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Optimal reinsurance and investment strategy with two piece utility function
Hidden Markov models with threshold effects and their applications to oil price forecasting
1. | School of Economics and Management, Southeast University, Nanjing, China |
2. | Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China |
3. | Centre for Applied Financial Studies, University of South Australia, Adelaide 5001, Australia |
4. | Haskayne School of Business, University of Calgary, Canada T3A 6A4 |
5. | Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia |
6. | School of Management and Engineering, Nanjing University, Nanjing, China |
In this paper, we propose a Hidden Markov Model (HMM) which incorporates the threshold effect of the observation process. Simulated examples are given to show the accuracy of the estimated model parameters. We also give a detailed implementation of the model by using a dataset of crude oil price in the period 1986-2011. The prediction of crude oil spot price is an important and challenging issue for both government policy makers and industrial investors as most of the world's energy comes from the consumption of crude oil. However, many random events and human factors may lead the crude oil price to a strongly fluctuating and highly non-linear behavior. To capture these properties, we modulate the mean and the variance of log-returns of commodity prices by a finite-state Markov chain. The $h$-day ahead forecasts generated from our model are compared with regular HMM and the Autoregressive Moving Average model (ARMA). The results indicate that our proposed HMM with threshold effect outperforms the other models in terms of predicting ability.
References:
[1] |
W. K. Ching, T. K. Siu, L. M. Li, T. Li and W. K. Li,
Modeling default data via an interactive hidden Markov model, Computational Economics, 34 (2009), 1-19.
doi: 10.1007/s10614-009-9183-5. |
[2] |
E. G. De Souza e Silva, L. Legey and E. A. De Souza e Silva, Forecasting oil price trends using wavelet and hidden Markov models, Energy Economics, 32 (2010), 1507-1519. Google Scholar |
[3] |
F. X. Diebold and R. Mariano, Comparing predictive accuracy, Journal of Business and Economic Statistics, 13 (1995), 253-265. Google Scholar |
[4] |
R. J. Elliott, L. Aggoun and J. Moore,
Hidden Markov Models: Estimation and Control, Springer, New York, 1995. |
[5] |
R. J. Elliott, L. L. Chan and T. K. Siu,
Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432.
doi: 10.1007/s10436-005-0013-z. |
[6] |
R. J. Elliott, C. C. Liew and T. K. Siu,
On filtering and estimation of a threshold stochastic volatility model, Applied Mathematics dand Computation, 218 (2011), 61-75.
doi: 10.1016/j.amc.2011.05.052. |
[7] |
R. J. Elliott, T. K. Siu and A. Badescu,
On mean-variance portfolio selection under a hidden Markovian regime-switching model, Economic Modelling, 27 (2010), 678-686.
doi: 10.1016/j.econmod.2010.01.007. |
[8] |
R. J. Elliott, T. K. Siu and A. Badescu,
On pricing and hedging options in regime-switching models with feedback effect, Journal of Economic Dynamics and Control, 35 (2011), 694-713.
doi: 10.1016/j.jedc.2010.12.014. |
[9] |
R. J. Elliott, T. K. Siu and J. W. Lau,
Filtering a double threshold model with regime switching, IEEE Transactions on Automatic Control, 58 (2013), 3185-3190.
doi: 10.1109/TAC.2013.2261186. |
[10] |
R. J. Elliott, T. K. Siu and H. L. Yang,
Ruin theory in a hidden Markov-modulated risk model, Stochastic Models, 27 (2011), 474-489.
doi: 10.1080/15326349.2011.593408. |
[11] |
R. J. Elliott and J. van der Hoek,
An application of hidden Markov models to asset allocation problems, Finance and Stochastic, 1 (1997), 229-238.
doi: 10.1007/s007800050022. |
[12] |
C. Erlwein and R. Mamon,
An online estimation scheme for a Hull and White model with HMM-driven parameters, Statistical Methods and Applications, 18 (2009), 87-107.
doi: 10.1007/s10260-007-0082-4. |
[13] |
C. Erlwein, R. Mamon and M. Davison,
An examination of HMM-based investment strategies for asset allocation, Applied Stchastic Models in Business and Industry, 27 (2011), 204-221.
doi: 10.1002/asmb.820. |
[14] |
G. Frey, M. Manera, A. Markandya and E. Scarpa, Econometric models for oil price forecasting, A Critical Survey CESinfo Forum, 10 (2009), 29-44. Google Scholar |
[15] |
D. Harding and A. R. Pagan, Synchronisation of Cycles, mimeo., Australian National University. Google Scholar |
[16] |
H. G. Huntington,
Oil price forecasting in the 1980s: What went wrong?, The Energy Journal, 15 (1994), 1-22.
