American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Auction and contracting mechanisms for channel coordination with consideration of participants' risk attitudes
  • JIMO Home
  • This Issue
  • Next Article
    Optimal reinsurance and investment strategy with two piece utility function
April  2017, 13(2): 757-773. doi: 10.3934/jimo.2016045

Hidden Markov models with threshold effects and their applications to oil price forecasting

Dong-Mei Zhu 1, , Wai-Ki Ching 2, , Robert J. Elliott 3,4, , Tak-Kuen Siu 5, and  Lianmin Zhang 6,,

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

* Corresponding author

Received  September 2015 Revised  June 2016 Published  August 2016

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.

Keywords: Hidden Markov Model, filtering, threshold effect, oil price, forecasting.
Mathematics Subject Classification: Primary: 90C40, 90C15; Secondary: 60J05.
Citation: Dong-Mei Zhu, Wai-Ki Ching, Robert J. Elliott, Tak-Kuen Siu, Lianmin Zhang. Hidden Markov models with threshold effects and their applications to oil price forecasting. Journal of Industrial & Management Optimization, 2017, 13 (2) : 757-773. doi: 10.3934/jimo.2016045
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

M. Zakai, On the optimal filtering of diffusion processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 11 (1969), 230-243.  doi: 10.1007/BF00536382.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

M. Zakai, On the optimal filtering of diffusion processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 11 (1969), 230-243.  doi: 10.1007/BF00536382.  Google Scholar

Figure 1.  µ 
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  σ 
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  A1 
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  A2 
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Prediction
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Year 2011
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Accuracy of estimated parameters
It=5, ${\rm It}_{\rm r}$=10 It=10, ${\rm It}_{\rm r}$=20 It=15, ${\rm It}_{\rm r}$=30
MAE RMSE SSE MAE RMSE SSE MAE RMSE SSE
$\mu_1$ 0.2105 0.3055 4.6652 0.2373 0.3232 5.2230 0.2583 0.3462 5.9937
$\mu_2$ 0.3028 0.3936 7.7456 0.3021 0.3773 7.1170 0.2821 0.3643 6.6375
$\mu_3$ 0.2372 0.3330 5.5452 0.2484 0.3258 5.3082 0.2781 0.3745 7.0111
$\sigma_1$ 0.1641 0.2285 2.6099 0.1804 0.2404 2.8899 0.2321 0.3385 5.7275
$\sigma_2$ 0.2396 0.3266 5.3347 0.2408 0.3287 5.4033 0.2494 0.3350 5.6124
$\sigma_3$ 0.2243 0.3114 4.8493 0.2462 0.3250 5.2813 0.2115 0.2788 3.8873
$a^1_{11}$ 0.3038 0.3637 6.6139 0.3768 0.4281 9.1632 0.2863 0.3344 5.5903
$a^1_{12}$ 0.3456 0.4060 8.2436 0.2979 0.3799 7.2170 0.3022 0.3526 6.2180
$a^1_{13}$ 0.2419 0.3059 4.6793 0.3467 0.4280 9.1571 0.3021 0.3642 6.6336
$a^1_{21}$ 0.2338 0.2981 4.4419 0.2488 0.3076 4.7312 0.2556 0.3086 4.7619
$a^1_{22}$ 0.2895 0.3597 6.4692 0.2979 0.3631 6.5927 0.3408 0.4053 8.2130
$a^1_{23}$ 0.2913 0.3469 6.0166 0.3329 0.3895 7.5851 0.3155 0.3786 7.1687
$a^1_{31}$ 0.2140 0.2742 3.7592 0.2687 0.3223 5.1940 0.3125 0.3757 7.0575
$a^1_{32}$ 0.3032 0.3512 6.1654 0.2646 0.3207 5.1414 0.2750 0.3268 5.3387
$a^1_{33}$ 0.2498 0.3233 5.2261 0.2997 0.3510 6.1592 0.3013 0.3629 6.5848
$a^2_{11}$ 0.2499 0.2963 4.3883 0.2597 0.3260 5.3139 0.2878 0.3671 6.7391
$a^2_{12}$ 0.3213 0.3809 7.2547 0.2946 0.3569 6.3691 0.2896 0.3631 6.5922
$a^2_{13}$ 0.2939 0.3453 5.9599 0.2746 0.3301 5.4489 0.3085 0.3638 6.6193
$a^2_{21}$ 0.2613 0.3335 5.5595 0.2789 0.3375 5.6964 0.3076 0.3629 6.5858
$a^2_{22}$ 0.2988 0.3511 6.1621 0.2791 0.3417 5.8363 0.3134 0.3695 6.8273
$a^2_{23}$ 0.3027 0.3638 6.6178 0.3301 0.3938 7.7553 0.3298 0.3947 7.7913
$a^2_{31}$ 0.3093 0.3620 6.5520 0.2088 0.2808 3.9438 0.2558 0.3121 4.8694
$a^2_{32}$ 0.2781 0.3270 5.3472 0.2779 0.3310 5.4777 0.2969 0.3636 6.6100
$a^2_{33}$ 0.3370 0.4104 8.4216 0.3540 0.4139 8.5662 0.3264 0.3925 7.7025
$r$ 1.3506 2.0633 212.8582 0.8303 1.4174 100.4480 0.5571 0.8851 39.1680
Table Options

