Local Lagged Adapted Generalized Method of Moments: An Innovative Estimation and Forecasting Approach and its Applications

Olusegun M. Otunuga; Gangaram S. Ladde; Nathan G. Ladde

doi:10.1515/jtse-2016-0024

Published by De Gruyter January 23, 2019

Local Lagged Adapted Generalized Method of Moments: An Innovative Estimation and Forecasting Approach and its Applications

Olusegun M. Otunuga , Gangaram S. Ladde and Nathan G. Ladde

From the journal Journal of Time Series Econometrics

https://doi.org/10.1515/jtse-2016-0024

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Abstract

In this work, an attempt is made to apply the Local Lagged Adapted Generalized Method of Moments (LLGMM) to estimate state and parameters in stochastic differential dynamic models. The development of LLGMM is motivated by parameter and state estimation problems in continuous-time nonlinear and non-stationary stochastic dynamic model validation problems in biological, chemical, engineering, energy commodity markets, financial, medical, military, physical sciences and social sciences. The byproducts of this innovative approach (LLGMM) are the balance between model specification and model prescription of continuous-time dynamic process and the development of discrete-time interconnected dynamic model of local sample mean and variance statistic process (DTIDMLSMVSP). Moreover, LLGMM is a dynamic non-parametric method. The DTIDMLSMVSP is an alternative approach to the GARCH(1,1) model, and it provides an iterative scheme for updating statistic coefficients in a system of generalized method of moment/observation equations. Furthermore, applications of LLGMM to energy commodities price, U.S. Treasury Bill interest rate and the U.S.–U.K. foreign exchange rate data strongly exhibit its unique role, scope and performance, in particular, in forecasting and confidence-interval problems in applied statistics.

Keywords: conceptual computational/theoretical parameter estimation scheme; method of moments; nonparametric; simulation; forecasting; mean square optimal procedure; reaction/response time delay

JEL Classification: 37M05; 37M10; 62M10; 62G05; 62P05

Article note

U.S. Patent Pending.

Acknowledgements

This research is supported by the Mathematical Sciences Division, the U.S. Army Office, under Grant Numbers W911NF-12-1-0090 and W911NF-15-1-0182.

Appendix

For Δt = 1, ε = 0.001, p = 2, the ε-best sub-optimal estimates of parameters a, µ and σ2 in (8) for four energy commodity data sets using r = 10 and r = 20 are outlined in Appendix A. The AGMM approach generated by the idea in Remark 5 is fully outlined, applied and compared with four energy data sets in Appendix B. A detailed comparison regarding the theoretical, graphical and performance of the LLGMM and OCBGMM methods are presented in Appendix C. In addition, a comparison of LLGMM with a few nonparametric statistical methods is also outlined in Appendix D.

A State and Parameter Estimates of Daily Natural Gas, Daily Crude Oil, Daily Coal, and Weekly Ethanol Data for Initial Delays r = 10 and r = 20

An initial choice of r and p in Section 3 plays a very significant role in computational coordination, parameter and state estimation and state simulation. The ACF (Box, Jenkins, and Reinsel 1994; Casella and Berger 2002) is used to determine the value p. An initial discrete-time delay r is used based on the increasing sequential choice r = 5,10,20 for the best graphical simulation result. The results are outlined in Appendix A.1 and Appendix A.2.

A.1 State and Parameter Estimates for Daily Natural Gas, Crude Oil, Coal, and Weekly Ethanol Data Using Initial delay r = 10

Table 13:

Estimates mˆk, σmˆk,k2, μmˆk,k and amˆk,k for initial delay r = 10.

tk	Natural gas				tk	Crude oil				tk	Coal				tk	Ethanol
	mˆk	σmkˆ,k2	μmkˆ,k	amkˆ,k		mˆk	σmkˆ,k2	μmkˆ,k	amkˆ,k		mˆk	σmkˆ,k2	μmkˆ,k	amkˆ,k		mˆk	σmkˆ,k2	μmkˆ,k	amkˆ,k
11	8	0.0003	2.0015	0.1718	11	4	0.0003	24.3532	0.0100	11	6	0.0015	8.5931	0.0245	11	6	0.0009	1.1830	0.8082
12	6	0.0003	2.1346	0.0131	12	4	0.0001	25.8537	−0.0157	12	10	0.0011	9.2573	0.0208	12	6	0.0009	1.2087	0.3843
13	7	0.0004	2.5701	0.0630	13	3	0.0003	25.8786	−0.0152	13	2	0.0029	7.6663	−0.0520	31	9	0.0013	4.0236	0.0040
14	9	0.0007	2.6746	0.0461	14	10	0.0010	24.0633	0.0084	14	5	0.0053	9.7962	0.0481	14	2	0.0009	1.1073	0.0509
15	7	0.0012	2.4425	0.4071	15	10	0.0009	22.7352	0.0025	15	10	0.0041	9.4047	0.0496	15	9	0.0024	1.0755	−0.1896
16	3	0.0013	2.5549	0.4621	16	4	0.0002	23.8665	0.0423	16	5	0.0050	9.4886	0.0694	16	2	0.0025	2.8800	0.0289
17	8	0.0015	2.5576	0.1934	17	7	0.0005	24.0777	0.0194	17	10	0.0048	9.1694	0.0598	17	9	0.0023	0.9139	−0.1012
18	8	0.0014	2.5628	0.2495	18	9	0.0008	24.2210	0.0138	18	4	0.0016	9.0681	0.1119	18	2	0.0018	0.7387	−0.0826
19	7	0.0015	2.5705	0.3522	19	7	0.0006	24.1147	0.0268	19	4	0.0043	9.0152	0.1527	19	7	0.0017	2.0655	0.0896
20	9	0.0011	2.5943	0.2946	20	6	0.0004	24.2748	0.0256	20	3	0.0039	9.0801	0.1613	20	8	0.0023	2.2742	0.0690
21	9	0.0010	2.6947	0.0775	21	7	0.0005	24.2175	0.0258	21	3	0.0030	8.7421	0.0946	21	7	0.0014	2.4094	0.0554
22	9	0.0010	2.6464	0.1883	22	4	0.0002	23.9993	0.0317	22	8	0.0085	8.8853	0.0944	22	6	0.0029	2.0457	0.1327
23	3	0.0009	2.7139	0.6983	23	10	0.0008	23.8479	0.0130	23	3	0.0010	8.6669	0.1055	23	7	0.0016	2.0441	0.1332
24	10	0.0013	2.6421	0.2966	24	10	0.0009	24.7657	−0.0087	24	6	0.0060	8.7592	0.0967	24	9	0.0020	1.3966	−0.2082
25	9	0.0018	2.6387	0.2382	25	4	0.0001	21.8903	0.0115	25	7	0.0064	8.8440	0.0908	25	6	0.0200	2.4981	0.1465
26	2	0.0015	2.5223	0.6595	26	4	0.0003	22.2871	0.0258	26	8	0.0067	8.8464	0.0895	26	7	0.0173	2.3356	0.1927
27	4	0.0018	2.5464	0.3474	27	10	0.0011	35.7200	−0.0010	27	3	0.0012	9.0667	0.1633	27	9	0.0143	2.3860	0.1416
28	3	0.0008	2.5780	0.2807	28	4	0.0003	22.1582	0.0391	28	8	0.0053	8.9557	0.0539	28	8	0.0138	2.3919	0.2196
29	2	0.0011	2.6588	−0.1271	29	6	0.0004	22.2194	0.0401	29	4	0.0007	9.0561	0.1246	29	7	0.0152	2.4087	0.3983
30	7	0.0031	2.5610	0.3718	30	7	0.0005	22.2596	0.0394	30	8	0.0041	8.9685	0.1025	30	10	0.0106	2.3164	0.2386
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
1,145	4	0.0002	5.7203	0.1225	2,440	6	0.0004	58.4990	0.0149	2,865	4	0.0001	29.6070	0.0339	375	5	0.0008	2.1469	0.9842
1,146	4	0.0003	5.6485	0.0951	2,441	6	0.0004	57.7330	0.0070	2,866	6	0.0005	29.3520	0.0213	376	4	0.0009	2.0689	0.2666
1,147	4	0.0003	5.6704	0.2152	2,442	8	0.0006	58.1010	0.0086	2,867	7	0.0008	29.8620	−0.0231	377	6	0.0011	2.0999	0.2756
1,148	7	0.0007	5.7138	0.1245	2,443	8	0.0006	58.2670	0.0105	2,868	3	0.0002	27.4300	0.0253	378	7	0.0014	2.0924	0.2551
1,149	4	0.0004	5.6800	0.2544	2,444	6	0.0004	60.6030	0.0027	2,869	7	0.0016	26.8240	0.0056	379	10	0.0044	2.0941	0.2867
1,150	6	0.0007	5.6331	0.1455	2,445	6	0.0003	70.6110	0.0005	2,870	3	0.0010	27.0540	0.0542	380	5	0.0007	2.0731	0.8434
1,151	4	0.0007	5.5648	0.0971	2,446	7	0.0003	58.6010	0.0072	2,871	6	0.0009	26.7590	0.0182	381	6	0.0017	2.0214	−0.4677
1,152	10	0.0026	5.5382	0.0588	2,447	9	0.0009	58.7720	0.0077	2,872	3	0.0006	26.4340	0.0220	382	6	0.0024	1.4504	−0.0549
1,153	5	0.0006	5.4049	0.1000	2,448	4	0.0006	58.9640	0.0115	2,873	3	0.0004	26.6850	−0.1453	383	6	0.0017	1.6343	−0.0794
1,154	5	0.0004	5.4155	0.1569	2,449	10	0.0011	58.4730	0.0073	2,874	9	0.0023	25.9970	0.0131	384	10	0.0057	2.7780	0.0309
1,155	8	0.0010	5.4718	0.0725	2,450	4	0.0003	58.5010	0.0344	2,875	3	0.0014	25.5990	0.0532	385	8	0.0039	2.7055	0.0750
1,156	7	0.0007	5.4528	0.1645	2,451	3	0.0003	59.6250	0.0077	2,876	4	0.0010	25.5580	0.0543	386	6	0.0018	2.6000	0.3021
1,157	8	0.0009	5.4395	0.2011	2,452	5	0.0003	58.9550	0.0143	2,877	10	0.0027	25.2940	0.0067	387	8	0.0031	2.6118	0.1997
1,158	5	0.0007	5.4183	0.1614	2,453	10	0.0014	59.3090	0.0137	2,878	6	0.0012	25.3300	0.0391	388	6	0.0027	2.6058	0.6130
1,159	7	0.0008	5.3905	0.1281	2,454	10	0.0013	59.4310	0.0108	2,879	9	0.0019	25.2960	0.0155	389	8	0.0035	2.5973	0.4169
1,160	9	0.0011	5.3367	0.0973	2,455	10	0.0012	59.2480	0.0133	2,880	9	0.0017	25.4620	0.0264	390	5	0.0024	2.5947	0.5364
1,161	8	0.0008	4.9339	0.0155	2,456	9	0.0010	59.3460	0.0112	2,881	7	0.0012	25.3400	0.0369	391	5	0.0019	2.6500	−0.2801
1,162	8	0.0007	5.0020	0.0210	2,457	6	0.0005	59.2690	0.0106	2,882	9	0.0018	25.4310	0.0416	392	5	0.0017	2.6321	−0.3394
1,163	7	0.0004	5.0947	0.0732	2,458	4	0.0002	58.4790	0.0103	2,883	7	0.0011	25.3550	0.0445	393	6	0.0020	3.0563	−0.0442
1,164	5	0.0001	4.9554	0.0671	2,459	3	0.0004	58.4160	0.0976	2,884	9	0.0016	25.3400	0.0445	394	9	0.0055	2.4093	0.0868
1,165	9	0.0009	4.0877	0.0148	2,460	10	0.0014	57.0380	0.0026	2,885	4	0.0005	25.5440	0.0675	395	4	0.0027	2.3140	0.4706