doi: 10.5547/ISSN0195-6574-EJ-Vol15-No2-1. |
[17] |
R. K. Kaufman, S. Dees, P. Karadeloglou and M. Sanchez,
Does OPEC matter? An econometric analysis of oil prices, The Energy Journal, 25 (2004), 67-90.
doi: 10.5547/ISSN0195-6574-EJ-Vol25-No4-4. |
[18] |
M. W. Korolkiewica and R. J. Elliott, Smoothed parameter estimation for a Markov model of credit quality, Hidden Markov Models in Finance, Springer, New York, 104 (2007), 69–90.
doi: 10.1007/0-387-71163-5_5. |
[19] |
C. C. Liew and T. K. Siu,
A hidden Markov regime-switching model for option valuation, Insurance: Mathematics and Economics, 47 (2010), 374-384.
doi: 10.1016/j.insmatheco.2010.08.003. |
[20] |
R. S. Mamon, C. Erlwein and R. B. Gopaluni,
Adaptive signal processing of asset price dynamics with predictability analysis, Information Sciences, 178 (2008), 203-219.
doi: 10.1016/j.ins.2007.05.021. |
[21] |
T. K. Siu, W. K. Ching, E. Fung, M. Ng and X. Li,
A high-order Markov-switching model for risk measurement, Computers and Mathematics with Applications, 58 (2009), 1-10.
doi: 10.1016/j.camwa.2008.10.099. |
[22] |
T. K. Siu, W. K. Ching, E. S. Fung and M. K. Ng, Extracting information from spot interest rates and credit ratings using double higher-order hidden Markov models, Computational Economics, 26 (2005), 251-284. Google Scholar |
[23] |
S. J. Taylor,
Asset Price Dynamics, Volatility, and Prediction, Princeton University Press, Princeton, 2005.
doi: 10.1515/9781400839254. |
[24] |
H. Tong,
Determination of the order of a Markov chain by Akaike's information criterion, Journal of applied probability, 12 (1975), 488-497.
doi: 10.1017/S0021900200048294. |
[25] |
H. Tong, On a threshold model, Pattern Recognition and signal processing, NATO ASI Series E: Applied Sc. No. 29, ed. C.H.Chen. The Netherlands: Sijthoff & Noordhoff, (1978), 575-586. Google Scholar |
[26] |
H. Tong, A view on non-linear time series model building, Time Series, ed. O. D. Anderson,
Amsterdam: North-Holland, 1980, 41–56. |
[27] |
H. Tong,
Threshold Models in Non-linear Time Series Analysis, Lecture Notes in Statistics, No. 21, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4684-7888-4. |
[28] |
H. Tong, Non-linear Time Series: A Dynamical System Approach, Oxford University Press, Oxford, 1990. Google Scholar |
[29] |
W. Xie, L. Yu, L. Xu and S. Wang,
A new method for crude oil price forecasting based on support vector machines, International Conference on Computational Science, (Part Ⅳ), 3994 (2006), 444-451.
doi: 10.1007/11758549_63. |
[30] |
M. Ye, J. Zyren and J. Shore, Forecasting crude oil spot price using OECD petroleum invenvory levels, International Advances in Economic Research, 8 (2002), 324-333. Google Scholar |
[31] |
M. Ye, J. Zyren and J. Shore,
A monthly crude oil spot price forecasting model using relative inventories, International Journal of Forecasting, 21 (2005), 491-501.
doi: 10.1016/j.ijforecast.2005.01.001. |
[32] |
M. Zakai,
On the optimal filtering of diffusion processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 11 (1969), 230-243.
doi: 10.1007/BF00536382. |
show all references
References:
[1] |
W. K. Ching, T. K. Siu, L. M. Li, T. Li and W. K. Li,
Modeling default data via an interactive hidden Markov model, Computational Economics, 34 (2009), 1-19.
doi: 10.1007/s10614-009-9183-5. |
[2] |
E. G. De Souza e Silva, L. Legey and E. A. De Souza e Silva, Forecasting oil price trends using wavelet and hidden Markov models, Energy Economics, 32 (2010), 1507-1519. Google Scholar |
[3] |
F. X. Diebold and R. Mariano, Comparing predictive accuracy, Journal of Business and Economic Statistics, 13 (1995), 253-265. Google Scholar |
[4] |
R. J. Elliott, L. Aggoun and J. Moore,
Hidden Markov Models: Estimation and Control, Springer, New York, 1995. |
[5] |
R. J. Elliott, L. L. Chan and T. K. Siu,
Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432.
