. X. Tang, “GARCH Prediction Using Spline Wavelet Support Vector Machine,” Journal of Neural Computing and Application, Vol. 18, No. 8, 2009, pp. 913-917.

[21] M. Bildirici and ?. ?. Ersin, “Improving Forecasts of GARCH Family Models with the Artificial Neural Networks: An Application to the Daily Returns in Istanbul Stock Exchange,” Expert Systems with Applications, Vol. 36, No. 4, 2009, pp. 7355-7362. doi:10.1016/j.eswa.2008.09.051

[22] L. J. Cao and F. Tay, “Application of Support Vector Machines in Financial Time Series Forecasting,” International Journal of Management Science, Vol. 29, No. 4, 2001, pp. 309-317.

[23] V. N. Vapnik, “The Nature of Statistical Learning Theory,” 2nd Edition, Sringer-Verlag, New York, 1995.

[24] L. J. Cao and F. Tay, “Modified Support Vector Machines in Financial Time Series Forecasting,” Journal of Neurocomputing, Vol. 48, No. 1-4, 2002, pp. 847-861.

[25] K. J. Kim, “Financial Time Series Forecasting Using Support Vector Machines,” Journal of Neurocomputing, Vol. 55, No. 1-2, 2003, pp. 307-319.

[26] W. Huang, Y. Nakamori and S. Y. Wang, “Forecasting Stock Market Movement Direction with Support Vector Machine,” Journal of Computers & Operational Research, Vol. 32, No. 10, 2005, pp. 513-522.

[27] C. J. Lu, T. S. Lee and C. C. Chiu, “Financial Time Series Forecasting Using Independent Component Analysis and Support Vector Regression,” Journal of Decision Support Systems, Vol. 47, No. 2, 2009, pp. 115-125.

[28] H. S. Kim and S. Y. Sohn, “Support Vector Machines for Default Prediction of SMEs Based on Technology Credit,” European Journal of Operational Research, Vol. 201, No. 3, 2010, pp. 938-846.

[29] A. J. Smola and B. Scholkopf, “A Tutorial on Support Vector Regression,” Journal of Statistics and Computing, Vol. 14, No. 3, 2004, pp. 199-222.

[30] M. E. Tipping, “Sparse Bayesian Learning and the Relevance Vector Machine,” Journal of Machine Learning Research, Vol. 1, 2001, pp. 211-244.

[31] C. M. Bishop and M. E. Tipping, “Variational Relevance Vector Machine,” In: C. Boutilier and M. Goldszmidt, Eds., Uncertainty in Artificial Intelligence, Morgan Kaufmann, Waltham, 2000, pp. 46-53.

[32] S. Ghosh and P. P. Mujumdar, “Statistical Downscaling of GCM Simulations to Stream Flow Using Relevance Vector Machine,” Advances in Water Resources, Vol. 31, No. 1, 2008, pp. 132-146.

[33] D. Porro, N. Hdez, I. Talavera, O. Nunez, A. Dago and R. J. Biscay, “Performance Evaluation of Relevance Vector Machines as a Nonlinear Regression method in Real World Chemical Spectroscopic Data,” 19th International Conference on Pattern Recognition (ICPR 2008), Tampa, 8-11 December 2008, pp. 1-4.

[34] S. Chen, K. Jeong and W. H?rdle, “Support Vector Regression Based GARCH Model with Application to Forecasting Volatility of Financial Returns,” SFB 649 Discussion Paper 2008-014. http://edoc.hu-berlin.de/series/sfb-649-papers/2008-14/PDF/14.pdf

[35] S. Chen, K. Jeong and W. H?rdle, “Recurrent Support Vector Regression for a Nonlinear ARMA Model with Applications to Forecasting Financial Returns,” SFB 649 Discussion Paper 2008-051.

[36] J. A. K. Suykens and J. Vandewalle, “Recurrent Least Squares Support Vector Machines,” IEEE Transactions on Circuits and Systems I, Vol. 47, No. 7, 2000, pp. 1109-1114. doi:10.1109/81.855471

[37] P. H. Ou and H. Wang, “Predict GARCH Based Volatility of Shanghai Composite Index by Recurrent Relevant Vector Machines and Recurrent Least Square Support Vector Machines,” Journal of Mathematics Research, Vol. 2, No. 2, 2010.

