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

Dong-Mei Zhu; Wai-Ki Ching; Robert J. Elliott; Tak-Kuen Siu; Lianmin Zhang

doi:10.3934/jimo.2016045

Article Contents

2017, Volume 13, Issue 2: 757-773. Doi: 10.3934/jimo.2016045

This issue Previous Article Optimal reinsurance and investment strategy with two piece utility function Next Article Auction and contracting mechanisms for channel coordination with consideration of participants' risk attitudes

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

* Corresponding author
* Corresponding author

Received: August 31, 2015

Revised: May 31, 2016

Published: August 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 90C40, 90C15; Secondary: 60J05.

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Figure 1. µ

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Figure 2. σ

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Figure 3. A¹

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Figure 4. A²

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Figure 5. Prediction

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Figure 6. Year 2011

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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

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Table 2. Accuracy of predictions

Step	MAE			MAPE			RMSE
Step	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

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Table 3. Directional Forecasts

Step	Harding-Pagan Test
Step	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

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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

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