doi: 10.1007/s10436-005-0013-z. |
[6] |
R. J. Elliott, C. C. Liew and T. K. Siu,
On filtering and estimation of a threshold stochastic volatility model, Applied Mathematics dand Computation, 218 (2011), 61-75.
doi: 10.1016/j.amc.2011.05.052. |
[7] |
R. J. Elliott, T. K. Siu and A. Badescu,
On mean-variance portfolio selection under a hidden Markovian regime-switching model, Economic Modelling, 27 (2010), 678-686.
doi: 10.1016/j.econmod.2010.01.007. |
[8] |
R. J. Elliott, T. K. Siu and A. Badescu,
On pricing and hedging options in regime-switching models with feedback effect, Journal of Economic Dynamics and Control, 35 (2011), 694-713.
doi: 10.1016/j.jedc.2010.12.014. |
[9] |
R. J. Elliott, T. K. Siu and J. W. Lau,
Filtering a double threshold model with regime switching, IEEE Transactions on Automatic Control, 58 (2013), 3185-3190.
doi: 10.1109/TAC.2013.2261186. |
[10] |
R. J. Elliott, T. K. Siu and H. L. Yang,
Ruin theory in a hidden Markov-modulated risk model, Stochastic Models, 27 (2011), 474-489.
doi: 10.1080/15326349.2011.593408. |
[11] |
R. J. Elliott and J. van der Hoek,
An application of hidden Markov models to asset allocation problems, Finance and Stochastic, 1 (1997), 229-238.
doi: 10.1007/s007800050022. |
[12] |
C. Erlwein and R. Mamon,
An online estimation scheme for a Hull and White model with HMM-driven parameters, Statistical Methods and Applications, 18 (2009), 87-107.
doi: 10.1007/s10260-007-0082-4. |
[13] |
C. Erlwein, R. Mamon and M. Davison,
An examination of HMM-based investment strategies for asset allocation, Applied Stchastic Models in Business and Industry, 27 (2011), 204-221.
doi: 10.1002/asmb.820. |
[14] |
G. Frey, M. Manera, A. Markandya and E. Scarpa, Econometric models for oil price forecasting, A Critical Survey CESinfo Forum, 10 (2009), 29-44. Google Scholar |
[15] |
D. Harding and A. R. Pagan, Synchronisation of Cycles, mimeo., Australian National University. Google Scholar |
[16] |
H. G. Huntington,
Oil price forecasting in the 1980s: What went wrong?, The Energy Journal, 15 (1994), 1-22.
doi: 10.5547/ISSN0195-6574-EJ-Vol15-No2-1. |
[17] |
R. K. Kaufman, S. Dees, P. Karadeloglou and M. Sanchez,
Does OPEC matter? An econometric analysis of oil prices, The Energy Journal, 25 (2004), 67-90.
doi: 10.5547/ISSN0195-6574-EJ-Vol25-No4-4. |
[18] |
M. W. Korolkiewica and R. J. Elliott, Smoothed parameter estimation for a Markov model of credit quality, Hidden Markov Models in Finance, Springer, New York, 104 (2007), 69–90.
doi: 10.1007/0-387-71163-5_5. |
[19] |
C. C. Liew and T. K. Siu,
A hidden Markov regime-switching model for option valuation, Insurance: Mathematics and Economics, 47 (2010), 374-384.
doi: 10.1016/j.insmatheco.2010.08.003. |
[20] |
R. S. Mamon, C. Erlwein and R. B. Gopaluni,
Adaptive signal processing of asset price dynamics with predictability analysis, Information Sciences, 178 (2008), 203-219.
doi: 10.1016/j.ins.2007.05.021. |
[21] |
T. K. Siu, W. K. Ching, E. Fung, M. Ng and X. Li,
A high-order Markov-switching model for risk measurement, Computers and Mathematics with Applications, 58 (2009), 1-10.
doi: 10.1016/j.camwa.2008.10.099. |
[22] |
T. K. Siu, W. K. Ching, E. S. Fung and M. K. Ng, Extracting information from spot interest rates and credit ratings using double higher-order hidden Markov models, Computational Economics, 26 (2005), 251-284. Google Scholar |
[23] |
S. J. Taylor,
Asset Price Dynamics, Volatility, and Prediction, Princeton University Press, Princeton, 2005.