[38] G. Bekaert and C. R. Harvey, “Emerging Equity Market Volatility,” Journal of Financial Economics, Vol. 43, No. 1, 1997, pp. 29-78. doi:10.1016/S0304-405X(96)00889-6

[39] R. Aggarwal, C. Inclan and R. Leal, “Volatility in Emerging Stock Markets,” Journal of Financial and Quantitative Analysis, Vol. 34, No. 1, 1999, pp. 33-55. doi:10.2307/2676245

[40] J. Ledolter, “The Effects of Additive Outliers on the Fore- casts from ARIMA Models,” International Journal of Forecasting, Vol. 5, No. 2, 1989, pp. 231-240. doi:10.1016/0169-2070(89)90090-3

[41] R. F. Engle and G. G. J. Lee, “A Permanent and Transitory Component Model of Stock Returns Volatility,” Discussion Paper 92-44R, University of California, San Diego, 1993.

[42] P. H. Franses, D. van Dijk and A. Locas, “Short Patches of Outliers, ARCH and Volatility Modeling,” Discussion Paper 98-057/4, Tinbergen Institute, Erasmus University, Rotterdam, 1998.

[43] P. H. Franses and H. Ghijsels, “Additive Outliers, GARCH and Forecasting Volatility,” International Journal of Forecasting, Vol. 15, No. 1, 1999, pp. 1-9. doi:10.1016/S0169-2070(98)00053-3

[44] A. J. Fox, “Outliers in Time Series,” Journal of the Royal Statistical Society, Series B, Vol. 34, No. 3, 1972, pp. 350-363.

[45] C. Chen and L. Liu, “Joint Estimation of Model Parameters and Outlier Effects in Time Series,” Journal of American Statistical Association, Vol. 88, No. 421, 1993, pp. 284-297. doi:10.2307/2290724

[46] M. E. Tipping, “Relevance Vector Machine,” Microsoft Research, Cambridge, 2000.

[47] M. E. Tipping, “Bayesian Inference: An Introduction to Principles and Practice in Machine Learning,” Advanced Lectures on Machine Learning, Vol. 3176/2004, 2004, pp. 41-62. doi:10.1007/978-3-540-28650-9_3

[48] P. R. Hansen and A. Lunde, “A Forecast Comparison of Volatility Models: Does Anything Beat a GARCH (1, 1)?” Journal of Applied Econometrics, Vol. 20, No. 7, 2005, pp. 873-889.

[49] J. D. Hamilton, “Time Series Analysis,” Princeton University Press, Saddle River, 1997.

[50] W. Enders, “Applied Econometric Time Series,” 2nd Edi- tion, John Wiley & Sons, New York, 2004. doi:10.1016/S0305-0483(01)00026-3

[51] F. E. H. Tay and L. Cao, “Application of Support Vector Machines in Financial Time-Series Forecasting,” Omega, Vol. 29, No. 4, 2001, pp. 309-317.

[52] M. Thomason, “The Practitioner Method and Tools: A Basic Neural Network-Based Trading System Project Revisited (Parts 1 and 2),” Journal of Computational Intelligence in Finance, Vol. 7, No. 3, 1999, pp. 36-45.

[53] M. Thomason, “The Practitioner Method and Tools: A Basic Neural Network-Based Trading System Project Revisited (Parts 3 and 4),” Journal of Computational Intelligence in Finance, Vol. 7, No. 3, 1999, pp. 35-48.

[54] C. Brooks, “Predicting Stock Index Volatility: Can Market Volume Help?” Journal of Forecasting, Vol. 17, No. 1, 1998, pp. 59-80. doi:10.1002/(SICI)1099-131X(199801)17:1<59::AID-FOR676>3.0.CO;2-H

[55] I. A. Moosa, “Exchange Rate Forecasting: Techniques and Applications,” Macmillan Press LTD, Lonton, 2000.

[56] H. Theil, “Principles of Econometrics,” Wiley, New York, 1971.

[57] A. K. Bera and C. M. Jarque, “An Efficient Large-Sample Test for Normality of Observations and Regression Residuals,” Australian National University Working Papers in Econometrics, 40, Canberra, 1981.

