Co-Jumps, Co-Jump Tests, and Volatility Forecasting: Monte Carlo and Empirical Evidence
Abstract
:1. Introduction
2. Co-Jump Identification
2.1. Setup
2.2. Co-Jump Tests
2.2.1. BLT Co-Jump Test
2.2.2. JT Co-Jump Test
2.2.3. Univariate Jump Co-Exceedance Rule
3. Monte Carlo Experiments
4. Forecasting Methods
4.1. Forecasting Models
4.2. Co-Jump and Idiosyncratic Jump Variations
5. Setup of the Forecasting Experiment
- Set .
- Test for the null hypothesis of equal predictive ability among candidate models in Q with significance level . If the null hypothesis cannot be rejected, then the set SSM is the same as the set of all models. Otherwise, determine the worst model in the set Q.
- Remove the worst model from Q, and proceed to step 2.
6. Data Description
7. Empirical Findings
8. Policy Implications
9. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
1 | For literature in this area, see (Barndorff-Nielsen and Shephard 2006; Lee and Mykland 2007; Jiang and Oomen 2008; Aït-Sahalia et al. 2009; Huang and Tauchen 2005; Mancini 2009; Podolskij and Ziggel 2010; Corradi et al. 2018; Boswijk et al. 2018; Mukherjee et al. 2020, and the references cited therein). |
2 | Bandi and Russell (2008) show in the presence of market microstructure noise, realized variance does not identify daily integrated variance of the frictionless equilibrium. They propose a more general treatment of the effect of market microstructure noise on realized variance estimates. |
3 | We follow the parameter settings from Jacod and Todorov (2009). |
4 | Among 10 assets, co-jumps occur more frequently among a small portion of all assets (e.g., ) than a large portion of all assets (e.g., ) at each time. |
5 | Gilder et al. (2014) also implement the same approach. |
6 | One approach to measure jump variation is to use the difference between realized volatility and bipower variation, as shown in Equations (34) and (35). Another approach to examine the jump power variation is formed using power transformation of the instantaneous return, i.e., . Key papers in this area include (Ding et al. 1993; Ding and Granger 1996; Todorov and Tauchen 2010; Barndorff-Nielsen et al. 2008, and the references cited there in). |
7 | The co-exceedance rule of the LM test is utilized to identify co-jumps and idiosyncratic jumps among assets. |
8 | The set of models that consists of the best model(s). |
9 | Overall, the mean absolute forecasting errors in Table 13 are comparatively smaller with respect to results in the relevant literature (e.g., see Qiu et al. 2019). |
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Jump Intensity | |
Leverage Effect | |
Jump Distribution | , where |
Sampling Frequency | |
Other Parameters |
N | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
DGP1 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||||
DGP2 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||||
DGP3 | 0.1 | 0.3 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP4 | 0.2 | 0.3 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP5 | 0.1 | 0.4 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP6 | 0.2 | 0.4 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP7 | 0.1 | 0.3 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP8 | 0.2 | 0.3 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP9 | 0.1 | 0.4 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP10 | 0.2 | 0.4 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP11 | 0.1 | 0.3 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP12 | 0.2 | 0.3 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP13 | 0.1 | 0.4 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP14 | 0.2 | 0.4 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP15 | 0.1 | 0.3 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP16 | 0.2 | 0.3 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP17 | 0.1 | 0.4 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP18 | 0.2 | 0.4 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |||
DGP19 | 0.1 | 0.1 | 0.3 | 0.3 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |
DGP20 | 0.2 | 0.2 | 0.3 | 0.3 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |
DGP21 | 0.1 | 0.1 | 0.4 | 0.4 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |
DGP22 | 0.2 | 0.2 | 0.4 | 0.4 | 0 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |
DGP23 | 0.1 | 0.1 | 0.3 | 0.3 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |
DGP24 | 0.2 | 0.2 | 0.3 | 0.3 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |
DGP25 | 0.1 | 0.1 | 0.4 | 0.4 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 | |
DGP26 | 0.2 | 0.2 | 0.4 | 0.4 | −0.5 | 0.05 | 5 | 0.0144 | 0.5 | 10 |
Case | BLT | JT | LM | LM–BNS |
---|---|---|---|---|
DGP1 | 0.0687 | 0.0220 | 0.0426 | 0.0014 |
DGP2 | 0.0718 | 0.0143 | 0.0435 | 0.0010 |
Case | BLT | JT | LM | LM–BNS |
---|---|---|---|---|
DGP3 | 0.0565 | 0.1892 | 0.0478 | 0.0015 |
DGP4 | 0.0506 | 0.3751 | 0.0520 | 0.0022 |
DGP5 | 0.0541 | 0.2486 | 0.0480 | 0.0017 |
DGP6 | 0.0476 | 0.4510 | 0.0523 | 0.0028 |
DGP7 | 0.0554 | 0.1771 | 0.0466 | 0.0019 |
DGP8 | 0.0480 | 0.3982 | 0.0512 | 0.0027 |
DGP9 | 0.0526 | 0.2178 | 0.0468 | 0.0022 |
DGP10 | 0.0452 | 0.4543 | 0.0515 | 0.0034 |
Case | BLT | JT | LM | LM–BNS |
---|---|---|---|---|
DGP11 | 0.9255 | 0.9451 | 0.9362 | 0.1702 |
DGP12 | 0.9167 | 0.9836 | 0.9495 | 0.2121 |
DGP13 | 0.9362 | 0.9670 | 0.9521 | 0.2606 |
DGP14 | 0.9242 | 0.9891 | 0.9697 | 0.3409 |
DGP15 | 0.9202 | 0.9451 | 0.9415 | 0.2340 |
DGP16 | 0.9242 | 0.9836 | 0.9672 | 0.2020 |
DGP17 | 0.9309 | 0.9560 | 0.9628 | 0.3085 |
DGP18 | 0.9293 | 0.9891 | 0.9722 | 0.3409 |
Case | BLT | JT | LM | LM–BNS |
---|---|---|---|---|
DGP19 | 0.9126 | 0.9394 | 0.9029 | 0.2087 |
DGP20 | 0.8456 | 0.9798 | 0.9332 | 0.2166 |
DGP21 | 0.9175 | 0.9697 | 0.9126 | 0.3155 |
DGP22 | 0.8687 | 0.9798 | 0.9516 | 0.3433 |
DGP23 | 0.8932 | 0.9596 | 0.9223 | 0.2524 |
DGP24 | 0.8687 | 0.9747 | 0.9424 | 0.2097 |
DGP25 | 0.9126 | 0.9697 | 0.9515 | 0.3350 |
DGP26 | 0.8825 | 0.9798 | 0.9516 | 0.3295 |
Model | Description |
---|---|
HAR-RV-C | Future RV depends on the lags of the continuous component of the RV. |
HAR-RV-C-CJ | Future RV depends on the lags of the continuous component of RV and the co-jump variation. |
HAR-RV-C-IJ | Future RV depends on the lags of the continuous component of the RV and the idiosyncratic jump variation. |
HAR-RV-C-CJ-IJ | Future RV depends on the lags of the continuous component of RV, co-jump variation, and idiosyncratic jump variation. |
Target Name | Description | Transformation | Frequency |
---|---|---|---|
XLF | Financial Sector SPDR Fund | Daily | |
XLK | Technology Sector SPDR Fund | Daily | |
XLP | Consumer Staples Select Sector SPDR Fund | Daily | |
XLU | Utilities Select Sector SPDR Fund | Daily | |
XLY | Consumer Discretionary Select Sector SPDR Fund | Daily |
Sector | XLB | XLE | XLF | XLI | XLK | XLP | XLU | XLV | XLY | |
---|---|---|---|---|---|---|---|---|---|---|