doi: 10.1515/9781400839254. |
[24] |
H. Tong,
Determination of the order of a Markov chain by Akaike's information criterion, Journal of applied probability, 12 (1975), 488-497.
doi: 10.1017/S0021900200048294. |
[25] |
H. Tong, On a threshold model, Pattern Recognition and signal processing, NATO ASI Series E: Applied Sc. No. 29, ed. C.H.Chen. The Netherlands: Sijthoff & Noordhoff, (1978), 575-586. Google Scholar |
[26] |
H. Tong, A view on non-linear time series model building, Time Series, ed. O. D. Anderson,
Amsterdam: North-Holland, 1980, 41–56. |
[27] |
H. Tong,
Threshold Models in Non-linear Time Series Analysis, Lecture Notes in Statistics, No. 21, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4684-7888-4. |
[28] |
H. Tong, Non-linear Time Series: A Dynamical System Approach, Oxford University Press, Oxford, 1990. Google Scholar |
[29] |
W. Xie, L. Yu, L. Xu and S. Wang,
A new method for crude oil price forecasting based on support vector machines, International Conference on Computational Science, (Part Ⅳ), 3994 (2006), 444-451.
doi: 10.1007/11758549_63. |
[30] |
M. Ye, J. Zyren and J. Shore, Forecasting crude oil spot price using OECD petroleum invenvory levels, International Advances in Economic Research, 8 (2002), 324-333. Google Scholar |
[31] |
M. Ye, J. Zyren and J. Shore,
A monthly crude oil spot price forecasting model using relative inventories, International Journal of Forecasting, 21 (2005), 491-501.
doi: 10.1016/j.ijforecast.2005.01.001. |
[32] |
M. Zakai,
On the optimal filtering of diffusion processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 11 (1969), 230-243.
doi: 10.1007/BF00536382. |




It=5, |
It=10, |
It=15, |
|||||||||
MAE | RMSE | SSE | MAE | RMSE | SSE | MAE | RMSE | SSE | |||
0.2105 | 0.3055 | 4.6652 | 0.2373 | 0.3232 | 5.2230 | 0.2583 | 0.3462 | 5.9937 | |||
0.3028 | 0.3936 | 7.7456 | 0.3021 | 0.3773 | 7.1170 | 0.2821 | 0.3643 | 6.6375 | |||
0.2372 | 0.3330 | 5.5452 | 0.2484 | 0.3258 | 5.3082 | 0.2781 | 0.3745 | 7.0111 | |||
0.1641 | 0.2285 | 2.6099 | 0.1804 | 0.2404 | 2.8899 | 0.2321 | 0.3385 | 5.7275 | |||
0.2396 | 0.3266 | 5.3347 | 0.2408 | 0.3287 | 5.4033 | 0.2494 | 0.3350 | 5.6124 | |||
0.2243 | 0.3114 | 4.8493 | 0.2462 | 0.3250 | 5.2813 | 0.2115 | 0.2788 | 3.8873 | |||
0.3038 | 0.3637 | 6.6139 | 0.3768 | 0.4281 | 9.1632 | 0.2863 | 0.3344 | 5.5903 | |||
0.3456 | 0.4060 | 8.2436 | 0.2979 | 0.3799 | 7.2170 | 0.3022 | 0.3526 | 6.2180 | |||
0.2419 | 0.3059 | 4.6793 | 0.3467 | 0.4280 | 9.1571 | 0.3021 | 0.3642 | 6.6336 | |||