Download as excel

It=5, ${\rm It}_{\rm r}$=10 It=10, ${\rm It}_{\rm r}$=20 It=15, ${\rm It}_{\rm r}$=30
MAE RMSE SSE MAE RMSE SSE MAE RMSE SSE
$\mu_1$ 0.2105 0.3055 4.6652 0.2373 0.3232 5.2230 0.2583 0.3462 5.9937
$\mu_2$ 0.3028 0.3936 7.7456 0.3021 0.3773 7.1170 0.2821 0.3643 6.6375
$\mu_3$ 0.2372 0.3330 5.5452 0.2484 0.3258 5.3082 0.2781 0.3745 7.0111
$\sigma_1$ 0.1641 0.2285 2.6099 0.1804 0.2404 2.8899 0.2321 0.3385 5.7275
$\sigma_2$ 0.2396 0.3266 5.3347 0.2408 0.3287 5.4033 0.2494 0.3350 5.6124
$\sigma_3$ 0.2243 0.3114 4.8493 0.2462 0.3250 5.2813 0.2115 0.2788 3.8873
$a^1_{11}$ 0.3038 0.3637 6.6139 0.3768 0.4281 9.1632 0.2863 0.3344 5.5903
$a^1_{12}$ 0.3456 0.4060 8.2436 0.2979 0.3799 7.2170 0.3022 0.3526 6.2180
$a^1_{13}$ 0.2419 0.3059 4.6793 0.3467 0.4280 9.1571 0.3021 0.3642 6.6336
$a^1_{21}$ 0.2338 0.2981 4.4419 0.2488 0.3076 4.7312 0.2556 0.3086 4.7619
$a^1_{22}$ 0.2895 0.3597 6.4692 0.2979 0.3631 6.5927 0.3408 0.4053 8.2130
$a^1_{23}$ 0.2913 0.3469 6.0166 0.3329 0.3895 7.5851 0.3155 0.3786 7.1687
$a^1_{31}$ 0.2140 0.2742 3.7592 0.2687 0.3223 5.1940 0.3125 0.3757 7.0575
$a^1_{32}$ 0.3032 0.3512 6.1654 0.2646 0.3207 5.1414 0.2750 0.3268 5.3387
$a^1_{33}$ 0.2498 0.3233 5.2261 0.2997 0.3510 6.1592 0.3013 0.3629 6.5848
$a^2_{11}$ 0.2499 0.2963 4.3883 0.2597 0.3260 5.3139 0.2878 0.3671 6.7391
$a^2_{12}$ 0.3213 0.3809 7.2547 0.2946 0.3569 6.3691 0.2896 0.3631 6.5922
$a^2_{13}$ 0.2939 0.3453 5.9599 0.2746 0.3301 5.4489 0.3085 0.3638 6.6193
$a^2_{21}$ 0.2613 0.3335 5.5595 0.2789 0.3375 5.6964 0.3076 0.3629 6.5858
$a^2_{22}$ 0.2988 0.3511 6.1621 0.2791 0.3417 5.8363 0.3134 0.3695 6.8273
$a^2_{23}$ 0.3027 0.3638 6.6178 0.3301 0.3938 7.7553 0.3298 0.3947 7.7913
$a^2_{31}$ 0.3093 0.3620 6.5520 0.2088 0.2808 3.9438 0.2558 0.3121 4.8694
$a^2_{32}$ 0.2781 0.3270 5.3472 0.2779 0.3310 5.4777 0.2969 0.3636 6.6100
$a^2_{33}$ 0.3370 0.4104 8.4216 0.3540 0.4139 8.5662 0.3264 0.3925 7.7025
$r$ 1.3506 2.0633 212.8582 0.8303 1.4174 100.4480 0.5571 0.8851 39.1680
Table 2.  Accuracy of predictions
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
Table Options