[1] R. F. Engle and A. J. Patton, “What Good Is a Volatility Model?” Journal of Quantitative Finance, Vol. 1, No. 2, 2001, pp. 237-245. doi:10.1088/1469-7688/1/2/305

[2] R. C. Merton, “On Estimating the Expected Return on the Market: An Exploratory Investigation,” Journal of Financial Economics, Vol. 8, 1980, pp. 323-361. doi:10.1016/0304-405X(80)90007-0

[3] R. F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, Vol. 50, No. 2, 1982, pp. 987-1007. doi:10.2307/1912773

[4] T. Bollerslev, “Generalized Autoregressive Conditional Heteroscedasticity,” Journal of Econometric, Vol. 31, No. 3, 1986, pp. 307-327. doi:10.1016/0304-4076(86)90063-1

[5] S. H. Poon and C. Granger, “Forecasting Volatility in Financial Markets: A Review,” Journal of Economic Literature, Vol. 41, No. 2, 2003, pp. 478-539. doi:10.1257/002205103765762743

[6] W. C. Wong, F. Yip and L. Xu, “Financial Prediction by Finite Mixture GARCH Model,” Proceedings of Fifth International Conference on Neural Information Processing, Kitakyushu, 21-23 October 1998, pp. 1351-1354.

[7] H. Tang, K. C. Chun and L. Xu, “Finite Mixture of ARMA-GARCH Model for Stock Price Prediction,” Proceedings of 3rd International Workshop on Computational Intelligence in Economics and Finance (CIEF 2003), North Carolina, 26-30 September 2003, pp. 1112-1119.

[8] A. Hossain and M. Nasser, “Comparison of Finite Mixture of ARMA-GARCH, Back Propagation Neural Net-works and Support-Vector Machines in Forecasting Financial Returns,” Journal of Applied Statistics, Vol. 38, No. 3, 2011, pp. 533-551.

[9] T. G. Andersen and T. Bollerslev, “Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts,” International Economic Review, Vol. 39, No. 4, 1998, pp. 885-905. doi:10.2307/2527343

[10] T. J. Brailsford and R. W. Faff, “An Evaluation of Volatility Forecasting Techniques,” Journal of Banking and Finance, Vol. 20, No. 3, 1996, pp. 419-438. doi:10.1016/0378-4266(95)00015-1

[11] R. Cumby, S. Figlewski and J. Hasbrouck, “Forecasting Volatility and Correlations with EGARCH Models,” Jour- nal of Derivatives Winter, Vol. 1, No. 2, 1993, pp. 51-63.

[12] S. Figlewski, “Forecasting Volatility,” Financial Markets, Institutions and Instruments, Vol. 6, No. 1, 1997, pp. 1-88. doi:10.1111/1468-0416.00009

[13] P. Jorion, “Predicting Volatility in the Foreign Exchange Market,” Journal of Finance, Vol. 50, No. 2, 1995, pp. 507-528. doi:10.2307/2329417

[14] P. Jorion, “Risk and Turnover in the Foreign Exchange Market,” In: J. A. Franke, G. Galli and A. Giovannini, Eds., The Microstructure of Foreign Exchange Markets, Chicago University Press, Chicago, 1996.

[15] D. G. McMillan, A. E. H. Speight and O. Gwilym, “Forecasting UK Stock Market Volatility: A Comparative Analysis of Alternate Methods,” Applied Financial Economics, Vol. 10, No. 4, 2000, pp. 435-448. doi:10.1080/09603100050031561

[16] R. G. Donaldson and M. Kamstra, “An Artificial Neural Network-GARCH Model for International Stock Returns Volatility,” Journal of Empirical Finance, Vol. 4, No. 1, 1997, pp. 17-46. doi:10.1016/S0927-5398(96)00011-4

[17] F. Perez-Cruz, J. A. Afonso-Rodriguez and J. Giner, “Estimating GARCH Models Using Support Vector Machines,” Journal of Quantitative Finance, Vol. 3, 2003, pp. 163-172.

[18] P. H. Ou and H. Wang, “Predicting GARCH, EGARCH, GJR Based Volatility by the Relevance Vector Machine: Evidence from the Hang Seng Index,” International Research Journal of Finance and Economics, No. 39, 2010, pp. 46-63.

[19] L. B. Tang, H. Y. Sheng and L. X. Tang, “Forecasting Volatility Based on Wavelet Support Vector Machine,” Ex- pert Systems with Applications, Vol. 36, No. 2, 2009, pp. 2901-2909.

[20] L. B. Tang, H. Y. Sheng and L. X. Tang, “GARCH Prediction Using Spline Wavelet Support Vector Machine,” Journal of Neural Computing and Application, Vol. 18, No. 8, 2009, pp. 913-917.

[21] M. Bildirici and ?. ?. Ersin, “Improving Forecasts of GARCH Family Models with the Artificial Neural Networks: An Application to the Daily Returns in Istanbul Stock Exchange,” Expert Systems with Applications, Vol. 36, No. 4, 2009, pp. 7355-7362. doi:10.1016/j.eswa.2008.09.051

[22] L. J. Cao and F. Tay, “Application of Support Vector Machines in Financial Time Series Forecasting,” International Journal of Management Science, Vol. 29, No. 4, 2001, pp. 309-317.

[23] V. N. Vapnik, “The Nature of Statistical Learning Theory,” 2nd Edition, Sringer-Verlag, New York, 1995.

[24] L. J. Cao and F. Tay, “Modified Support Vector Machines in Financial Time Series Forecasting,” Journal of Neurocomputing, Vol. 48, No. 1-4, 2002, pp. 847-861.

[25] K. J. Kim, “Financial Time Series Forecasting Using Support Vector Machines,” Journal of Neurocomputing, Vol. 55, No. 1-2, 2003, pp. 307-319.

[26] W. Huang, Y. Nakamori and S. Y. Wang, “Forecasting Stock Market Movement Direction with Support Vector Machine,” Journal of Computers & Operational Research, Vol. 32, No. 10, 2005, pp. 513-522.

[27] C. J. Lu, T. S. Lee and C. C. Chiu, “Financial Time Series Forecasting Using Independent Component Analysis and Support Vector Regression,” Journal of Decision Support Systems, Vol. 47, No. 2, 2009, pp. 115-125.

[28] H. S. Kim and S. Y. Sohn, “Support Vector Machines for Default Prediction of SMEs Based on Technology Credit,” European Journal of Operational Research, Vol. 201, No. 3, 2010, pp. 938-846.

[29] A. J. Smola and B. Scholkopf, “A Tutorial on Support Vector Regression,” Journal of Statistics and Computing, Vol. 14, No. 3, 2004, pp. 199-222.

[30] M. E. Tipping, “Sparse Bayesian Learning and the Relevance Vector Machine,” Journal of Machine Learning Research, Vol. 1, 2001, pp. 211-244.

[31] C. M. Bishop and M. E. Tipping, “Variational Relevance Vector Machine,” In: C. Boutilier and M. Goldszmidt, Eds., Uncertainty in Artificial Intelligence, Morgan Kaufmann, Waltham, 2000, pp. 46-53.

[32] S. Ghosh and P. P. Mujumdar, “Statistical Downscaling of GCM Simulations to Stream Flow Using Relevance Vector Machine,” Advances in Water Resources, Vol. 31, No. 1, 2008, pp. 132-146.

[33] D. Porro, N. Hdez, I. Talavera, O. Nunez, A. Dago and R. J. Biscay, “Performance Evaluation of Relevance Vector Machines as a Nonlinear Regression method in Real World Chemical Spectroscopic Data,” 19th International Conference on Pattern Recognition (ICPR 2008), Tampa, 8-11 December 2008, pp. 1-4.

[34] S. Chen, K. Jeong and W. H?rdle, “Support Vector Regression Based GARCH Model with Application to Forecasting Volatility of Financial Returns,” SFB 649 Discussion Paper 2008-014. http://edoc.hu-berlin.de/series/sfb-649-papers/2008-14/PDF/14.pdf

[35] S. Chen, K. Jeong and W. H?rdle, “Recurrent Support Vector Regression for a Nonlinear ARMA Model with Applications to Forecasting Financial Returns,” SFB 649 Discussion Paper 2008-051.

[36] J. A. K. Suykens and J. Vandewalle, “Recurrent Least Squares Support Vector Machines,” IEEE Transactions on Circuits and Systems I, Vol. 47, No. 7, 2000, pp. 1109-1114. doi:10.1109/81.855471

[37] P. H. Ou and H. Wang, “Predict GARCH Based Volatility of Shanghai Composite Index by Recurrent Relevant Vector Machines and Recurrent Least Square Support Vector Machines,” Journal of Mathematics Research, Vol. 2, No. 2, 2010.