0.2338 | 0.2981 | 4.4419 | 0.2488 | 0.3076 | 4.7312 | 0.2556 | 0.3086 | 4.7619 | |||
0.2895 | 0.3597 | 6.4692 | 0.2979 | 0.3631 | 6.5927 | 0.3408 | 0.4053 | 8.2130 | |||
0.2913 | 0.3469 | 6.0166 | 0.3329 | 0.3895 | 7.5851 | 0.3155 | 0.3786 | 7.1687 | |||
0.2140 | 0.2742 | 3.7592 | 0.2687 | 0.3223 | 5.1940 | 0.3125 | 0.3757 | 7.0575 | |||
0.3032 | 0.3512 | 6.1654 | 0.2646 | 0.3207 | 5.1414 | 0.2750 | 0.3268 | 5.3387 | |||
0.2498 | 0.3233 | 5.2261 | 0.2997 | 0.3510 | 6.1592 | 0.3013 | 0.3629 | 6.5848 | |||
0.2499 | 0.2963 | 4.3883 | 0.2597 | 0.3260 | 5.3139 | 0.2878 | 0.3671 | 6.7391 | |||
0.3213 | 0.3809 | 7.2547 | 0.2946 | 0.3569 | 6.3691 | 0.2896 | 0.3631 | 6.5922 | |||
0.2939 | 0.3453 | 5.9599 | 0.2746 | 0.3301 | 5.4489 | 0.3085 | 0.3638 | 6.6193 | |||
0.2613 | 0.3335 | 5.5595 | 0.2789 | 0.3375 | 5.6964 | 0.3076 | 0.3629 | 6.5858 | |||
0.2988 | 0.3511 | 6.1621 | 0.2791 | 0.3417 | 5.8363 | 0.3134 | 0.3695 | 6.8273 | |||
0.3027 | 0.3638 | 6.6178 | 0.3301 | 0.3938 | 7.7553 | 0.3298 | 0.3947 | 7.7913 | |||
0.3093 | 0.3620 | 6.5520 | 0.2088 | 0.2808 | 3.9438 | 0.2558 | 0.3121 | 4.8694 | |||
0.2781 | 0.3270 | 5.3472 | 0.2779 | 0.3310 | 5.4777 | 0.2969 | 0.3636 | 6.6100 | |||
0.3370 | 0.4104 | 8.4216 | 0.3540 | 0.4139 | 8.5662 | 0.3264 | 0.3925 | 7.7025 | |||
1.3506 | 2.0633 | 212.8582 | 0.8303 | 1.4174 | 100.4480 | 0.5571 | 0.8851 | 39.1680 |
It=5, |
It=10, |
It=15, |
|||||||||
MAE | RMSE | SSE | MAE | RMSE | SSE | MAE | RMSE | SSE | |||
0.2105 | 0.3055 | 4.6652 | 0.2373 | 0.3232 | 5.2230 | 0.2583 | 0.3462 | 5.9937 | |||
0.3028 | 0.3936 | 7.7456 | 0.3021 | 0.3773 | 7.1170 | 0.2821 | 0.3643 | 6.6375 | |||
0.2372 | 0.3330 | 5.5452 | 0.2484 | 0.3258 | 5.3082 | 0.2781 | 0.3745 | 7.0111 | |||
0.1641 | 0.2285 | 2.6099 | 0.1804 | 0.2404 | 2.8899 | 0.2321 | 0.3385 | 5.7275 | |||
0.2396 | 0.3266 | 5.3347 | 0.2408 | 0.3287 | 5.4033 | 0.2494 | 0.3350 | 5.6124 | |||
0.2243 | 0.3114 | 4.8493 | 0.2462 | 0.3250 | 5.2813 | 0.2115 | 0.2788 | 3.8873 | |||
0.3038 | 0.3637 | 6.6139 | 0.3768 | 0.4281 | 9.1632 | 0.2863 | 0.3344 | 5.5903 | |||
0.3456 | 0.4060 | 8.2436 | 0.2979 | 0.3799 | 7.2170 | 0.3022 | 0.3526 | 6.2180 | |||
0.2419 | 0.3059 | 4.6793 | 0.3467 | 0.4280 | 9.1571 | 0.3021 | 0.3642 | 6.6336 | |||
0.2338 | 0.2981 | 4.4419 | 0.2488 | 0.3076 | 4.7312 | 0.2556 | 0.3086 | 4.7619 | |||
0.2895 | 0.3597 | 6.4692 | 0.2979 | 0.3631 | 6.5927 | 0.3408 | 0.4053 | 8.2130 | |||
0.2913 | 0.3469 | 6.0166 | 0.3329 | 0.3895 | 7.5851 | 0.3155 | 0.3786 | 7.1687 | |||