Download as excel

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
Table 3.  Directional Forecasts
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
Table Options

Download as excel

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
Table 4.  The Diebold-Mariano Test
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
Table Options

Download as excel

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
[1]

Yang Shen, Tak Kuen Siu. Consumption-portfolio optimization and filtering in a hidden Markov-modulated asset price model. Journal of Industrial & Management Optimization, 2017, 13 (1) : 23-46. doi: 10.3934/jimo.2016002
[2]

Shichen Zhang, Jianxiong Zhang, Jiang Shen, Wansheng Tang. A joint dynamic pricing and production model with asymmetric reference price effect. Journal of Industrial & Management Optimization, 2019, 15 (2) : 667-688. doi: 10.3934/jimo.2018064
[3]

Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132
[4]

Roberto C. Alamino, Nestor Caticha. Bayesian online algorithms for learning in discrete hidden Markov models. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 1-10. doi: 10.3934/dcdsb.2008.9.1
[5]

H. Thomas Banks, Shuhua Hu, Zackary R. Kenz, Hien T. Tran. A comparison of nonlinear filtering approaches in the context of an HIV model. Mathematical Biosciences & Engineering, 2010, 7 (2) : 213-236. doi: 10.3934/mbe.2010.7.213
[6]

Min Li, Jiahua Zhang, Yifan Xu, Wei Wang. Effect of disruption risk on a supply chain with price-dependent demand. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2019095
[7]

Xinli Hu. Threshold dynamics for a Tuberculosis model with seasonality. Mathematical Biosciences & Engineering, 2012, 9 (1) : 111-122. doi: 10.3934/mbe.2012.9.111
[8]

Elham Mardaneh, Ryan Loxton, Qun Lin, Phil Schmidli. A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1601-1623. doi: 10.3934/jimo.2017009
[9]

Hans-Otto Walther. Convergence to square waves for a price model with delay. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1325-1342. doi: 10.3934/dcds.2005.13.1325
[10]

Ábel Garab, Veronika Kovács, Tibor Krisztin. Global stability of a price model with multiple delays. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6855-6871. doi: 10.3934/dcds.2016098
[11]

Zhenguo Bai, Yicang Zhou. Threshold dynamics of a bacillary dysentery model with seasonal fluctuation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 1-14. doi: 10.3934/dcdsb.2011.15.1
[12]

Mitali Sarkar, Young Hae Lee. Optimum pricing strategy for complementary products with reservation price in a supply chain model. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1553-1586. doi: 10.3934/jimo.2017007
[13]

Yang Yang, Kaiyong Wang, Jiajun Liu, Zhimin Zhang. Asymptotics for a bidimensional risk model with two geometric Lévy price processes. Journal of Industrial & Management Optimization, 2019, 15 (2) : 481-505. doi: 10.3934/jimo.2018053
[14]

Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207
[15]

Gerasimos G. Rigatos, Efthymia G. Rigatou, Jean Daniel Djida. Change detection in the dynamics of an intracellular protein synthesis model using nonlinear Kalman filtering. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1017-1035. doi: 10.3934/mbe.2015.12.1017
[16]

Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039
[17]

Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080
[18]

Toshikazu Kuniya, Yoshiaki Muroya, Yoichi Enatsu. Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1375-1393. doi: 10.3934/mbe.2014.11.1375
[19]

Hui Meng, Fei Lung Yuen, Tak Kuen Siu, Hailiang Yang. Optimal portfolio in a continuous-time self-exciting threshold model. Journal of Industrial & Management Optimization, 2013, 9 (2) : 487-504. doi: 10.3934/jimo.2013.9.487
[20]

Yanan Zhao, Daqing Jiang, Xuerong Mao, Alison Gray. The threshold of a stochastic SIRS epidemic model in a population with varying size. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1277-1295. doi: 10.3934/dcdsb.2015.20.1277

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (77)
  • HTML views (420)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences