A Proposal to Fix the Number of Factors on Modeling the Dynamics of Futures Contracts on Commodity Prices †
Abstract
:1. Introduction
2. Theoretical Model
2.1. Theoretical Model
2.2. A General Procedure to Determine the Stochastic Factors
- Compute .
- Compute the rank of . Let us assume that this is 3.
- As a result, we have three eigenvalues and . It is usual to assume that as the futures process is not stationary, but can nevertheless be estimated. If we do assume it, however, we obtain that is a linear combination of the products of , and . Therefore, we obtain the general equation which can be estimated numerically as:
- Select an initial estimate of .
- Regress and compute the error.
- Iteratively select another estimate of and get back to b.
3. Data and Main Results
3.1. Data
3.2. Main Results
4. Summary and Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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WTI Crude Oil | Gasoline | Natural Gas | Heating Oil | |||||
---|---|---|---|---|---|---|---|---|
Mean ($/bbl) | Volatility (%) | Mean ($/bbl) | Volatility (%) | Mean ($/MMBtu) | Volatility (%) | Mean ($/bbl) | Volatility (%) | |
F1 | 43.1 | 30% | 96.7 | 32% | 4.2 | 45% | 63.6 | 28% |
F2 | 43.3 | 27% | 96.5 | 30% | 4.3 | 40% | 63.8 | 26% |
F3 | 43.3 | 25% | 96.4 | 29% | 4.3 | 36% | 64.0 | 24% |
F4 | 43.4 | 24% | 96.3 | 27% | 4.4 | 32% | 64.1 | 23% |
F5 | 43.3 | 23% | 96.1 | 27% | 4.4 | 29% | 64.1 | 22% |
F6 | 43.3 | 22% | 95.9 | 26% | 4.4 | 27% | 64.2 | 21% |
F7 | 43.3 | 21% | 95.7 | 25% | 4.5 | 26% | 64.2 | 20% |
F8 | 43.2 | 20% | 95.6 | 26% | 4.5 | 24% | 64.1 | 19% |
F9 | 43.2 | 20% | 95.6 | 25% | 4.5 | 24% | 64.1 | 19% |
F10 | 43.1 | 19% | 95.4 | 26% | 4.5 | 22% | 64.1 | 18% |
F11 | 43.1 | 19% | 95.5 | 25% | 4.5 | 22% | 64.1 | 18% |
F12 | 43.0 | 18% | 95.5 | 25% | 4.5 | 21% | 64.0 | 17% |
F13 | 43.0 | 18% | 4.5 | 20% | 63.7 | 17% | ||
F14 | 42.9 | 18% | 4.5 | 20% | 63.4 | 17% | ||
F15 | 42.9 | 17% | 4.5 | 20% | 63.0 | 17% | ||
F16 | 42.8 | 17% | 4.5 | 20% | ||||
F17 | 42.8 | 17% |
WTI Crude Oil | Gasoline | Natural Gas | Heating oil | |||||
---|---|---|---|---|---|---|---|---|
Mean ($/bbl) | Volatility (%) | Mean ($/bbl) | Volatility (%) | Mean ($/MMBtu) | Volatility (%) | Mean ($/bbl) | Volatility (%) | |
F1 | 50.9 | 30% | 101.1 | 32% | 4.9 | 45% | 53.3 | 30% |
F2 | 51.2 | 28% | 100.6 | 30% | 5.0 | 40% | 53.5 | 28% |
F3 | 51.3 | 26% | 100.3 | 30% | 5.1 | 38% | 53.6 | 26% |
F4 | 51.3 | 25% | 100.1 | 28% | 5.1 | 33% | 53.7 | 25% |
F5 | 51.3 | 24% | 99.8 | 28% | 5.2 | 31% | 53.7 | 24% |
F6 | 51.3 | 23% | 99.6 | 27% | 5.2 | 28% | 53.7 | 23% |
F7 | 51.3 | 22% | 99.3 | 26% | 5.3 | 27% | 53.7 | 22% |
F8 | 51.3 | 21% | 99.2 | 26% | 5.3 | 26% | 53.7 | 21% |
F9 | 51.2 | 21% | 99.1 | 25% | 5.3 | 25% | 53.7 | 21% |
F10 | 51.2 | 20% | 99.1 | 27% | 5.3 | 24% | 53.6 | 20% |
F11 | 51.1 | 20% | 99.1 | 26% | 5.3 | 22% | 53.6 | 19% |
F12 | 51.1 | 19% | 99.1 | 26% | 5.3 | 21% | 53.5 | 19% |
F13 | 51.0 | 19% | 99.1 | 26% | 5.3 | 21% | 53.3 | 19% |
F14 | 50.9 | 19% | 99.0 | 25% | 5.3 | 21% | 53.0 | 19% |
F15 | 50.8 | 18% | 98.9 | 25% | 5.3 | 21% | 53.0 | 18% |
F16 | 50.8 | 18% | 98.8 | 23% | 5.3 | 20% | 53.5 | 18% |
F17 | 50.7 | 18% | 98.5 | 24% | 5.3 | 20% | 55.4 | 18% |
F18 | 50.7 | 18% | 98.3 | 23% | 5.3 | 19% | 58.4 | 18% |
F19 | 50.6 | 17% | 98.1 | 23% | 5.3 | 20% | ||
F20 | 50.5 | 17% | 98.0 | 23% | 5.3 | 19% | ||
F21 | 50.5 | 17% | 98.0 | 23% | 5.3 | 19% | ||
F22 | 50.4 | 17% | 97.9 | 24% | 5.3 | 20% | ||
F23 | 50.4 | 17% | 97.9 | 23% | 5.2 | 18% | ||
F24 | 50.3 | 17% | 97.9 | 24% | 5.2 | 18% | ||
F25 | 50.3 | 16% | 97.9 | 24% | 5.2 | 18% | ||
F26 | 50.2 | 16% | 97.8 | 23% | 5.2 | 18% | ||
F27 | 50.2 | 16% | 97.8 | 24% | 5.2 | 18% | ||
F28 | 50.1 | 16% | 97.7 | 23% | 5.2 | 18% | ||
F29 | 97.6 | 23% | 5.2 | 18% | ||||
F30 | 97.5 | 22% | 5.2 | 18% | ||||
F31 | 97.4 | 22% | 5.2 | 18% | ||||
F32 | 97.4 | 23% | 5.2 | 19% | ||||
F33 | 97.3 | 22% | 5.2 | 18% | ||||
F34 | 97.1 | 23% | 5.2 | 19% | ||||
F35 | 97.0 | 23% | 5.2 | 18% | ||||
F36 | 97.0 | 23% | 5.2 | 17% |
Dataset 1 | Dataset 2 | ||||
---|---|---|---|---|---|
Eigenvalues | Percentage of Total Variance | Cumulative Variance (%) | Eigenvalues | Percentage of Total Variance | Cumulative Variance (%) |
100 | 99.6713 | 99.6713 | 100 | 99.5448 | 99.5448 |
0.3202 | 0.3191 | 99.9904 | 0.4428 | 0.4408 | 99.9855 |
0.0084 | 0.0084 | 99.9988 | 0.0126 | 0.0125 | 99.9980 |
0.0010 | 0.0010 | 99.9998 | 0.0017 | 0.0017 | 99.9997 |
0.0001 | 0.0001 | 99.9999 | 0.0002 | 0.0002 | 99.9998 |
2.9905 × 10−5 | 2.9806 × 10−5 | 100 | 0.0001 | 0.0001 | 99.9999 |
1.2209 × 10−5 | 1.2169 × 10−5 | 100 | 3.7318 × 10−5 | 3.7148 × 10−5 | 100 |
5.4907 × 10−6 | 5.4727 × 10−6 | 100 | 1.6898 × 10−5 | 1.6821 × 10−5 | 100 |
2.7838 × 10−6 | 2.7746 × 10−6 | 100 | 8.7477 × 10−6 | 8.7079 × 10−6 | 100 |
1.5250 × 10−6 | 1.5200 × 10−6 | 100 | 4.4713 × 10−6 | 4.4509 × 10−6 | 100 |
7.5290 × 10−7 | 7.5043 × 10−7 | 100 | 3.0355 × 10−6 | 3.0217 × 10−6 | 100 |
4.3460 × 10−7 | 4.3317 × 10−7 | 100 | 2.3004 × 10−6 | 2.2899 × 10−6 | 100 |
3.3010 × 10−7 | 3.2901 × 10−7 | 100 | 1.3628 × 10−6 | 1.3566 × 10−6 | 100 |
1.8100 × 10−7 | 1.8041 × 10−7 | 100 | 8.7940 × 10−7 | 8.7540 × 10−7 | 100 |
1.1490 × 10−7 | 1.1452 × 10−7 | 100 | 4.7100 × 10−7 | 4.6886 × 10−7 | 100 |
1.0350 × 10−7 | 1.0316 × 10−7 | 100 | 3.6450 × 10−7 | 3.6284 × 10−7 | 100 |
3.9300 × 10−8 | 3.9171 × 10−8 | 100 | 2.3880 × 10−7 | 2.3771 × 10−7 | 100 |
1.8840 × 10−7 | 1.8754 × 10−7 | 100 | |||
1.4180 × 10−7 | 1.4115 × 10−7 | 100 | |||
1.1480 × 10−7 | 1.1428 × 10−7 | 100 | |||
1.0070 × 10−7 | 1.0024 × 10−7 | 100 | |||
7.7800 × 10−8 | 7.7446 × 10−8 | 100 | |||
5.4200 × 10−8 | 5.3953 × 10−8 | 100 | |||
4.7800 × 10−8 | 4.7582 × 10−8 | 100 | |||
3.8600 × 10−8 | 3.8424 × 10−8 | 100 | |||
2.3000 × 10−8 | 2.2895 × 10−8 | 100 | |||
2.1600 × 10−8 | 2.1502 × 10−8 | 100 | |||
8.9000 × 10−9 | 8.8595 × 10−9 | 100 |
Dataset 1 | Dataset 2 | ||||
---|---|---|---|---|---|
Eigenvalues | Percentage of Total Variance | Cumulative Variance (%) | Eigenvalues | Percentage of Total Variance | Cumulative Variance (%) |
100 | 99.6133 | 99.6133 | 100 | 99.5365 | 99.5365 |
0.2698 | 0.2687 | 99.8820 | 0.3666 | 0.3649 | 99.9014 |
0.0658 | 0.0655 | 99.9475 | 0.0475 | 0.0472 | 99.9486 |
0.0474 | 0.0472 | 99.9947 | 0.0448 | 0.0446 | 99.9932 |
0.0028 | 0.0028 | 99.9975 | 0.0037 | 0.0037 | 99.9969 |
0.0013 | 0.0013 | 99.9988 | 0.0012 | 0.0012 | 99.9981 |
0.0009 | 0.0008 | 99.9997 | 0.0011 | 0.0011 | 99.9992 |
0.0001 | 0.0001 | 99.9998 | 0.0005 | 0.0005 | 99.9997 |
0.0001 | 0.0001 | 99.9999 | 0.0001 | 0.0001 | 99.9998 |
4.7937 × 10−5 | 4.7752 × 10−5 | 99.9999 | 0.0001 | 0.0001 | 99.9999 |
1.9734 × 10−5 | 1.9658 × 10−5 | 100 | 4.1450 × 10−5 | 4.1257 × 10−5 | 99.9999 |
1.1626 × 10−5 | 1.1581 × 10−5 | 100 | 2.8767 × 10−5 | 2.8633 × 10−5 | 99.9999 |
1.0482 × 10−5 | 1.0441 × 10−5 | 100 | 1.5784 × 10−5 | 1.5711 × 10−5 | 100 |
9.6273 × 10−6 | 9.5901 × 10−6 | 100 | 1.3478 × 10−5 | 1.3416 × 10−5 | 100 |
6.3346 × 10−6 | 6.3101 × 10−6 | 100 | 9.3425 × 10−6 | 9.2992 × 10−6 | 100 |
7.9877 × 10−6 | 7.9507 × 10−6 | 100 | |||
6.1859 × 10−6 | 6.1572 × 10−6 | 100 | |||
5.7507 × 10−6 | 5.7240 × 10−6 | 100 |
Dataset 1 | Dataset 2 | ||||
---|---|---|---|---|---|
Eigenvalues | Percentage of Total Variance | Cumulative Variance (%) | Eigenvalues | Percentage of Total Variance | Cumulative Variance (%) |
100 | 96.8591 | 96.8591 | 100 | 95.2473 | 95.2473 |
1.6748 | 1.6222 | 98.4813 | 2.3570 | 2.2450 | 97.4924 |
1.2901 | 1.2496 | 99.7308 | 1.3050 | 1.2429 | 98.7353 |
0.1334 | 0.1292 | 99.8600 | 1.0762 | 1.0250 | 99.7603 |
0.0558 | 0.0540 | 99.9140 | 0.0639 | 0.0608 | 99.8212 |
0.0386 | 0.0374 | 99.9515 | 0.0599 | 0.0570 | 99.8782 |
0.0217 | 0.0210 | 99.9724 | 0.0437 | 0.0416 | 99.9198 |
0.0156 | 0.0151 | 99.9876 | 0.0217 | 0.0206 | 99.9405 |
0.0093 | 0.0090 | 99.9966 | 0.0208 | 0.0198 | 99.9602 |
0.0022 | 0.0022 | 99.9988 | 0.0171 | 0.0163 | 99.9765 |
0.0009 | 0.0009 | 99.9997 | 0.0074 | 0.0070 | 99.9835 |
0.0003 | 0.0003 | 100 | 0.0049 | 0.0047 | 99.9882 |
0.0030 | 0.0029 | 99.9910 | |||
0.0025 | 0.0024 | 99.9935 | |||
0.0019 | 0.0018 | 99.9953 | |||
0.0012 | 0.0011 | 99.9964 | |||
0.0008 | 0.0008 | 99.9972 | |||
0.0006 | 0.0006 | 99.9978 | |||
0.0004 | 0.0004 | 99.9982 | |||
0.0004 | 0.0003 | 99.9986 | |||
0.0003 | 0.0003 | 99.9989 | |||
0.0003 | 0.0003 | 99.9991 | |||
0.0002 | 0.0002 | 99.9993 | |||
0.0001 | 0.0001 | 99.9995 | |||
0.0001 | 0.0001 | 99.9996 | |||
0.0001 | 0.0001 | 99.9997 | |||
0.0001 | 0.0001 | 99.9997 | |||
0.0001 | 0.0001 | 99.9998 | |||
0.0001 | 0.0000 | 99.9999 | |||
3.8439 × 10−5 | 3.6612 × 10−5 | 99.9999 | |||
2.8300 × 10−5 | 2.6955 × 10−5 | 99.9999 | |||
2.4205 × 10−5 | 2.3055 × 10−5 | 99.9999 | |||
1.9530 × 10−5 | 1.8602 × 10−5 | 100 | |||
1.5016 × 10−5 | 1.4303 × 10−5 | 100 | |||
1.2694 × 10−5 | 1.2091 × 10−5 | 100 | |||
9.9475 × 10−6 | 9.4747 × 10−6 | 100 |
Dataset 1 | Dataset 2 | ||||
---|---|---|---|---|---|
Eigenvalues | Percentage of Total Variance | Cumulative Variance (%) | Eigenvalues | Percentage of Total Variance | Cumulative Variance (%) |
100 | 97.8179 | 97.8179 | 100 | 95.8957 | 95.8957 |
1.1564 | 1.1311 | 98.9491 | 2.7972 | 2.6824 | 98.5782 |
0.4785 | 0.4681 | 99.4172 | 0.5993 | 0.5747 | 99.1529 |
0.3960 | 0.3874 | 99.8046 | 0.5221 | 0.5007 | 99.6535 |