0.2140 | 0.2742 | 3.7592 | 0.2687 | 0.3223 | 5.1940 | 0.3125 | 0.3757 | 7.0575 | |||
0.3032 | 0.3512 | 6.1654 | 0.2646 | 0.3207 | 5.1414 | 0.2750 | 0.3268 | 5.3387 | |||
0.2498 | 0.3233 | 5.2261 | 0.2997 | 0.3510 | 6.1592 | 0.3013 | 0.3629 | 6.5848 | |||
0.2499 | 0.2963 | 4.3883 | 0.2597 | 0.3260 | 5.3139 | 0.2878 | 0.3671 | 6.7391 | |||
0.3213 | 0.3809 | 7.2547 | 0.2946 | 0.3569 | 6.3691 | 0.2896 | 0.3631 | 6.5922 | |||
0.2939 | 0.3453 | 5.9599 | 0.2746 | 0.3301 | 5.4489 | 0.3085 | 0.3638 | 6.6193 | |||
0.2613 | 0.3335 | 5.5595 | 0.2789 | 0.3375 | 5.6964 | 0.3076 | 0.3629 | 6.5858 | |||
0.2988 | 0.3511 | 6.1621 | 0.2791 | 0.3417 | 5.8363 | 0.3134 | 0.3695 | 6.8273 | |||
0.3027 | 0.3638 | 6.6178 | 0.3301 | 0.3938 | 7.7553 | 0.3298 | 0.3947 | 7.7913 | |||
0.3093 | 0.3620 | 6.5520 | 0.2088 | 0.2808 | 3.9438 | 0.2558 | 0.3121 | 4.8694 | |||
0.2781 | 0.3270 | 5.3472 | 0.2779 | 0.3310 | 5.4777 | 0.2969 | 0.3636 | 6.6100 | |||
0.3370 | 0.4104 | 8.4216 | 0.3540 | 0.4139 | 8.5662 | 0.3264 | 0.3925 | 7.7025 | |||
1.3506 | 2.0633 | 212.8582 | 0.8303 | 1.4174 | 100.4480 | 0.5571 | 0.8851 | 39.1680 |
Step | MAE | MAPE | RMSE | ||||||||
THMM | HMM | ARMA | THMM | HMM | ARMA | THMM | HMM | ARMA | |||
1 | 0.6571 | 0.6772 | 38.5056 | 70.1171 | 70.1162 | 70.6075 | 1.2772 | 1.2900 | 44.6933 | ||
2 | 0.8618 | 0.8767 | 38.4588 | 70.1171 | 70.1157 | 70.6071 | 1.5165 | 1.5291 | 44.6933 | ||
3 | 1.0943 | 1.1075 | 38.2271 | 70.1170 | 70.1152 | 70.6048 | 1.7544 | 1.7672 | 44.3661 | ||
4 | 1.3337 | 1.3252 | 38.5154 | 70.1191 | 70.1168 | 70.6107 | 2.1103 | 2.1094 | 44.4763 | ||
5 | 1.5809 | 1.5720 | 38.1649 | 70.1200 | 70.1171 | 70.6049 | 2.4751 | 2.4744 | 44.2075 | ||
6 | 1.7653 | 1.7512 | 39.0297 | 70.1190 | 70.1157 | 70.6191 | 2.6917 | 2.6858 | 44.8844 | ||
7 | 1.9091 | 1.9004 | 37.9088 | 70.1182 | 70.1143 | 70.6022 | 2.8747 | 2.8849 | 43.8580 | ||
8 | 2.0631 | 2.0395 | 36.8175 | 70.1197 | 70.1153 | 70.5854 | 3.1950 | 3.1885 | 42.8849 | ||
9 | 2.3027 | 2.2690 | 37.0833 | 70.1225 | 70.1176 | 70.5901 | 3.4312 | 3.4062 | 43.0634 | ||
10 | 2.4098 | 2.3912 | 37.0323 | 70.1221 | 70.1168 | 70.5893 | 3.6031 | 3.6021 | 43.0214 | ||
11 | 2.4520 | 2.4237 | 38.0120 | 70.1213 | 70.1153 | 70.6041 | 3.6323 | 3.6301 | 43.9126 | ||
12 | 2.7095 | 2.6502 | 38.3527 | 70.1256 | 70.1192 | 70.6088 | 3.9464 | 3.9046 | 44.2681 | ||
13 | 2.7674 | 2.7020 | 38.2748 | 70.1234 | 70.1166 | 70.6077 | 3.9701 | 3.9416 | 44.1929 | ||
14 | 3.1003 | 3.0394 | 38.7028 | 70.1207 | 70.1132 | 70.6143 | 4.6993 | 4.7048 | 44.5676 | ||