Table 13 shows the ε-best sub-optimal local admissible sample size mˆk and the parameter estimates amˆk,k, μmˆk,k and σmˆk,k2 for four energy commodities price at time tk. This is based on the value of p and the initial real data time delay r = 10. We further note that the range of the ε-best sub-optimal local admissible sample size mˆk for any time tk∈[11,30]⋃[1,145,1,165], tk∈[11,30]⋃[2,440,2,460], tk∈[11,30]⋃[2,865,2,885], and tk∈[11,30]⋃[375,395] for natural gas, crude oil, coal and ethanol data, respectively, is 2≤mˆk≤10. Moreover, all comments (Remark 18) that are made with regard to Table 2 regarding the four energy commodities remain valid with regard to Table 13

Table 14:

Real, Simulation using LLGMM method, and absolute error of simulation using starting delay r = 10.

tk	Natural gas			tk	Crude oil			tk	Coal			tk	Ethanol
	Real	Simulated	\|Error\|		Real	Simulated	\|Error\|		Real	Simulated	\|Error\|		Real	Simulated	\|Error\|
	yk	ymˆk,ks	\|yk−ymˆk,ks\|			ymˆk,ks	\|yk−ymˆk,ks\|			ymˆk,ks	\|yk−ymˆk,ks\|			ymˆk,ks	\|yk−ymˆk,ks\|
		(LLGMM)				(LLGMM)				(LLGMM)				(LLGMM)
10	2.3830	2.3830	0.0000	10	25.1000	25.1000	0.0000	10	9.0600	9.0600	0.0000	10	1.1900	1.1900	0.0000
11	2.4170	2.4179	0.0009	11	24.8000	25.0181	0.2181	11	8.8800	8.8800	0.0000	11	1.2250	1.2249	0.0001
12	2.5590	2.4935	0.0655	12	24.4000	24.3221	0.0779	12	9.4400	9.4216	0.0184	12	1.2200	1.2425	0.0225
13	2.4850	2.4949	0.0099	13	23.8500	23.7260	0.1240	13	10.3100	10.0621	0.2479	13	1.2900	1.2278	0.0622
14	2.5280	2.5123	0.0157	14	23.8500	24.4203	0.5703	14	9.8100	9.8058	0.0042	14	1.4100	1.5339	0.1239
15	2.6160	2.6158	0.0002	15	23.8500	23.8174	0.0326	15	9.0600	8.8075	0.2525	15	1.4700	1.3390	0.1310
16	2.5230	2.5233	0.0003	16	23.9000	23.8845	0.0155	16	8.7500	8.4774	0.2726	16	1.5300	1.5745	0.0445
17	2.6100	2.6314	0.0214	17	24.5000	24.0924	0.4076	17	8.8200	8.7839	0.0361	17	1.6300	1.5996	0.0304
18	2.6100	2.5852	0.0248	18	24.8000	24.3340	0.4660	18	9.5600	9.3610	0.1990	18	1.7500	1.6320	0.1180
19	2.6100	2.6130	0.0030	19	24.1500	24.1566	0.0066	19	8.8200	8.6667	0.1533	19	1.7500	1.7495	0.0005
20	2.6990	2.6728	0.0262	20	24.2000	24.5277	0.3277	20	8.8200	8.7833	0.0367	20	1.8400	1.8586	0.0186
21	2.7590	2.7601	0.0011	21	24.0000	23.7803	0.2197	21	8.6900	8.5498	0.1402	21	1.8950	1.8874	0.0076
22	2.6590	2.6427	0.0163	22	23.9000	24.1935	0.2935	22	8.6300	8.7065	0.0765	22	1.9500	1.9257	0.0243
23	2.7420	2.7365	0.0055	23	23.0500	23.0564	0.0064	23	8.6900	8.7620	0.0720	23	1.9740	1.9548	0.0192
24	2.5620	2.5610	0.0010	24	22.3000	23.2208	0.9208	24	8.9400	8.9706	0.0306	24	2.7000	2.1431	0.5569
25	2.4950	2.5455	0.0505	25	22.4500	23.1640	0.7140	25	9.3100	8.8231	0.4869	25	2.5150	2.6941	0.1791
26	2.5400	2.5245	0.0155	26	22.3500	22.7275	0.3775	26	8.9400	8.9945	0.0545	26	2.2900	2.2753	0.0147
27	2.5920	2.5996	0.0076	27	21.7500	21.5907	0.1593	27	8.9400	8.9676	0.0276	27	2.4400	2.3645	0.0755
28	2.5700	2.5849	0.0149	28	22.1000	22.0868	0.0132	28	9.1300	9.1741	0.0441	28	2.4150	2.4019	0.0131
29	2.5410	2.5403	0.0007	29	22.4000	22.4301	0.0301	29	9.1900	9.1766	0.0134	29	2.3000	2.2440	0.0560
30	2.6180	2.6151	0.0029	30	22.5000	22.6614	0.1614	30	8.5700	8.4567	0.1133	30	2.1000	2.2048	0.1048
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
1,145	5.712	5.7533	0.0413	2,440	57.35	57.762	0.412	2,865	29.31	29.518	0.2083	375	2.073	2.0662	0.0068
1,146	5.588	5.5892	0.0012	2,441	56.74	56.743	0.0028	2,866	28.68	28.495	0.1851	376	2.02	2.0267	0.0067
1,147	5.693	5.7143	0.0213	2,442	57.55	57.739	0.189	2,867	26.77	28.727	1.9571	377	2.073	2.0731	0.0001
1,148	5.791	5.8127	0.0217	2,443	59.09	58.925	0.1646	2,868	27.45	26.979	0.471	378	2.065	2.0709	0.0059
1,149	5.614	5.5940	0.0200	2,444	60.27	59.663	0.607	2,869	27.00	26.879	0.121	379	2.055	2.0232	0.0318
1150	5.442	5.6266	0.1846	2,445	60.75	61.161	0.4109	2,870	26.67	27.32	0.6499	380	2.209	2.2109	0.0019
1,151	5.533	5.5122	0.0208	2,446	58.41	58.011	0.3994	2,871	26.51	25.468	1.0415	381	2.44	2.296	0.144
1,152	5.378	5.3971	0.0191	2,447	58.72	58.762	0.042	2,872	26.48	26.263	0.2174	382	2.517	2.4074	0.1096
1,153	5.373	5.3496	0.0234	2,448	58.64	58.409	0.2309	2,873	25.15	25.395	0.2445	383	2.718	2.6839	0.0341
1,154	5.382	5.3735	0.0085	2,449	57.87	57.762	0.1081	2,874	25.57	25.555	0.0153	384	2.541	2.5246	0.0164
1,155	5.507	5.5360	0.0290	2,450	59.13	59.243	0.1135	2,875	25.88	26.08	0.2003	385	2.566	2.5629	0.0031
1,156	5.552	5.5507	0.0013	2,451	60.11	60.068	0.0419	2,876	25.24	25.528	0.2879	386	2.626	2.6248	0.0012
1,157	5.310	5.3019	0.0081	2,452	58.94	58.956	0.0155	2,877	25	25.337	0.3375	387	2.587	2.5871	0.0001
1,158	5.338	5.3884	0.0504	2,453	59.93	59.924	0.0062	2,878	25.08	24.685	0.3951	388	2.628	2.6363	0.0083
1,159	5.298	5.2554	0.0426	2,454	61.18	62.168	0.9876	2,879	25.05	24.848	0.2024	389	2.587	2.5332	0.0538
1,160	5.189	5.1644	0.0146	2,455	59.66	59.381	0.2786	2,880	25.89	25.638	0.2518	390	2.536	2.5374	0.0014
1,161	5.082	5.0874	0.0054	2,456	58.59	58.468	0.1224	2,881	25.23	25.405	0.1749	391	2.42	2.3401	0.0799
1,162	5.082	5.0977	0.0157	2,457	58.28	58.487	0.2067	2,882	25.94	25.739	0.2007	392	2.247	2.1792	0.0678
1,163	5.082	5.1334	0.0514	2,458	58.79	58.896	0.1058	2,883	25.26	24.858	0.4025	393	2.223	2.1661	0.0569
1,164	4.965	5.0340	0.0690	2,459	56.23	57.202	0.9715	2,884	25.25	25.147	0.1028	394	2.39	2.5122	0.1222
1,165	4.767	4.9143	0.1473	2,460	55.9	56.87	0.9701	2,885	26.06	25.613	0.4475	395	2.38	2.3583	0.0217

In Table 14, the real and LLGMM simulated price values for each of the four energy commodities: natural gas, crude oil, coal and ethanol are recorded in columns 2–3, 6–7, 10–11, and 14–15, respectively. The absolute error of each of the energy commodity’s simulated value is shown in columns 4, 8, 12, 16, respectively.Add a table footnote here

Table 15:

Estimates mˆk, σmˆk,k2, μmˆk,k and amˆk,k for initial delay r = 20.

tk	Natural gas				tk	Crude oil				tk	Coal				tk	Ethanol
	mˆk	σmkˆ,k2	μmkˆ,k	amkˆ,k		mˆk	σmkˆ,k2	μmkˆ,k	amkˆ,k		mˆk	σmkˆ,k2	μmkˆ,k	amkˆ,k		mˆk	σmkˆ,k2	μmkˆ,k	amkˆ,k
21	13	0.0011	2.7056	0.0816	21	11	0.0003	24.115	0.0204	21	19	0.0042	9.1915	0.0255	21	18	0.0024	0.7591	0.0467
22	5	0.0009	2.6748	0.233	22	7	0.0003	24.215	0.0278	22	15	0.0044	9.0773	0.0601	22	4	0.0015	0.7929	-0.0272
23	3	0.0013	2.7139	0.6983	23	2	0.0006	24.013	-0.314	23	19	0.0038	9.1073	0.0319	23	8	0.0004	2.1528	0.0888
24	12	0.0021	2.6197	0.2119	24	15	0.0007	14.246	0.0009	24	10	0.0035	8.8762	0.0924	24	15	0.0025	1.0048	-0.1078
25	10	0.0022	2.6201	0.2199	25	19	0.0011	18.542	0.001	25	14	0.0049	9.1783	0.0517	25	20	0.0094	-0.4372	-0.0208
26	5	0.0015	2.567	0.2063	26	19	0.001	21.738	0.0031	26	9	0.003	8.9447	0.1	26	19	0.0094	3.1726	0.0251
27	9	0.0021	2.6295	0.1919	27	4	0.0001	22.135	0.0355	27	10	0.0031	8.9442	0.1	27	7	0.0205	2.3915	0.2198
28	17	0.0031	2.6074	0.2204	28	14	0.0007	20.045	0.0015	28	6	0.0013	9.0358	0.0767	28	17	0.0087	2.6208	0.0553
29	11	0.0022	2.6099	0.1688	29	14	0.0007	22.096	0.0034	29	3	0.0006	9.4379	0.0213	29	3	0.0218	2.3857	0.634
30	8	0.0014	2.5821	0.2593	30	9	0.0004	22.249	0.0154	30	8	0.0019	8.9685	0.1025	30	19	0.0161	2.3086	0.0752
31	7	0.0013	2.5605	0.3999	31	3	0.0002	22.739	0.0203	31	4	0.0014	8.8837	0.0869	31	18	0.0162	2.2442	0.1049
32	9	0.0016	2.5738	0.3887	32	6	0.0004	22.226	0.0427	32	15	0.0096	8.9287	0.0972	32	9	0.0279	2.3519	0.4089
33	16	0.0035	2.6195	0.2084	33	7	0.0005	22.084	0.0296	33	5	0.0013	8.7634	0.0932	33	12	0.0193	2.2912	0.2631
34	20	0.0041	2.6078	0.2483	34	11	0.001	21.683	0.0138	34	7	0.0018	8.8238	0.0869	34	6	0.0186	2.1259	0.2733
35	16	0.0033	2.6031	0.2024	35	10	0.0009	20.446	0.0041	35	8	0.0021	8.7923	0.0823	35	20	0.0218	2.2078	0.1261
36	5	0.0007	2.579	0.2816	36	3	0	21.027	0.0489	36	9	0.0023	8.7282	0.0671	36	10	0.0199	1.9158	0.0549
37	9	0.0013	2.5814	0.3453	37	4	0.0002	20.962	0.0465	37	13	0.0062	8.7653	0.0502	37	7	0.0146	1.9215	0.088
38	10	0.0014	2.5836	0.3371	38	3	0.0002	21.267	-0.0327	38	7	0.001	8.6612	0.1378	38	7	0.0127	2.0226	0.1587
39	3	0.0015	2.603	0.3923	39	13	0.0014	15.485	0.0012	39	20	0.0151	8.8225	0.0644	39	19	0.0413	2.1885	0.1729
40	18	0.0048	2.6026	0.2551	40	5	0.0004	20.617	0.028	40	17	0.0101	8.8585	0.0667	40	8	0.0112	1.9751	0.1655
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
1,145	3	0.0001	5.7243	0.1464	2,440	8	0.0007	58.338	0.0143	2,865	4	0.0002	29.607	0.034	375	6	0.0013	2.1486	0.7096
1,146	17	0.0033	5.7831	0.0272	2,441	20	0.0033	58.546	0.0028	2,866	12	0.0023	29.257	0.0209	376	3	0.0009	2.0699	0.2808
1,147	15	0.0025	5.8662	0.0337	2,442	10	0.0008	58.056	0.0098	2,867	20	0.0054	26.256	0.0021	377	5	0.0011	2.0858	0.3308
1,148	8	0.0006	5.7271	0.0741	24,43	8	0.0006	58.267	0.0106	2,868	14	0.0028	28.678	0.009	378	11	0.007	2.1286	0.2103
1,149	5	0.0004	5.6834	0.2598	24,44	7	0.0005	58.414	0.0079	2,869	11	0.0019	27.482	0.0052	379	3	0.0007	2.0623	0.6096
1,150	18	0.0034	5.6161	0.0138	2,445	7	0.0005	65.583	0.001	2,870	14	0.0026	26.136	0.0023	380	16	0.0137	2.1586	0.1983
1,151	16	0.0026	5.6048	0.0268	2,446	8	0.0005	58.733	0.0078	2,871	12	0.0019	25.376	0.0021	381	19	0.0185	2.2115	0.1503
1,152	18	0.0031	5.3059	0.0099	2,447	9	0.0007	58.772	0.0078	2,872	9	0.0011	26.067	0.0064	382	11	0.0066	1.7644	-0.0401
1,153	9	0.0008	5.4937	0.0517	2,448	20	0.0033	58.727	0.0079	2,873	4	0.0003	27.22	-0.0313	383	3	0.0025	2.9233	0.1347
1,154	7	0.0006	5.4044	0.0549	2,449	13	0.0013	58.371	0.0087	2,874	10	0.0016	25.744	0.0095	384	4	0.0025	2.5937	0.3073
1,155	5	0.0003	5.4342	0.2005	2,450	3	0.0001	58.48	0.0345	2,875	3	0.0012	25.599	0.0532	385	5	0.0039	2.5887	0.3099
1,156	7	0.0006	5.4528	0.1646	2,451	9	0.0008	59.324	0.013	2,876	3	0.0008	25.559	0.0541	386	3	0.006	2.5861	0.4792
1,157	8	0.0006	5.4395	0.2012	2,452	5	0.0005	58.955	0.0144	2,877	5	0.0006	25.415	0.0446	387	4	0.0039	2.5882	0.4761
1,158	14	0.002	5.4704	0.0583	2,453	9	0.001	59.171	0.0135	2,878	4	0.0005	25.193	0.0206	388	11	0.0087	2.6964	0.077
1,159	10	0.0009	5.4035	0.1412	2,454	15	0.002	59.298	0.0063	2,879	3	0.0002	25.059	0.0528	389	6	0.0038	2.5952	0.4921
1,160	14	0.0018	5.3501	0.0373	2,455	13	0.0015	59.512	0.0126	2,880	5	0.0004	25.256	0.0431	390	10	0.0075	2.5899	0.3122
1,161	11	0.001	5.174	0.0277	2,456	11	0.0011	59.169	0.0137	2,881	5	0.0005	25.254	0.0435	391	9	0.0062	2.5817	0.4568
1,162	18	0.0029	5.1069	0.016	2,457	12	0.0012	59.072	0.0128	2,882	9	0.002	25.431	0.0417	392	7	0.0038	2.6222	-0.3162
1,163	18	0.0027	5.1426	0.0213	2,458	8	0.0006	59.427	0.0112	2,883	13	0.0033	25.507	0.0243	393	15	0.0142	2.5051	0.1102
1,164	16	0.002	5.0554	0.0297	2,459	15	0.0018	58.808	0.0092	2,884	20	0.006	25.52	0.0094	394	12	0.01	2.4881	0.1156
1,165	15	0.0016	5.7431	-0.0195	2,460	14	0.0015	58.187	0.0042	2,885	5	0.0007	25.538	0.069	395	3	0.0036	2.355	0.2939

Table 15 shows the ε-best sub-optimal local admissible sample size mˆk and the parameter estimates amˆk,k, μmˆk,k and σmˆk,k2 for four energy commodities price at time tk. This was based on the value of p and the initial real data time delay r = 20. We further note that the range of the ε-best sub-optimal local admissible sample size mˆk for any time tk∈[21,40]⋃[1,145,1,165], tk∈[21,40]⋃[2,440,2,460], tk∈[21,40]⋃[2,865,2,885], and tk∈[21,40]⋃[375,395] for natural gas, crude oil, coal and ethanol data, respectively, is 3≤mˆk≤20. Moreover, all comments (Remark 18) that are made with regard to Table 2 regarding the four energy commodities remain valid with regard to Table 15

A.2 State and Parameter Estimates of Daily Natural Gas, Daily Crude Oil, Daily Coal, and Weekly Ethanol Data for Initial Delay r = 20

In Table 16, the real and the LLGMM simulated price values for each of the four energy commodities: natural gas, crude oil, coal and ethanol are exhibited in columns 2–3, 6–7, 10–11, and 14–15, respectively. The absolute error of each of the energy commodity’s simulated value is shown in columns 4, 8, 12, 16, respectively.

Table 16:

Real, Simulation using the LLGMM method, and absolute error of simulation using starting delay r = 20.

tk	Natural gas			tk	Crude oil			tk	Coal			tk	Ethanol
	Real	Simulated	\|Error\|		Real	Simulated	\|Error\|		Real	Simulated	\|Error\|		Real	Simulated	\|Error\|
	yk	ymˆk,ks	\|yk−ymˆk,ks\|			ymˆk,ks	\|yk−ymˆk,ks\|			ymˆk,ks	\|yk−ymˆk,ks\|			ymˆk,ks	\|yk−ymˆk,ks\|
		(LLGMM)				(LLGMM)				(LLGMM)				(LLGMM)
21	2.759	2.7718	0.0128	21	24	24.025	0.025	21	8.69	8.6747	0.0153	21	1.895	1.9024	0.0074
22	2.659	2.6566	0.0024	22	23.9	24.093	0.193	22	8.63	8.6175	0.0125	22	1.95	1.9315	0.0185
23	2.742	2.7353	0.0067	23	23.05	23.051	0.001	23	8.69	8.6862	0.0038	23	1.974	1.9788	0.0048
24	2.562	2.5757	0.0137	24	22.3	22.887	0.587	24	8.94	8.9184	0.0216	24	2.7	2.5529	0.1471
25	2.495	2.5332	0.0382	25	22.45	22.126	0.324	25	9.31	9.3069	0.0031	25	2.515	2.5134	0.0016
26	2.54	2.5336	0.0064	26	22.35	22.409	0.059	26	8.94	8.8992	0.0408	26	2.29	2.3306	0.0406
27	2.592	2.5631	0.0289	27	21.75	22.12	0.37	27	8.94	8.8745	0.0655	27	2.44	2.3718	0.0682
28	2.57	2.5797	0.0097	28	22.1	22.137	0.037	28	9.13	9.1162	0.0138	28	2.415	2.3927	0.0223
29	2.541	2.4846	0.0564	29	22.4	22.315	0.085	29	9.19	9.234	0.044	29	2.3	2.3311	0.0311
30	2.618	2.6245	0.0065	30	22.5	22.531	0.031	30	8.57	8.5495	0.0205	30	2.1	2.072	0.028
31	2.564	2.5469	0.0171	31	22.65	22.712	0.062	31	8.69	8.7241	0.0341	31	2.04	2.0323	0.0077
32	2.667	2.6763	0.0093	32	21.95	22.003	0.053	32	8.88	8.8866	0.0066	32	2.16	2.1561	0.0039
33	2.633	2.6308	0.0022	33	21.6	21.853	0.253	33	8.57	8.5084	0.0616	33	2.13	2.0796	0.0504
34	2.515	2.5021	0.0129	34	21	21.099	0.099	34	8.75	8.7447	0.0053	34	2.155	2.2141	0.0591
35	2.53	2.5136	0.0164	35	20.95	21.012	0.062	35	8.63	8.6003	0.0297	35	2.01	1.9687	0.0413
36	2.549	2.5458	0.0032	36	21.1	20.971	0.129	36	8.44	8.412	0.028	36	1.93	1.8762	0.0538
37	2.603	2.5835	0.0195	37	20.8	20.786	0.014	37	8.44	8.4465	0.0065	37	1.9	1.9186	0.0186
38	2.603	2.5822	0.0208	38	20.3	20.048	0.252	38	8.94	8.9538	0.0138	38	1.975	1.9052	0.0698
39	2.603	2.6075	0.0045	39	20.25	20.244	0.006	39	9	9.0064	0.0064	39	1.98	2.019	0.039
40	2.815	2.8728	0.0578	40	20.75	20.734	0.016	40	8.94	8.8655	0.0745	40	2	1.9385	0.0615
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
1,145	5.712	5.7577	0.0457	2,440	57.35	57.376	0.026	2,865	29.31	29.291	0.019	375	2.073	2.09	0.017
1,146	5.588	5.6488	0.0608	2,441	56.74	56.447	0.293	2,866	28.68	28.8	0.12	376	2.02	2.0589	0.0389
1,147	5.693	5.7062	0.0132	2,442	57.55	57.523	0.027	2,867	26.77	26.891	0.121	377	2.073	2.0601	0.0129
1,148	5.791	5.7917	0.0007	2,443	59.09	58.968	0.122	2,868	27.45	27.316	0.134	378	2.065	2.0312	0.0338
1,149	5.614	5.5799	0.0341	2,444	60.27	60.278	0.008	2,869	27	27.189	0.189	379	2.055	2.0725	0.0175
1,150	5.442	5.4099	0.0321	2,445	60.75	60.737	0.013	2,870	26.67	26.812	0.142	380	2.209	2.2254	0.0164
1,151	5.533	5.5035	0.0295	2,446	58.41	58.494	0.084	2,871	26.51	26.709	0.199	381	2.44	2.462	0.022
1,152	5.378	5.407	0.029	2,447	58.72	58.614	0.106	2,872	26.48	26.54	0.06	382	2.517	2.51	0.007
1,153	5.373	5.3682	0.0048	2,448	58.64	58.95	0.31	2,873	25.15	25.313	0.163	383	2.718	2.6979	0.0201
1,154	5.382	5.3827	0.0007	2,449	57.87	57.865	0.005	2,874	25.57	25.47	0.1	384	2.541	2.5164	0.0246
1,155	5.507	5.4896	0.0174	2,450	59.13	58.967	0.163	2,875	25.88	26.078	0.198	385	2.566	2.5328	0.0332
1,156	5.552	5.5423	0.0097	2,451	60.11	59.937	0.173	2,876	25.24	25.208	0.032	386	2.626	2.5831	0.0429
1,157	5.31	5.318	0.008	2,452	58.94	59.068	0.128	2,877	25	25.138	0.138	387	2.587	2.5606	0.0264
1,158	5.338	5.3794	0.0414	2,453	59.93	60.141	0.211	2,878	25.08	25.306	0.226	388	2.628	2.6322	0.0042
1,159	5.298	5.3541	0.0561	2,454	61.18	61.53	0.35	2,879	25.05	25.16	0.11	389	2.587	2.5651	0.0219
1,160	5.189	5.1838	0.0052	2,455	59.66	59.792	0.132	2,880	25.89	25.509	0.381	390	2.536	2.53	0.006
1,161	5.082	5.3804	0.2984	2,456	58.59	58.481	0.109	2,881	25.23	25.278	0.048	391	2.42	2.4268	0.0068
1,162	5.082	4.9802	0.1018	2,457	58.28	58.224	0.056	2,882	25.94	25.961	0.021	392	2.247	2.2228	0.0242
1,163	5.082	5.1933	0.1113	2,458	58.79	58.928	0.138	2,883	25.26	25.255	0.005	393	2.223	2.2072	0.0158
1,164	4.965	5.1925	0.2275	2,459	56.23	56.329	0.099	2,884	25.25	25.298	0.048	394	2.39	2.4141	0.0241
1,165	4.767	4.7917	0.0247	2,460	55.9	54.676	1.224	2,885	26.06	25.882	0.178	395	2.38	2.4265	0.0465

B Formulation of Aggregated Generalized Method of Moment (AGMM)

In this section, using the theoretical basis of the LLGMM and Remark 5 (Section 2), we generated aggregated state and parameter estimates based on the method for state and parameter estimation problems. The generalized method is then applied to energy commodity dynamic model (8). The results are compared with the LLGMM method.

B.1 AGMM Method Applied to Energy Commodities

In this section, using the aggregated parameter estimates aˉ, μˉ, and σ2‾ described by the mean value of the estimated samples {amˆi,i}i=0N, {μmˆi,i}i=0N and {σmˆi,i2}i=0N (Remark 5), respectively, we discuss the simulated price values for the four energy commodities. aˉ, μˉ, and σ2‾ defined in (19) are referred to as aggregated parameter estimates of a, µ, and σ2 over the given entire finite interval of time. These estimates are derived using the following discretized system:

(52)yiag=yi−1ag+aˉ(μˉ−yi−1ag)yi−1agΔt+σ2‾1/2yi−1agΔWi

where ykag denotes the simulated value for yk at time tk. The overall descriptive data statistic regarding the four energy commodities price and estimated parameters are recorded in Table 17.

Table 17:

Descriptive statistics for a, µ and σ2 using initial delay r = 20.

Data set Y	Yˉ	Std(Y)	Δln(Y)‾	var(Δln(Y))	aˉ	Std(a)	μˉ	Std(µ)	σ2‾	std(σ2)	95%C. I.μˉ
Natural gas	4.5504	1.5090	0.0008	0.0015	0.1867	0.3013	4.5538	2.3565	0.0013	0.0017	(4.4196, 4.6880)
Crude oil	54.0093	31.0248	0.0003	0.0006	0.0215	0.0517	54.0307	37.4455	0.0005	0.0008	(51.8978, 56.1636)
Coal	27.1441	17.8394	0.0003	0.0015	0.0464	0.0879	27.0567	21.3506	0.0014	0.0022	(25.8405, 28.2729)
Ethanol	2.1391	0.4455	0.0011	0.0020	0.3167	0.8745	2.1666	0.7972	0.0018	0.0030	(2.0919,2.2414)

Table 17 shows the descriptive statistics for a, µ and σ2 with time delay r = 20. Moreover, μˉ is approximately close to the overall descriptive statistics of the mean Yˉ of the real data for each of the energy commodity shown in column 2. Also, σ2‾ is approximately close to the overall descriptive statistics of the variance of Δln(Y)=ln(Yi)−ln(Yi−1) in Column 5. Moreover, column 12 shows that the mean of the actual data set in Column 2 falls within the 95% confidence interval of μˉ. This exhibits that the parameter μmˆk,k is the mean level of yk at time tk.

Using the aggregated parameter estimates aˉ, μˉ, and σ2‾ in Table 17 (Column 6, 8, and 10), the simulated price values for the four energy commodities are shown in columns 3, 6, 9 and 12 of Table 18.

Table 18:

Real, Simulation value using AGMM with r = 20.

tk	Natural gas		tk	Crude oil		tk	Coal		tk	Ethanol
	Real	Simulated		Real	Simulated		Real	Simulated		Real	Simulated
		ykag			ykag			ykag			ykag
		(AGMM)			(AGMM)			(AGMM)			(AGMM)
21	2.759	2.649	21	24.00	23.974	21	8.690	9.111	21	1.895	1.834
22	2.659	2.651	22	23.900	24.204	22	8.630	9.028	22	1.950	1.854
23	2.742	2.636	23	23.050	25.229	23	8.690	9.192	23	1.974	1.798
24	2.562	2.625	24	22.300	25.586	24	8.940	9.032	24	2.700	1.858
25	2.495	2.593	25	22.450	26.470	25	9.310	8.938	25	2.515	1.830
26	2.54	2.525	26	22.350	25.953	26	8.940	8.792	26	2.290	1.954
27	2.592	2.513	27	21.750	26.229	27	8.940	9.035	27	2.440	1.926
28	2.57	2.399	28	22.100	26.555	28	9.130	9.255	28	2.415	1.939
29	2.541	2.485	29	22.400	26.402	29	9.190	9.018	29	2.300	1.883
30	2.618	2.506	30	22.500	27.34	30	8.570	8.687	30	2.100	1.880
31	2.564	2.460	31	22.650	26.24	31	8.690	8.985	31	2.040	1.817
32	2.667	2.295	32	21.950	26.765	32	8.880	9.339	32	2.160	1.810
33	2.633	2.534	33	21.600	26.358	33	8.570	9.359	33	2.130	1.774
34	2.515	2.514	34	21.000	26.87	34	8.750	9.310	34	2.155	1.717
35	2.53	2.573	35	20.950	26.835	35	8.630	9.302	35	2.010	1.658
36	2.549	2.592	36	21.100	26.725	36	8.440	9.543	36	1.930	1.607
37	2.603	2.456	37	20.800	26.439	37	8.440	9.288	37	1.900	1.645
38	2.603	2.428	38	20.300	26.916	38	8.940	9.155	38	1.975	1.635
39	2.603	2.505	39	20.250	26.989	39	9.000	8.469	39	1.980	1.629
40	2.815	2.526	40	20.750	26.759	40	8.940	8.899	40	2.00	1.745
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
1,145	5.712	5.218	2,440	57.350	48.179	2,865	29.310	17.839	375	2.073	2.625
1,146	5.588	5.414	2,441	56.740	48.239	2,866	28.680	18.563	376	2.02	2.784
1,147	5.693	5.460	2,442	57.550	46.984	2,867	26.770	19.577	377	2.073	2.558
1,148	5.791	5.464	2,443	59.090	47.418	2,868	27.450	19.841	378	2.065	2.670
1,149	5.614	5.544	2,444	60.270	48.137	2,869	27.000	18.876	379	2.055	2.565
1,150	5.442	5.700	2,445	60.750	49.185	2,870	26.670	18.465	380	2.209	2.796
1,151	5.533	5.710	2,446	58.410	48.271	2,871	26.510	18.139	381	2.44	2.783
1,152	5.378	5.936	2,447	58.720	48.384	2,872	26.480	17.963	382	2.517	2.659
1,153	5.373	5.869	2,448	58.640	47.509	2,873	25.150	18.151	383	2.718	2.739
1,154	5.382	5.778	2,449	57.870	48.654	2,874	25.570	17.987	384	2.541	2.681
1,155	5.507	5.732	2,450	59.130	46.883	2,875	25.880	18.393	385	2.566	2.631
1,156	5.552	5.816	2,451	60.110	46.403	2,876	25.240	18.492	386	2.626	2.638
1157	5.31	6.000	2,452	58.940	45.564	2,877	25.000	18.621	387	2.587	2.542
1,158	5.338	6.162	2,453	59.930	44.177	2,878	25.080	18.806	388	2.628	2.491
1,159	5.298	5.899	2,454	61.180	43.112	2,879	25.050	19.384	389	2.587	2.392
1,160	5.189	6.008	2,455	59.660	43.47	2,880	25.890	20.131	390	2.536	2.393
1,161	5.082	6.175	2,456	58.590	41.531	2,881	25.230	21.099	391	2.42	2.534
1,162	5.082	6.191	2,457	58.280	40.452	2,882	25.940	21.499	392	2.247	2.687
1,163	5.082	5.814	2,458	58.790	41.968	2,883	25.260	21.38	393	2.223	2.701
1,164	4.965	5.701	2,459	56.230	44.359	2,884	25.250	20.786	394	2.39	2.703
1165	4.767	5.871	2,460	55.90	44.679	2,885	26.060	20.892	395	2.38	2.655
1,166	4.675	5.998	2,461	56.420	43.081	2,886	26.030	21.269	396	2.366	2.559
1,167	4.79	5.952	2,462	58.010	44.235	2,887	26.660	20.371	397	2.335	2.575
1,168	4.631	5.782	2,463	57.280	43.199	2,888	27.120	19.822	398	2.428	2.466
1169	4.658	5.673	2,464	60.30	42.655	2,889	26.400	19.644	399	2.409	2.369
1,170	4.57	5.936	2,465	60.970	43.498	2,890	26.940	20.602	400	2.29	2.222

Figure 20 shows a comparison between the real data set, simulated price using LLGMM and AGMM methods.

$Figure 20: Real, Simulated Prices Using LLGMM, And Simulated Prices Using AGMM: Initial Delay r = 20.Figure 20(a), (b), (c) and (d) shows the graphs of the real, simulated prices using the local lagged adaptive generalized method (LLGMM), and the simulated price using the average of the parameters for Henry Hub natural gas data (Natural Gas 2000–2004), daily crude oil data (Crude Oil 1997–2008), daily coal data (Coal 2000–2013) and weekly ethanol data (Ethanol 2005–2013), respectively, for r = 20. The red line represents the real data set yky_k, the blue line represent the simulated prices using the LLGMM method, while the black line represent the simulated price (AGMM) using the aggregated parameter estimates aˉ\bar{a}, μˉ\bar{\mu}, and σ2‾\overline{\sigma^2} in Table 17, Columns 6, 8, and 10, respectively. From these simulated graphs, it is clear that the LLGMM simulation results are more realistic than the AGMM simulation results. This exhibits the superiority of LLGMM over AGMM.$

Figure 20:

Real, Simulated Prices Using LLGMM, And Simulated Prices Using AGMM: Initial Delay r = 20.

Figure 20(a), (b), (c) and (d) shows the graphs of the real, simulated prices using the local lagged adaptive generalized method (LLGMM), and the simulated price using the average of the parameters for Henry Hub natural gas data (Natural Gas 2000–2004), daily crude oil data (Crude Oil 1997–2008), daily coal data (Coal 2000–2013) and weekly ethanol data (Ethanol 2005–2013), respectively, for r = 20. The red line represents the real data set yk, the blue line represent the simulated prices using the LLGMM method, while the black line represent the simulated price (AGMM) using the aggregated parameter estimates aˉ, μˉ, and σ2‾ in Table 17, Columns 6, 8, and 10, respectively. From these simulated graphs, it is clear that the LLGMM simulation results are more realistic than the AGMM simulation results. This exhibits the superiority of LLGMM over AGMM.

Comparison of goodness-of-fit measures for the LLGMM and AGMM methods using initial delay r = 20.

Table 19:

Comparison of goodness-of-fit measures for the LLGMM and AGMM methods using initial delay r = 20.

Goodness-of-fit measure	LLGMM				AGMM
	Natural gas	Crude oil	Coal	Ethanol	Natural gas	Crude oil	Coal	Ethanol
RAMSEˆ	0.0674	0.4625	0.4794	0.0375	1.4968	30.7760	17.7620	0.4356
AMADˆ	1.1318	24.5010	9.4009	0.3213	0.0068	0.0857	0.0833	0.0035
AMBˆ	1.1371	27.2707	12.8370	0.3566	1.2267	27.3050	13.1060	0.3579

B.2 Formulation of Aggregated Generalized Method of Moment (AGMM) for U.S. Treasury Bill and U.S.–U.K. Foreign Exchange Rate

The overall descriptive statistics of data sets regarding U.S. Treasury Bill interest rate and U.S.–U.K. foreign exchange rate are recorded in the following table for initial delay r = 20.

Table 20:

Descriptive statistics for βˉ, μˉ, δˉ, σ‾, and γ‾ for interest rate data using initial delay r = 20.

Yˉ	Std(Y)	βˉ	Std(β)	μˉ	Std(µ)	δˉ	Std(δ)	σˉ	std(σ)	γˉ	Std(γ)
0.05667	0.0268	0.8739	1.8129	-3.8555	8.7608	1.4600	0.00	0.3753	0.5197	1.4877	0.1357

Table 21:

Descriptive statistics for βˉ, μˉ, δˉ, σ‾, and γ‾ for U.S.–U.K. foreign exchange rate data using initial delay r = 20 .

Yˉ	Std(Y)	βˉ	Std(β)	μˉ	Std(µ)	δˉ	Std(δ)	σˉ	std(σ)	γˉ	Std(γ)
1.6249	0.1337	1.5120	2.1259	−1.1973	1.6811	1.4892	0.00	0.0243	0.0180	1.08476	1.0050

Table 20 and Table 21 show the descriptive statistics for βˉ, μˉ, δˉ, σ‾, and γ‾ for the U.S. TBYIR and the U.S.–U.K. FER data, respectively.

In Table 22, the real and the LLGMM simulated rates of the US-TBYIR and the U.S.–U.K. foreign exchange rate (U.S.–U.K. FER) are exhibited in the first and second columns, respectively. Using the aggregated parameter estimates βˉ, μˉ, δˉ, σ‾ and γˉ in the respective Table 20 (columns 3, 5, 7, 9 and 11) and Table 21 (columns 3, 5, 7, 9, and 11), the simulated rates for the U.S. TYBIR and the U.S.–U.K. FER are shown in column 3 of Table 22. These estimates are derived using the following discretized system:

(53)yiag=yi−1ag+(βˉyi−1ag+μˉ(yi−1ag)δˉ)+σˉ(yi−1ag)γˉΔWi

where AGMM, ykag, yk at time tk are defined in (52).

Table 22:

Estimates for real, simulated value using LLGMM and AGMM methods for U.S. TYBIR and the U.S.–U.K. FER, respectively for initial delay r = 20.

tk	Interest rate data			tk	U.S.–U.K. rate data
	Real	Simulated	Simulated		Real	Simulated	Simulated
		LLGMM	AGMM		Real	LLGMM	AGMM
21	0.0465	0.0459	0.0326	21	1.7448	1.6732	1.655
22	0.0459	0.0467	0.0299	22	1.7465	1.7711	1.6588
23	0.0462	0.0463	0.0342	23	1.7638	1.7588	1.6096
24	0.0464	0.0463	0.034	24	1.874	1.8423	1.6251
25	0.045	0.0457	0.0365	25	1.7902	1.7971	1.6221
26	0.048	0.048	0.0447	26	1.7635	1.7668	1.5984
27	0.0496	0.0496	0.0449	27	1.74	1.7362	1.6368
28	0.0537	0.053	0.0538	28	1.7763	1.7755	1.5795
29	0.0535	0.0529	0.0535	29	1.8219	1.8224	1.5708
30	0.0532	0.0536	0.0489	30	1.8985	1.9002	1.6174
31	0.0496	0.0495	0.0575	31	1.9166	1.8897	1.6403
32	0.0472	0.0479	0.0548	32	1.992	1.9361	1.6425
33	0.0456	0.0453	0.0385	33	1.7741	1.7738	1.6409
34	0.0426	0.0423	0.042	34	1.5579	1.5601	1.6759
35	0.0384	0.0413	0.0339	35	1.5138	1.5017	1.5287
36	0.036	0.0363	0.0384	36	1.5102	1.5028	1.5445
37	0.0354	0.0358	0.0457	37	1.4832	1.5171	1.6334
38	0.0421	0.0434	0.0321	38	1.4276	1.4353	1.6666
39	0.0427	0.043	0.023	39	1.51	1.4972	1.606
40	0.0442	0.044	0.0299	40	1.5734	1.588	1.662
41	0.0456	0.0463	0.0301	41	1.5633	1.5556	1.6305
42	0.0473	0.0462	0.0365	42	1.4966	1.4856	1.5987
43	0.0497	0.0512	0.0341	43	1.4868	1.4914	1.5832
44	0.05	0.0505	0.042	44	1.4864	1.4785	1.621
45	0.0498	0.0497	0.0451	45	1.4965	1.4854	1.6208
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
420	0.045	0.0449	0.0337	155	1.5581	1.5635	1.6326
421	0.0457	0.045	0.0309	156	1.6097	1.6195	1.574
422	0.0455	0.0459	0.0389	157	1.6435	1.6089	1.6232
423	0.0472	0.047	0.0306	158	1.5793	1.5817	1.6669
424	0.0468	0.0464	0.0385	159	1.5782	1.5826	1.649
425	0.0486	0.0481	0.0179	160	1.6108	1.6206	1.5725
426	0.0507	0.0499	0.0191	161	1.6368	1.6256	1.6879
427	0.052	0.0514	0.0257	162	1.662	1.6444	1.6681
428	0.0532	0.0539	0.029	163	1.6115	1.6156	1.6534
429	0.0555	0.0546	0.0379	164	1.571	1.5708	1.6387
430	0.0569	0.0588	0.0404	165	1.6692	1.6912	1.6243
431	0.0566	0.056	0.0487	166	1.6766	1.6832	1.5822
432	0.0579	0.0587	0.0432	167	1.7188	1.7224	1.5764
433	0.0569	0.0571	0.0436	168	1.7856	1.7785	1.6206
434	0.0596	0.0602	0.0393	169	1.8225	1.7952	1.6044
435	0.0609	0.0601	0.04	170	1.8699	1.8896	1.6792
436	0.06	0.0601	0.0483	171	1.8562	1.8964	1.5417
437	0.0611	0.0604	0.0292	172	1.772	1.7717	1.6087
438	0.0617	0.0617	0.031	173	1.8398	1.8372	1.5426
439	0.0577	0.0583	0.0379	174	1.8207	1.8214	1.6147
440	0.0515	0.0509	0.0464	175	1.8248	1.8242	1.6544
441	0.0488	0.05	0.0476	176	1.7934	1.7795	1.5929
442	0.0442	0.0441	0.0516	177	1.7982	1.8056	1.5845
443	0.0387	0.0445	0.0675	178	1.8335	1.835	1.6625
444	0.0362	0.0313	0.0484	179	1.934	1.9301	1.5832
445	0.0349	0.0386	0.0484	180	1.9054	1.8939	1.5472

In Table 22, we show a side by side comparison of the estimates for the simulated value using LLGMM and AGMM methods for U.S. Treasury Bill interest rate and U.S.–U.K. foreign exchange rate using initial delay r = 20.

Figure 21:

Real, Simulated paths using LLGMM and AGMM methods for U.S. Treasury Bill interest rate and U.S.–U.K. foreign exchange rate for initial delay r = 20.

C Comparative study of the LLGMM with OCBGMM methods

In this appendix, an additional detailed comparisons regarding the theoretical, graphical and performance of the LLGMM and OCBGMM methods are presented in Appendix C.1., Appendix C.2., and Appendix C.3., respectively. In fact, by employing three statistical goodness-of-fit measures (Czella, Karolyi, and Ronchetti 2007), a comparative performance analysis of forecasting and ranking of the LLGMM and OCBGMM based methods are presented in Appendix C.3.

C.1 Theoretical comparison between LLGMM and OCBGMM

Based on the foundations of the analytical, conceptual, computational, mathematical, practical, statistical and theoretical motivations and developments outlined in Sections 2, 3, 4, 5 and 6, we summarize the comparison between the innovative approach LLGMM with the existing and newly developed OCBGMM methods in separate tables in a systematic manner.

In the following, we state the differences between the LLGMM method and existing orthogonality condition based GMM/IRGMM-Algebraic and the newly formulated GMM/IRGMM-Analytic methods together with AGMM.

Table 23:

Mathematical comparison between the LLGMM and OCBGMM.

Feature	LLGMM	OCBGMM-Algebraic	OCGMM-Analytic	Justifications
Composition:	Seven components	Five components	Five components	Sections 1, 2
Model:	Development	Selection	Development/ selection	Sections 1, 2
Goal:	Validation	Specification/testing	Validation/testing	Sections 1, 2
Discrete-time Scheme:	Constructed from SDE	Using econometric specification	Constructed from SDE	Remark 8
Formation of Orthogonality Vector:	Using stochastic calculus	Formed using algebraic manipulation	Using Stochastic calculus	Remarks 2, 7, 8

Table 24:

Intercomponent interaction comparison between LLGMM and OCBGMM.

Feature	LLGMM	OCBGMM-Algebraic	OCGMM-Analytic	Justifications
Moment Equations:	Local Lagged adaptive process	Single/ global system	Single/ global system	Remarks 16a, and 16b
Type of Moment Equations:	Local lagged adaptive process	Single-shot	Single-shot	Remarks 6, 8, and 9
Component Interconnections:	Strongly connected	Weakly connected	Weakly connected	Remarks 6, 7, 8, 9, and 16
Dynamic and Static:	Discrete-time Dynamic	Static	Static	Remarks 16 and Lemma 1 (Section 2)

Table 25:

Conceptual computational comparison between LLGMM and OCBGMM.

Feature	LLGMM	OCBGMM-Algebraic	OCGMM-Analytic	Justifications
Local admissible Lagged Data Size:	Multi-choice	Single-choice/data size	Single-choice/data size	Definition 13, Remark 16, Subsection 3.2
Local admissible class of lagged finite restriction sequences	Multi-choice	Single-choice data sequence	Single-choice data sequence	Adapted finite restricted sample data: Definition 14, Remark 16, Subsection 3.2
Local admissible finite sequence parameter estimates:	Multi-choice	Single-shot estimate	Single-shot estimates	Subsection 3.2
Local admissible sequence of finite state simulation values:	Multi-choice	Single-choice	Single-choice	Remark 16, Subsection 3.3
Quadratic Mean Square ε-sub-optimal errors:	Multi-choice	Singe-error	Single-error	Remark 16, Subsection 3.3
ε-sub-optimal local lagged sample size:	Multi-choice	Single-choice	Single-choice	Definition 12, Remark 16, Subsection 3.3
ε-best sub optimal sample size:	ε-best sub optimal choice	No-choice	No-choice	Remark 16, Subsection 3.3
ε-best sub optimal parameter estimated:	ε-best estimators	No-choice	No-choice	Remark 16, Subsection 3.3
ε-best sub optimal state estimate:	ε-best sub optimal choice	No-choice	No-choice	Remark 16, Subsection 3.3

Table 26:

Theoretical performance comparison between LLGMM and OCBGMM.

Feature	LLGMM	OCBGMM-Algebraic	OCGMM-Analytic	Justifications
Data Size:	Reasonable Size	Large Data Size	Large Data Size	For Respectable results
Stationary Condition:	Not required	Need Ergotic/ Asymptotic stationary	Need Ergodic / Asymptotic	For Reasonable results
Multi-level optimization:	At least 2 level hierarchical optimization	Single-shot	Single-shot	Not comparable
Admissible Strategies:	Multi-choices	Single-shot	Single-shot	Not comparable
Computational Stability:	Algorithm Converges in single / double digit trials	Single-choice	Single-choice	Simulation results
Significance of lagged adaptive process:	Stabilizing agent	Non-existence of the feature	Non-existence	Not comparable
Operation:	Operates like Discrete time Dynamic Process	Operates like static dynamic process	Operates like static process	Obvious, details see Sections 3, 4, 5, 6 and 7

C.2 Graphical comparison of the LLGMM with OCBGMM methods

Parameter estimates of (46) using LLGMM and OCBGMM methods: Using the LLGMM method, the parameter estimates αmˆk,k, βmˆk,k, σmˆk,k, and γmˆk,k of (7) in USTBIR are shown in Table 27. Here, we use ε = 0.001, p = 2, and initial delay r = 20.

Table 27:

Estimates for mˆk, αmˆk,k, βmˆk,k, σmˆk,k, γmˆk,k for U.S. Treasury Bill interest rate data using LLGMM.

tk	Interest rate
	mˆk	αmkˆ,k	βmkˆ,k	σmkˆ,k	γmkˆ,k
21	2	0.0334	−0.7143	0.0446	1.5
22	3	0.0427	−0.9254	0.0766	1.5
23	4	0.0425	−0.9198	0.0914	1.5
24	5	0.0413	−0.8937	0.09	1.5
25	4	0.1042	−2.2619	0.1003	1.5
26	19	0.0002	0.0083	0.1043	1.5
27	14	0.0024	−0.0359	0.1281	1.5
28	5	−0.023	0.5207	0.3501	1.5
29	13	0.0037	−0.0573	0.1652	1.5
30	18	0.0008	0.001	0.1447	1.5
31	3	−0.3827	7.1316	0.26	1.5
32	19	0.006	−0.1213	0.1828	1.5
33	6	0.0063	−0.1359	0.343	1.5
34	19	0.0081	−0.1705	0.1993	1.5
35	4	−0.0166	0.2984	0.3509	1.5
36	4	−0.0059	0.0721	0.2318	1.5
37	9	−0.0035	0.0324	0.3114	1.5
38	14	0.0051	−0.1186	0.3385	1.5
39	20	0.0059	−0.1294	0.282	1.5
40	12	0.0075	−0.185	0.3447	1.5
41	12	0.0099	−0.2379	0.3579	1.5
42	4	−0.0089	0.2335	0.3562	1.5
43	7	0.0074	−0.1289	0.4654	1.5
44	7	0.0182	−0.3677	0.4206	1.5
45	6	0.0106	−0.2031	0.2356	1.5
⋯	⋯	⋯	⋯	⋯	⋯
⋯	⋯	⋯	⋯	⋯	⋯
420	3	0.0836	−1.9	0.1006	1.5
421	8	0.0428	−0.9671	0.783	1.5
422	3	0.0359	−0.7857	0.1702	1.5
423	8	0.0127	−0.2766	0.1719	1.5
424	6	0.0178	−0.3857	0.1636	1.5
425	6	0.0177	−0.3685	0.1829	1.5
426	18	0.0146	−0.3172	0.3871	1.5
427	8	0.0017	−0.012	0.1788	1.5
428	4	0.009	−0.1489	0.1341	1.5
429	9	−0.0059	0.1469	0.1616	1.5
430	13	−0.0046	0.116	0.191	1.5
431	9	0.0039	−0.0532	0.1369	1.5
432	9	0.0027	−0.0287	0.1109	1.5
433	3	0.0857	−1.5	0.0952	1.5
434	9	0.0102	−0.1661	0.1197	1.5
435	9	0.0075	−0.114	0.107	1.5
436	5	0.029	−0.485	0.1446	1.5
437	4	0.0476	−0.784	0.2163	1.5
438	9	0.0122	−0.1966	0.1054	1.5
439	4	0.1626	−2.6824	0.1248	1.5
440	20	0.0072	−0.1278	0.1916	1.5
441	19	0.0084	−0.1502	0.2016	1.5
442	17	0.0024	−0.0479	0.2369	1.5
443	7	−0.0153	0.2236	0.2687	1.5
444	3	0.0054	−0.2188	0.3887	1.5
445	16	−0.0076	0.1177	0.2528	1.5

Table 27 shows the parameter estimates of mˆk, αmˆk,k, βmˆk,k, σmˆk,k, γmˆk,k in the model (46) for U.S. Treasury Bill interest rate data. As noted before, the range of the ε-best sub-optimal local admissible sample size mˆk for ant time tk∈[21,45]⋃[420,445] is 2≤mˆk≤20. We also draw the similar conclusions (a) to (e) as outlined in Remark 18.

Figure 22:

Real and simulated path using LLGMM method.

Figure 22 shows the real and simulated path of the monthly interest rate data (U.S. Treasury 1964–2004) using the LLGMM method. The root mean square error of the simulated value is 0.0027.

Figure 23:

Comparison of simulation result using GMM-Analytic and GMM-Algebraic methods.

Figure 23(a) and (b) shows the real and simulated value of the monthly interest rate data (U.S. Treasury 1964–2004) using the GMM-Analytic and GMM-Algebraic methods, respectively. The root mean square errors of simulated values are shown in Table 10. Figure 23(c) shows the comparison between the real and simulated values of GMM-Analytic and GMM-Algebraic methods. The red, green, and blue line represent the real data path (U.S. Treasury 1964–2004), the simulated path using GMM-Algebraic, and the GMM-Analytic, respectively.

Figure 24:

Comparison of simulation result using IRGMM-Analytic and IRGMM-Algebraic methods.

Figure 24(a) and (b) shows the real and simulated value of the monthly interest rate data (U.S. Treasury 1964–2004) using the IRGMM-Analytic and IRGMM-Algebraic, respectively. The root mean square errors of simulated values are shown in Table 10. Figure 24(c) shows the comparison between the real and simulated values of IRGMM-Analytic and IRGMM-Algebraic method. The red, green, and blue curve represents the real data path (U.S. Treasury 1964–2004), simulated path using IRGMM-Algebraic, and GMM-Analytic, respectively.

Figure 25:

Comparison of simulation results of GMM-analytic, IRGMM-analytic, GMM-algebraic and IRGMM-algebraic methods together with AGMM method.

Figure 25(a) compares the simulation results using GMM-algebraic and IRGMM-algebraic. The blue denotes the GMM-algebraic simulation curve while the green line represents the IRGMM-algebraic simulation curve. Figure 25(b) compares the simulation results using the GMM-Analytic, and IRGMM-Analytic represented by the black, and green lines, respectively. Figure 25(c) compares the simulation results using the GMM-Algebra, IRGMM-Algebra, and LLGMM represented by the black, green, and blue lines, respectively.

Figure 26:

Comparison of simulation results for GMM-Analytic, IRGMM-analytic, GMM-Algebraic and IRGMM-Algebraic methods as well as the LLGMM and AGMM methods.

Figure 26(a) compares the simulation results using GMM-analytic, IRGMM-analytic and the LLGMM methods. The GMM-analytic, IRGMM-analytic and the LLGMM simulation results are exhibited by the black, green and blue lines, respectively. Figure 26(b) compares the simulation results using the GMM-Algebraic, IRGMM-Algebraic, LLGMM and AGMM methods represented by the black, green, blue and magenta lines, respectively. Figure 26(c) compares the simulation results using the GMM-Analytic, IRGMM-Analytic, LLGMM and AGMM methods represented by the black, green, blue and magenta curves, respectively.

Comparative analysis of forecasting with 95% confidence intervals: Using data set from June 1964 to December 1989, the parameters of model (46) are estimated. Using these parameter estimates, we forecasted the monthly interest rate for January 1, 1990 to December 31, 2004. Table 28 shows the parameter estimates in the context of the data from June 1964 to December 1989.

Table 28:

Parameter estimates in (46) in the context of data July 1964–December 1989.

Method	α	β	σ	γ
GMM-Algebraic	0.0033	−0.051	0.4121	1.5311
GMM-Analytic	0.0009	−0.0155	0.0197	0.4854
IRGMM-Algebraic	0.0023	−0.0421	0.3230	1.3112
IRGMM-Analytic	0.0084	−0.1436	0.1073	1.3641
AGMM	0.0154	−0.2497	0.2949	1.4414

Figure 27:

Real, simulation and Forecast state estimates using LLGMM method.

In Figure 27, region S shows the real, simulated value using the monthly interest rate data from June 30, 1964 to December 31, 1989 (U.S. Treasury 1964–2004). In the F region (forecasting region), the estimated parameters in the context of the data set (U.S. Treasury 1964–2004) are used to forecast interest rate from January 1, 1990 to December 31, 2004 using the LLGMM method.

Figure 28:

Real, simulation and forecast estimates using GMM-Algebraic, GMM-Analytic, IRGMM-Algebraic and IRGMM-Analytic methods.

Figure 28(a), (b), (c) and (d) exhibits the side-by-side comparison of the simulated forecasting results of the GMM-Analytic, GMM-Algebraic, IRGMM-Analytic, and IRGMM-Algebraic methods, respectively. The S region represents the simulation region based on the real data while the F region represents the forecasting region. In addition, the 95% confidence level of the simulation results are also shown (in black).

C.3 Performance Comparisons of LLGMM Method with Existing and Newly Introduced OCBGMM Methods Using Energy Commodity Stochastic Model

Using the stochastic dynamic model in (8) of energy commodity represented by stochastic differential equation:

(54)dy=ay(μ−y)dt+σ(t,yt)ydW(t), y(t0)=y0,

the orthogonality condition parameter vector (OCPV) is described in (13) in Remark (2). Based on discretized scheme using the econometric specification Chan et al. (1992), the orthogonality condition parameter vector in the context of algebraic manipulation is as Chan et al. (1992): OCBGMM looks like

(55)yt−yt−1−ayt−1(μ−yt−1)Δtyt−1yt−yt−1−ayt−1(μ−yt−1)Δtyt−yt−1−ayt−1(μ−yt−1)Δt)2−σ2yt−12

The goodness-of-fit measures are computed using pseudo-data sets of the same sample size as the real data set: (i) N = 1184 days for natural gas data, (ii) N = 4165 days for crude oil data, (iii) N = 3470 for coal data, and (iv) N = 438 weeks for ethanol data. The smallest value of RAMSEˆ for all method is italicized.

Table 29:

Comparison of parameter estimates of model (54) using GMM-Algebraic, GMM-Analytic, IRGMM-Algebraic, IRGMM-Analytic and AGMM for natural gas data.

Method	a	µ	σ2	RAMSEˆ	AMADˆ	AMBˆ
GMM-Algebraic	0.0023	5.3312	0.0019	1.5119	0.0663	1.1488
GMM-Analytic	0.0018	5.4106	0.0015	1.5014	0.0538	1.1677
IRGMM-Algebraic	0.2000	4.4996	0.0010	1.4985	0.0050	1.2299
IRGMM-Analytic	0.1998	4.4917	0.0011	1.4901	0.0044	1.2329
AGMM	0.1867	4.5538	0.0013	1.4968	0.0068	1.2267
LLGMM				0.0674∗	1.1318	1.1371

Table 30:

Comparison of parameter estimates of model (54) using GMM-Algebraic, GMM-Analytic, IRGMM-Algebraic, IRGMM-Analytic and AGMM for crude oil data.

Method	a	µ	σ2	RAMSEˆ	AMADˆ	AMBˆ
GMM-Algebraic	0.0023	54.4847	0.0005	39.2853	0.3577	29.1587
GMM-Analytic	0.0021	51.2145	0.0006	38.8007	0.5181	28.7414
IRGMM-Algebraic	0.0000	88.5951	0.0005	30.7511	0.0920	27.5791
IRGMM-Analytic	0.0021	51.2195	0.0005	28.9172	0.2496	27.3564
AGMM	0.0215	54.0307	0.0005	30.776	0.0857	27.3050
LLGMM				0.4625∗	24.501	27.2707

Table 31:

Comparison of parameter estimates of model (54) using GMM-Algebraic, GMM-Analytic, IRGMM-Algebraic, IRGMM-Analytic and AGMM for coal data.

Method	a	µ	σ2	RAMSEˆ	AMADˆ	AMBˆ
GMM-Algebraic	0.0000	94.4847	0.0006	22.6866	0.2015	16.3444
GMM-Analytic	0.0000	94.4446	0.0006	21.6564	0.2121	16.3264
IRGMM-Algebraic	0.0027	34.4838	0.0013	17.6894	0.3438	13.4981
IRGMM-Analytic	0.0021	23.1151	0.0005	17.6869	0.3448	13.4989
AGMM	0.0464	27.0567	0.0014	17.7620	0.0833	13.106
LLGMM				0.4794∗	9.4009	12.8370

Table 32:

Comparison of parameter estimates of model (54) using GMM-Algebraic, GMM-Analytic, IRGMM-Algebraic, IRGMM-Analytic and AGMM for ethanol data.

Method	a	µ	σ2	RAMSEˆ	AMADˆ	AMBˆ
GMM-Algebraic	0.0000	94.4847	0.0006	22.6866	0.2015	16.3444
GMM-Analytic	0.0000	94.4446	0.0006	21.6564	0.2121	16.3264
IRGMM-Algebraic	0.0014	3.4506	0.0026	0.5844	0.0322	0.4346
IRGMM-Analytic	0.0015	3.4506	0.0026	0.5813	0.0336	0.4303
AGMM	0.3167	2.166	0.0018	0.4356	0.0035	0.3579
LLGMM				0.0375∗	0.3213	0.3566

Table 29, Table 30, Table 31, and Table 32 show a comparison parameter estimates of model (54) and the goodness-of-fit measures RAMSEˆ, AMADˆ and AMBˆ using GMM-Algebraic, GMM-Analytic, IRGMM-Algebraic, IRGMM-Analytic, AGMM and LLGMM method for the daily natural gas data (Natural Gas 2000–2004), daily crude oil data (Crude Oil 1997–2008), daily coal data (Coal 2000–2013), and weekly ethanol data (Ethanol 2005–2013), respectively. The LLGMM estimates are derived using initial delay r = 20, p = 2 and ε = 0.001. Among all methods under study, the LLGMM method generates the smallest RAMSEˆ value. In fact, the RAMSEˆ value is smaller than the 1/22, 1/62, 1/36, and 1/10 of any other RAMSEˆ values regarding the natural gas, crude oil, coal and ethanol, respectively. This exhibits the superiority of the LLGMM method over all other methods. We further observe that the LLGMM approach yields the smallest AMBˆ and highest AMADˆ value regarding the natural gas, crude oil, coal and ethanol. The high value of AMADˆ for the LLGMM method signifies that LLGMM method captures the influence of random environmental fluctuations on the dynamic of energy commodity process. From Remark 20, the smallest RAMSEˆ, highest AMADˆ and smallest AMBˆ value under the LLGMM method exhibits the superior performance under the three goodness-of-fit measures.

Ranking of methods under goodness-of-fit measure

Table 33:

Ranking result for natural gas, crude oil, coal and ethanol under three statistical measures.

c	Rank of methods under goodness-of-fit measure
Method	Natural gas			Crude oil			Coal			Ethanol
	RAMSEˆ	AMADˆ	AMBˆ	RAMSEˆ	AMADˆ	AMBˆ	RAMSEˆ	AMADˆ	AMBˆ	RAMSEˆ	AMADˆ	AMBˆ
GMM-Algebraic	6	2	2	6	3	6	6	5	6	6	3	6
GMM-Analytic	5	3	3	5	2	5	5	4	5	5	2	5
IRGMM-Algebraic	4	5	5	3	5	4	4	3	3	4	5	4
IRGMM-Analytic	2	6	6	2	4	3	3	2	4	3	4	3
AGMM	3	4	4	4	6	2	2	6	2	2	6	2
LLGMM	1	1	1	1	1	1	1	1	1	1	1	1

Remark 24.

The ranking of LLGMM is top one in all three Goodness-of-fit statistical measures for all four energy commodity data sets. Moreover, one of the IRGMM-Analytic and AGMM is ranked either as top two or three under RAMSEˆ measure. This exhibits the influence of the usage of stochastic calculus based orthogonality condition parameter vectors (OCPV-Analytic).

D Comparative analysis of LLGMM with existing nonparametric statistical methods

In this section, we compare our LLGMM method with existing nonparametric methods. We consider the following existing nonparametric methods.

D.1 Nonparametric estimation of nonlinear dynamics by metric-based local linear approximation (LLA)

The LLA method Shoji (2013) assumes no functional form of a given model but estimates from experimental data by approximating the curve implied by the function of the tangent plane around the neighborhood of a tangent point. Suppose the state of interest xt at time t is differentiable with respect to t and satisfies dxt=f(xt)dt, where f:ℜk→ℜ is a smooth map, xt∈ℜk. The approximation of the curve f(xt) in a neighborhood Uϵ(x0)={x: d(x,x0)<ϵ} is defined by a tangent plane at x0:

yt=f(x0)+∑i=1k∂f∂xi(x0)(xi−x0),

where d is a metric on ℜk. Allowing error in the equation and assigning a weight w(xt) to each error terms ϵt, the method reduces to estimating parameters βi=∂f∂xi(x0), i = 1,2,…,k in the equation

w(t)yt=β0⋅w(xt)+∑i=1kβi⋅w(xt)(xt,i−x0,i).

Applying the standard linear regression approach, the least square estimate βˆ is given by

(56)βˆ=X˜TX˜−1X˜TY˜,

where

x˜i=w(xt1)(xt1,i−x0,i),…,w(xtn)(xtn,i−x0,i)T, i=1,…,kw˜=w(xt1),…,w(xtn)TY˜=w(xt1)yt1,…,w(xtn)ytnTX˜=w˜,x˜1,…,x˜k

Particularly, the trajectory f(xti) is estimated by choosing x0=xti, for each i = 1,2,…,n. As discussed in Shoji (2013), we use d(x,x0)=|x−x0|, where |.| is the standard Euclidean metric on ℜk, and w(x)=ϕ(d(x,x0)), where ϕ(u)=K(u/ϵ) and K is the Epanechnikov Kernel Shoji (2013)K(x)=0.75(1−x2)+.

D.2 Risk estimation and adaptation after coordinate transformation (REACT) method

Given n pairs of observations (x1,Y1), …, (xn,Yn). Using the REACT method Wasserman (2007), the response variable Y is related to the covariate x (called a feature) by the equation

(57)Yi=r(xi)+σϵi,

where ϵi∼N(0,1) are identically independently distributed, and xi=in, i = 1,2,…,n and the function r(x), approximated using orthogonal cosine basis ϕi, i=1,2,3,… of [0,1] described by

(58)ϕ1(x)≡1, ϕj(x)=2cos((j−1)πx), j≥2.

is expanded as

(59)r(x)=∑j=1∞θjϕj(x),

where θj=∫01ϕj(x)r(x)dx, is approximated. The function estimator rˆ(x)=∑j=1JˆZjϕj(x), where Zj=1n∑i=1nYiϕj(xi), j = 1,2,…,n and Jˆ is found so that the risk estimator Rˆ(J)=Jσˆ2n+∑j=J+1nZj2−σˆ2n is minimized, σˆ2 is the estimator of variance of Zj.

D.3 Exponential moving average (EMA) method

The EMA (Lucas and Saccucci 1990) for an observation yt at time t may be calculated recursively as

(60)St=αyt+(1−α)St−1, t=1,2,3,…,n

where 0 < α≤1 is a constant that determines the depth of memory of St.

D.4 Goodness-of-fit measures for the LLA, REACT, and EMA methods

In this subsection, we show the goodness-of-fit measures for the LLA, REACT, and EMA methods. We use Jˆ=183 for the REACT method and α = 0.5 for the EMA method.

Table 34:

Goodness-of-fit measures for the LLGMM, REACT, and EMA methods

Goodness-of-fit Measure	LLGMM method				LLA method				REACT method				EMA method
Goodness-of-fit Measure	Natural gas	Crude oil	Coal	Ethanol	Natural gas	Crude oil	Coal	Ethanol	Natural gas	Crude oil	Coal	Ethanol	Natural gas	Crude oil	Coal	Ethanol
RAMSEˆ	0.0674	0.4625	0.4794	0.0375	0.3114	1.9163	2.1645	0.2082	0.1895	2.0377	2.0162	0.0775	0.1222	0.7845	0.8233	0.0682
AMADˆ	1.1318	24.5010	9.4009	0.3213	1.1406	24.3266	9.4511	0.3290	1.1779	24.6967	9.3791	0.3291	1.1336	24.5858	9.4183	0.3159
AMBˆ	1.1371	27.2707	12.8370	0.3566	1.2375	27.2713	12.8388	0.3677	1.12352	27.2711	12.8369	0.3566	1.2352	27.2710	12.8370	0.3567

Comparison of the results derived using these non-parametric methods with the LLGMM method show that the results derived using the LLGMM method is far better than the results of the nonparametric methods.

Graphical comparison of the LLGMM with LLA, REACT, and EMA methods:

$Figure 29: Real and simulated curve using LLA method.Figure 29(a), (b), (c) and (d) shows the graphs of the Real and Simulated Spot Prices for the daily Henry Hub natural gas data (Natural Gas 2000–2004), daily crude oil data (Crude Oil 1997–2008), daily coal data (Coal 2000–2013), and weekly ethanol data (Ethanol 2005–2013), respectively, using LLA method. The red line represents the real data yky_{k} while the blue line represents the simulated value.$

Figure 29:

Real and simulated curve using LLA method.

Figure 29(a), (b), (c) and (d) shows the graphs of the Real and Simulated Spot Prices for the daily Henry Hub natural gas data (Natural Gas 2000–2004), daily crude oil data (Crude Oil 1997–2008), daily coal data (Coal 2000–2013), and weekly ethanol data (Ethanol 2005–2013), respectively, using LLA method. The red line represents the real data yk while the blue line represents the simulated value.

$Figure 30: Real and simulated curve using REACT method.Figure 30(a), (b), (c) and (d) shows the graphs of the Real and Simulated Spot Prices for the daily Henry Hub natural gas data (Natural Gas 2000–2004), daily crude oil data (Crude Oil 1997–2008), daily coal data (Coal 2000–2013) and weekly ethanol data (Ethanol 2005–2013), respectively, using the REACT method. The red line represents the real data yky_{k} while the blue line represents the simulated value.$

Figure 30:

Real and simulated curve using REACT method.

Figure 30(a), (b), (c) and (d) shows the graphs of the Real and Simulated Spot Prices for the daily Henry Hub natural gas data (Natural Gas 2000–2004), daily crude oil data (Crude Oil 1997–2008), daily coal data (Coal 2000–2013) and weekly ethanol data (Ethanol 2005–2013), respectively, using the REACT method. The red line represents the real data yk while the blue line represents the simulated value.

$Figure 31: Real and simulated curve using EMA method.Figure 31(a), (b), (c) and (d) shows the graphs of the Real and Simulated Spot Prices for the daily Henry Hub natural gas data (Natural Gas 2000–2004), daily crude oil data (Crude Oil 1997–2008), daily coal data Coal (2000–2013) and weekly ethanol data (Ethanol 2005–2013), respectively, using the EMA method. The red line represents the real data yky_{k} while the blue line represents the simulated value.$

Figure 31:

Real and simulated curve using EMA method.

Figure 31(a), (b), (c) and (d) shows the graphs of the Real and Simulated Spot Prices for the daily Henry Hub natural gas data (Natural Gas 2000–2004), daily crude oil data (Crude Oil 1997–2008), daily coal data Coal (2000–2013) and weekly ethanol data (Ethanol 2005–2013), respectively, using the EMA method. The red line represents the real data yk while the blue line represents the simulated value.

Figure 32:

Comparison of Real and simulated curve using LLGMM, LLA, REACT, and EMA method.

Figure 32(a), (b), (c) and (d) shows the graphs of the Real and Simulated Spot Prices for the daily Henry Hub natural gas data (Natural Gas 2000–2004), daily crude oil data (Crude Oil 1997–2008), daily coal data (Coal 2000–2013) and weekly ethanol data (Ethanol 2005–2013), respectively, using the LLGMM, LLA, REACT, and EMA method. The red line represents the real data.

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Published Online: 2019-01-23

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Journal of Time Series Econometrics

Volume 11 Issue 1

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Research Articles

Local Lagged Adapted Generalized Method of Moments: An Innovative Estimation and Forecasting Approach and its Applications

Modelling with Dispersed Bivariate Moving Average Processes

A Neural Network Method for Nonlinear Time Series Analysis

Finite-Sample Theory and Bias Correction of Maximum Likelihood Estimators in the EGARCH Model