0.0839 | 0.0821 | 99.8867 | 0.1178 | 0.1130 | 99.7665 |
0.0730 | 0.0714 | 99.9580 | 0.0993 | 0.0952 | 99.8617 |
0.0223 | 0.0218 | 99.9798 | 0.0782 | 0.0750 | 99.9367 |
0.0052 | 0.0050 | 99.9849 | 0.0166 | 0.0159 | 99.9527 |
0.0039 | 0.0038 | 99.9887 | 0.0074 | 0.0071 | 99.9598 |
0.0031 | 0.0030 | 99.9917 | 0.0070 | 0.0067 | 99.9665 |
0.0027 | 0.0027 | 99.9944 | 0.0060 | 0.0057 | 99.9722 |
0.0023 | 0.0023 | 99.9967 | 0.0051 | 0.0049 | 99.9772 |
0.0012 | 0.0012 | 99.9979 | 0.0048 | 0.0046 | 99.9818 |
0.0010 | 0.0010 | 99.9988 | 0.0038 | 0.0037 | 99.9854 |
0.0007 | 0.0007 | 99.9995 | 0.0032 | 0.0031 | 99.9886 |
0.0005 | 0.0005 | 100 | 0.0024 | 0.0023 | 99.9908 |
0.0020 | 0.0019 | 99.9927 | |||
0.0018 | 0.0017 | 99.9944 | |||
0.0017 | 0.0016 | 99.9961 | |||
0.0013 | 0.0012 | 99.9973 | |||
0.0010 | 0.0009 | 99.9983 | |||
0.0006 | 0.0006 | 99.9988 | |||
0.0003 | 0.0003 | 99.9991 | |||
0.0002 | 0.0002 | 99.9993 | |||
0.0002 | 0.0002 | 99.9995 | |||
0.0001 | 0.0001 | 99.9996 | |||
0.0001 | 0.0001 | 99.9997 | |||
0.0001 | 0.0001 | 99.9998 | |||
0.0001 | 0.0001 | 99.9998 | |||
0.0001 | 0.0001 | 99.9999 | |||
3.2345 × 10−5 | 3.1018 × 10−5 | 99.9999 | |||
2.8128 × 10−5 | 2.6974 × 10−5 | 100 | |||
1.5565 × 10−5 | 1.4926 × 10−5 | 100 | |||
1.2489 × 10−5 | 1.1977 × 10−5 | 100 | |||
9.3473 × 10−6 | 8.9637 × 10−6 | 100 | |||
7.6972 × 10−6 | 7.3813 × 10−6 | 100 |
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García-Mirantes, A.; Larraz, B.; Población, J. A Proposal to Fix the Number of Factors on Modeling the Dynamics of Futures Contracts on Commodity Prices . Mathematics 2020, 8, 973. https://doi.org/10.3390/math8060973
García-Mirantes A, Larraz B, Población J. A Proposal to Fix the Number of Factors on Modeling the Dynamics of Futures Contracts on Commodity Prices . Mathematics. 2020; 8(6):973. https://doi.org/10.3390/math8060973
Chicago/Turabian StyleGarcía-Mirantes, Andrés, Beatriz Larraz, and Javier Población. 2020. "A Proposal to Fix the Number of Factors on Modeling the Dynamics of Futures Contracts on Commodity Prices " Mathematics 8, no. 6: 973. https://doi.org/10.3390/math8060973
APA StyleGarcía-Mirantes, A., Larraz, B., & Población, J. (2020). A Proposal to Fix the Number of Factors on Modeling the Dynamics of Futures Contracts on Commodity Prices . Mathematics, 8(6), 973. https://doi.org/10.3390/math8060973