Co-jump | Frequency (%) | 30.10% | 24.45% | 25.31% | 30.78% | 24.48% | 32.73% | 29.55% | 29.49% | 29.03% |
Prop. (%) | 79.36% | 70.08% | 72.79% | 87.46% | 83.52% | 79.76% | 64.37% | 78.23% | 86.09% | |
upside jump mean (%) | 0.63% | 0.78% | 0.72% | 0.56% | 0.57% | 0.43% | 0.61% | 0.49% | 0.55% | |
downside jump mean (%) | −0.61% | −0.76% | −0.70% | −0.54% | −0.53% | −0.43% | −0.60% | −0.47% | −0.54% | |
St. dev. | 0.0074 | 0.0090 | 0.0090 | 0.0064 | 0.0064 | 0.0052 | 0.0072 | 0.0055 | 0.0067 | |
Ratio of JV to RV (%) | 27.96% | 22.71% | 23.50% | 28.58% | 22.74% | 30.40% | 27.45% | 27.39% | 26.97% | |
Idiosyncratic jump | Frequency (%) | 15.01% | 17.42% | 14.82% | 10.39% | 10.36% | 15.19% | 24.05% | 16.56% | 10.36% |
Prop. (%) | 20.64% | 29.92% | 27.21% | 12.54% | 16.48% | 20.24% | 35.63% | 21.77% | 13.91% | |
upside jump mean (%) | 0.50% | 0.59% | 0.67% | 0.42% | 0.38% | 0.31% | 0.43% | 0.39% | 0.40% | |
downside jump mean (%) | −0.51% | −0.59% | −0.62% | −0.43% | −0.39% | −0.31% | −0.45% | −0.40% | −0.38% | |
St. dev. | 0.0056 | 0.0064 | 0.0077 | 0.0048 | 0.0042 | 0.0034 | 0.0047 | 0.0043 | 0.0043 | |
Ratio of JV to RV (%) | 13.94% | 16.18% | 13.77% | 9.65% | 9.62% | 14.11% | 22.34% | 15.38% | 9.62% |
Sector | HAR-RV-C | HAR-RV-C-CJ | HAR-RV-C-IJ | HAR-RV-C-CJ-IJ |
---|---|---|---|---|
Forecasting Period: 2008:11–2019:12 | ||||
XLF | 0.6560 * | 1.0000 * | 0.6560 * | 0.6560 * |
XLK | 0.2165 * | 0.4195 * | 0.1415 * | 1.0000 * |
XLP | 0.1220 * | 0.1220 * | 0.0880 | 1.0000 * |
XLU | 0.3860 * | 0.3860 * | 0.2975 * | 1.0000 * |
XLY | 0.7780 * | 0.9385 * | 0.7780 * | 1.0000 * |
Sector | HAR-RV-C | HAR-RV-C-CJ | HAR-RV-C-IJ | HAR-RV-C-CJ-IJ |
---|---|---|---|---|
Forecasting Period: 2008:11–2019:12 | ||||
XLF | 0.7761 | 0.7765 | 0.7761 | 0.7768 |
XLK | 0.6748 | 0.6758 | 0.6758 | 0.6763 |
XLP | 0.6430 | 0.6441 | 0.6439 | 0.6443 |
XLU | 0.6157 | 0.6189 | 0.6175 | 0.6200 |
XLY | 0.7458 | 0.7470 | 0.7462 | 0.7470 |
Sector | HAR-RV-C | HAR-RV-C-CJ | HAR-RV-C-IJ | HAR-RV-C-CJ-IJ |
---|---|---|---|---|
Forecasting Period: 2008:11–2019:12 | ||||
XLF | 0.6765 | 0.6866 | 0.6823 | 0.6823 |
XLK | 0.5680 | 0.5772 | 0.5649 | 0.5817 |
XLP | 0.5567 | 0.5602 | 0.5556 | 0.5696 |
XLU | 0.4927 | 0.4970 | 0.4914 | 0.5047 |
XLY | 0.6804 | 0.6838 | 0.6802 | 0.6841 |
Sector | HAR-RV-C | HAR-RV-C-CJ | HAR-RV-C-IJ | HAR-RV-C-CJ-IJ |
---|---|---|---|---|
Forecasting Period: 2008:11–2019:12 | ||||
XLF | 0.2293 | 0.2251 | 0.2276 | 0.2267 |
XLK | 0.2371 | 0.2348 | 0.2360 | 0.2320 |
XLP | 0.2062 | 0.2056 | 0.2063 | 0.2031 |
XLU | 0.2070 | 0.2061 | 0.2069 | 0.2042 |
XLY | 0.2225 | 0.2211 | 0.2214 | 0.2205 |
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Peng, W.; Yao, C. Co-Jumps, Co-Jump Tests, and Volatility Forecasting: Monte Carlo and Empirical Evidence. J. Risk Financial Manag. 2022, 15, 334. https://doi.org/10.3390/jrfm15080334
Peng W, Yao C. Co-Jumps, Co-Jump Tests, and Volatility Forecasting: Monte Carlo and Empirical Evidence. Journal of Risk and Financial Management. 2022; 15(8):334. https://doi.org/10.3390/jrfm15080334
Chicago/Turabian StylePeng, Weijia, and Chun Yao. 2022. "Co-Jumps, Co-Jump Tests, and Volatility Forecasting: Monte Carlo and Empirical Evidence" Journal of Risk and Financial Management 15, no. 8: 334. https://doi.org/10.3390/jrfm15080334
APA StylePeng, W., & Yao, C. (2022). Co-Jumps, Co-Jump Tests, and Volatility Forecasting: Monte Carlo and Empirical Evidence. Journal of Risk and Financial Management, 15(8), 334. https://doi.org/10.3390/jrfm15080334