15 | 3.3073 | 3.2407 | 39.8606 | 70.1216 | 70.1138 | 70.6318 | 4.8768 | 4.8855 | 45.6269 | ||
16 | 3.0475 | 2.9143 | 39.8835 | 70.1256 | 70.1173 | 70.6321 | 4.2991 | 4.2164 | 45.6542 | ||
17 | 3.2624 | 3.2002 | 38.7757 | 70.1220 | 70.1131 | 70.6147 | 4.9488 | 4.9643 | 44.7068 | ||
18 | 3.8019 | 3.7099 | 38.6336 | 70.1287 | 70.1195 | 70.6121 | 5.3424 | 5.2526 | 44.6244 | ||
19 | 3.3387 | 3.2340 | 39.8483 | 70.1258 | 70.1160 | 70.6302 | 4.6491 | 4.5823 | 45.7631 | ||
20 | 3.5091 | 3.4241 | 39.9252 | 70.1265 | 70.1162 | 70.6313 | 5.0282 | 4.9773 | 45.8393 |
Step | MAE | MAPE | RMSE | ||||||||
THMM | HMM | ARMA | THMM | HMM | ARMA | THMM | HMM | ARMA | |||
1 | 0.6571 | 0.6772 | 38.5056 | 70.1171 | 70.1162 | 70.6075 | 1.2772 | 1.2900 | 44.6933 | ||
2 | 0.8618 | 0.8767 | 38.4588 | 70.1171 | 70.1157 | 70.6071 | 1.5165 | 1.5291 | 44.6933 | ||
3 | 1.0943 | 1.1075 | 38.2271 | 70.1170 | 70.1152 | 70.6048 | 1.7544 | 1.7672 | 44.3661 | ||
4 | 1.3337 | 1.3252 | 38.5154 | 70.1191 | 70.1168 | 70.6107 | 2.1103 | 2.1094 | 44.4763 | ||
5 | 1.5809 | 1.5720 | 38.1649 | 70.1200 | 70.1171 | 70.6049 | 2.4751 | 2.4744 | 44.2075 | ||
6 | 1.7653 | 1.7512 | 39.0297 | 70.1190 | 70.1157 | 70.6191 | 2.6917 | 2.6858 | 44.8844 | ||
7 | 1.9091 | 1.9004 | 37.9088 | 70.1182 | 70.1143 | 70.6022 | 2.8747 | 2.8849 | 43.8580 | ||
8 | 2.0631 | 2.0395 | 36.8175 | 70.1197 | 70.1153 | 70.5854 | 3.1950 | 3.1885 | 42.8849 | ||
9 | 2.3027 | 2.2690 | 37.0833 | 70.1225 | 70.1176 | 70.5901 | 3.4312 | 3.4062 | 43.0634 | ||
10 | 2.4098 | 2.3912 | 37.0323 | 70.1221 | 70.1168 | 70.5893 | 3.6031 | 3.6021 | 43.0214 | ||
11 | 2.4520 | 2.4237 | 38.0120 | 70.1213 | 70.1153 | 70.6041 | 3.6323 | 3.6301 | 43.9126 | ||
12 | 2.7095 | 2.6502 | 38.3527 | 70.1256 | 70.1192 | 70.6088 | 3.9464 | 3.9046 | 44.2681 | ||
13 | 2.7674 | 2.7020 | 38.2748 | 70.1234 | 70.1166 | 70.6077 | 3.9701 | 3.9416 | 44.1929 | ||
14 | 3.1003 | 3.0394 | 38.7028 | 70.1207 | 70.1132 | 70.6143 | 4.6993 | 4.7048 | 44.5676 | ||
15 | 3.3073 | 3.2407 | 39.8606 | 70.1216 | 70.1138 | 70.6318 | 4.8768 | 4.8855 | 45.6269 | ||
16 | 3.0475 | 2.9143 | 39.8835 | 70.1256 | 70.1173 | 70.6321 | 4.2991 | 4.2164 | 45.6542 | ||
17 | 3.2624 | 3.2002 | 38.7757 | 70.1220 | 70.1131 | 70.6147 | 4.9488 | 4.9643 | 44.7068 | ||
18 | 3.8019 | 3.7099 | 38.6336 | 70.1287 | 70.1195 | 70.6121 | 5.3424 | 5.2526 | 44.6244 | ||
19 | 3.3387 | 3.2340 | 39.8483 | 70.1258 | 70.1160 | 70.6302 | 4.6491 | 4.5823 | 45.7631 | ||
20 | 3.5091 | 3.4241 | 39.9252 | 70.1265 | 70.1162 | 70.6313 | 5.0282 | 4.9773 | 45.8393 |
Step | Harding-Pagan Test | ||