[38] G. Bekaert and C. R. Harvey, “Emerging Equity Market Volatility,” Journal of Financial Economics, Vol. 43, No. 1, 1997, pp. 29-78. doi:10.1016/S0304-405X(96)00889-6

[39] R. Aggarwal, C. Inclan and R. Leal, “Volatility in Emerging Stock Markets,” Journal of Financial and Quantitative Analysis, Vol. 34, No. 1, 1999, pp. 33-55. doi:10.2307/2676245

[40] J. Ledolter, “The Effects of Additive Outliers on the Fore- casts from ARIMA Models,” International Journal of Forecasting, Vol. 5, No. 2, 1989, pp. 231-240. doi:10.1016/0169-2070(89)90090-3

[41] R. F. Engle and G. G. J. Lee, “A Permanent and Transitory Component Model of Stock Returns Volatility,” Discussion Paper 92-44R, University of California, San Diego, 1993.

[42] P. H. Franses, D. van Dijk and A. Locas, “Short Patches of Outliers, ARCH and Volatility Modeling,” Discussion Paper 98-057/4, Tinbergen Institute, Erasmus University, Rotterdam, 1998.

[43] P. H. Franses and H. Ghijsels, “Additive Outliers, GARCH and Forecasting Volatility,” International Journal of Forecasting, Vol. 15, No. 1, 1999, pp. 1-9. doi:10.1016/S0169-2070(98)00053-3

[44] A. J. Fox, “Outliers in Time Series,” Journal of the Royal Statistical Society, Series B, Vol. 34, No. 3, 1972, pp. 350-363.

[45] C. Chen and L. Liu, “Joint Estimation of Model Parameters and Outlier Effects in Time Series,” Journal of American Statistical Association, Vol. 88, No. 421, 1993, pp. 284-297. doi:10.2307/2290724

[46] M. E. Tipping, “Relevance Vector Machine,” Microsoft Research, Cambridge, 2000.

[47] M. E. Tipping, “Bayesian Inference: An Introduction to Principles and Practice in Machine Learning,” Advanced Lectures on Machine Learning, Vol. 3176/2004, 2004, pp. 41-62. doi:10.1007/978-3-540-28650-9_3

[48] P. R. Hansen and A. Lunde, “A Forecast Comparison of Volatility Models: Does Anything Beat a GARCH (1, 1)?” Journal of Applied Econometrics, Vol. 20, No. 7, 2005, pp. 873-889.

[49] J. D. Hamilton, “Time Series Analysis,” Princeton University Press, Saddle River, 1997.

[50] W. Enders, “Applied Econometric Time Series,” 2nd Edi- tion, John Wiley & Sons, New York, 2004. doi:10.1016/S0305-0483(01)00026-3

[51] F. E. H. Tay and L. Cao, “Application of Support Vector Machines in Financial Time-Series Forecasting,” Omega, Vol. 29, No. 4, 2001, pp. 309-317.

[52] M. Thomason, “The Practitioner Method and Tools: A Basic Neural Network-Based Trading System Project Revisited (Parts 1 and 2),” Journal of Computational Intelligence in Finance, Vol. 7, No. 3, 1999, pp. 36-45.

[53] M. Thomason, “The Practitioner Method and Tools: A Basic Neural Network-Based Trading System Project Revisited (Parts 3 and 4),” Journal of Computational Intelligence in Finance, Vol. 7, No. 3, 1999, pp. 35-48.

[54] C. Brooks, “Predicting Stock Index Volatility: Can Market Volume Help?” Journal of Forecasting, Vol. 17, No. 1, 1998, pp. 59-80. doi:10.1002/(SICI)1099-131X(199801)17:1<59::AID-FOR676>3.0.CO;2-H

[55] I. A. Moosa, “Exchange Rate Forecasting: Techniques and Applications,” Macmillan Press LTD, Lonton, 2000.

[56] H. Theil, “Principles of Econometrics,” Wiley, New York, 1971.

[57] A. K. Bera and C. M. Jarque, “An Efficient Large-Sample Test for Normality of Observations and Regression Residuals,” Australian National University Working Papers in Econometrics, 40, Canberra, 1981.