THMM | HMM | ARMA | |
1 | 0.3475 | 0.5111 | 0.4737 |
2 | 0.2158 | 0.2786 | 0.4569 |
3 | 0.1403 | 0.1494 | 0.4443 |
4 | 0.1479 | 0.1515 | 0.4559 |
5 | 0.1378 | 0.1337 | 0.4382 |
6 | 0.1474 | 0.1489 | 0.4321 |
7 | 0.1312 | 0.1302 | 0.4311 |
8 | 0.1398 | 0.1383 | 0.4265 |
9 | 0.1398 | 0.1393 | 0.4235 |
10 | 0.1530 | 0.1570 | 0.4281 |
11 | 0.1555 | 0.1525 | 0.4326 |
12 | 0.1520 | 0.1459 | 0.4265 |
13 | 0.1575 | 0.1494 | 0.4306 |
14 | 0.1611 | 0.1631 | 0.4250 |
15 | 0.1601 | 0.1575 | 0.4250 |
16 | 0.1702 | 0.1636 | 0.4235 |
17 | 0.1499 | 0.1525 | 0.4159 |
18 | 0.1738 | 0.1651 | 0.4169 |
19 | 0.1651 | 0.1520 | 0.4139 |
20 | 0.1550 | 0.1550 | 0.4144 |
Step | Harding-Pagan Test | ||
THMM | HMM | ARMA | |
1 | 0.3475 | 0.5111 | 0.4737 |
2 | 0.2158 | 0.2786 | 0.4569 |
3 | 0.1403 | 0.1494 | 0.4443 |
4 | 0.1479 | 0.1515 | 0.4559 |
5 | 0.1378 | 0.1337 | 0.4382 |
6 | 0.1474 | 0.1489 | 0.4321 |
7 | 0.1312 | 0.1302 | 0.4311 |
8 | 0.1398 | 0.1383 | 0.4265 |
9 | 0.1398 | 0.1393 | 0.4235 |
10 | 0.1530 | 0.1570 | 0.4281 |
11 | 0.1555 | 0.1525 | 0.4326 |
12 | 0.1520 | 0.1459 | 0.4265 |
13 | 0.1575 | 0.1494 | 0.4306 |
14 | 0.1611 | 0.1631 | 0.4250 |
15 | 0.1601 | 0.1575 | 0.4250 |
16 | 0.1702 | 0.1636 | 0.4235 |
17 | 0.1499 | 0.1525 | 0.4159 |
18 | 0.1738 | 0.1651 | 0.4169 |
19 | 0.1651 | 0.1520 | 0.4139 |
20 | 0.1550 | 0.1550 | 0.4144 |
Step | DM Test based on THMM and HMM | DM Test based on THMM and ARMA |
1 | -7.3944 | -39.8539 |
2 | -3.5605 | -23.0956 |
3 | -1.9302 | -18.0701 |
4 | 0.0952 | -15.6151 |
5 | 0.0506 | -13.6254 |
6 | 0.3246 | -12.6574 |
7 | -0.4322 | -11.4710 |
8 | 0.2511 | -10.4746 |
9 | 0.8516 | -9.9699 |
10 | 0.0270 | -9.4191 |
11 | 0.0532 | -9.0919 |
12 | 0.9990 | -8.6963 |
13 | 0.5196 | -8.3548 |
14 | -0.0829 | -8.0797 |
15 | -0.1185 | -7.9669 |
16 | 1.2755 | -7.6941 |
17 | -0.1728 | -7.2728 |
18 | 1.1428 | -6.9931 |
19 | 0.8583 | -6.9341 |
20 | 0.5422 | -6.7681 |
Step | DM Test based on THMM and HMM | DM Test based on THMM and ARMA |
1 | -7.3944 | -39.8539 |
2 | -3.5605 | -23.0956 |
3 | -1.9302 | -18.0701 |
4 | 0.0952 | -15.6151 |
5 | 0.0506 | -13.6254 |
6 | 0.3246 | -12.6574 |
7 | -0.4322 | -11.4710 |
8 | 0.2511 | -10.4746 |
9 | 0.8516 | -9.9699 |
10 | 0.0270 | -9.4191 |
11 | 0.0532 | -9.0919 |
12 | 0.9990 | -8.6963 |
13 | 0.5196 | -8.3548 |
14 | -0.0829 | -8.0797 |
15 | -0.1185 | -7.9669 |
16 | 1.2755 | -7.6941 |
17 | -0.1728 | -7.2728 |
18 | 1.1428 | -6.9931 |
19 | 0.8583 | -6.9341 |
20 | 0.5422 | -6.7681 |
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