Selection Criteria in Regime Switching Conditional Volatility Models
Abstract
:1. Introduction
2. Theory: Models and Selection Criteria
2.1. Models
2.1.1. Univariate GARCH Model
2.1.2. Asymmetric Volatility Models
2.1.3. MS-GARCH Models
2.2. Selection Criteria: Information Criteria and Loss Functions
3. Design of the Experiments
3.1. Common Design: Starting Values and Numerical Method
3.2. Experiment 1: Simulation of MS-GARCH-K Processes
3.3. Experiment 2: Simulation of MS-GARCH-H Processes
3.4. Experiment 3: Simulation of LST-GARCH Processes
4. Results and Discussion
4.1. Results
4.1.1. Experiment 1
DGP | MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|---|
Experiment 1 and 2 with Gaussian innovations | |||||||||
MS-K | |||||||||
MS-H | x | x | x | x | x | ||||
MS-K | |||||||||
MS-H | x | x | x | x | x | ||||
MS-K | x | x | x | x | x | x | x | x | |
MS-H | x | x | x | x | x | ||||
MS-K | |||||||||
MS-H | x | x | x | x | x | x | x | x | |
Experiment 2 | |||||||||
LST-G | |||||||||
LST-G | |||||||||
LST-G |
MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|
GARCH | 7.2 | 3.8 | 0.5 | 0 | 0 | 0 | 16.6 | 67.8 |
GARCH-T | 4.2 | 2.1 | 0.5 | 0 | 0 | 0 | 32.2 | 31.1 |
MSG-GARCH-H | 8.0 | 4.9 | 13.2 | 33.5 | 52.2 | 59.2 | 6.6 | 0 |
MST-GARCH-H | 0.4 | 0.6 | 0.6 | 9 | 5.3 | 2.3 | 0.3 | 0 |
MSG-GARCH-K | 30.7 | 46.1 | 54.3 | 49.1 | 34.2 | 34.0 | 27.1 | 0 |
MST-GARCH-K | 40.6 | 40.6 | 29.4 | 8.4 | 8.3 | 4.5 | 1.1 | 0 |
LST-GARCH | 1.1 | 0.9 | 0.1 | 0 | 0 | 0 | 0.4 | 0 |
LST-GARCH-T | 1.3 | 0.2 | 0 | 0 | 0 | 0 | 1.3 | 0 |
GJR | 0.7 | 0.4 | 0.1 | 0 | 0 | 0 | 3.8 | 0.8 |
GJR-T | 0.7 | 0.3 | 0.1 | 0 | 0 | 0 | 5.5 | 0.1 |
EGARCH | 3.1 | 0.1 | 0.4 | 0 | 0 | 0 | 2.0 | 0.1 |
EGARCH-T | 2.0 | 0 | 0.8 | 0 | 0 | 0 | 2.1 | 0 |
MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|
GARCH | 0.5 | 0.1 | 0 | 0 | 0 | 0 | 0 | 1.8 |
GARCH-T | 0.2 | 0 | 0.1 | 0 | 0 | 0 | 8.2 | 82.3 |
MSG-GARCH-H | 24.0 | 17.3 | 23.8 | 13.3 | 21.6 | 19.3 | 21.4 | 5.2 |
MST-GARCH-H | 9.3 | 7.0 | 5.4 | 5.6 | 1.7 | 1.5 | 8.4 | 0.2 |
MSG-GARCH-K | 36.1 | 46.1 | 49.5 | 74.6 | 72.7 | 74.4 | 42.1 | 5.4 |
MST-GARCH-K | 21.9 | 29.4 | 19.6 | 6.5 | 4.0 | 4.8 | 12.7 | 0.1 |
LST-GARCH | 0 | 0.1 | 0 | 0 | 0 | 0 | 0 | 0 |
LST-GARCH-T | 0.4 | 0 | 0 | 0 | 0 | 0 | 0.2 | 0 |
GJR | 0.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0.1 |
GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 3.3 | 0.5 |
EGARCH | 6.1 | 0 | 1.6 | 0 | 0 | 0 | 0 | 0.1 |
EGARCH-T | 1.3 | 0 | 0 | 0 | 0 | 0 | 3.7 | 4.3 |
MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|
GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
MSG-GARCH-H | 5.1 | 5.5 | 4.4 | 1.9 | 0.1 | 6.4 | 7.9 | |
MST-GARCH-H | 3.2 | 4.3 | 1.3 | 0.1 | 0.1 | 0 | 1.7 | 0.2 |
MSG-GARCH-K | 48.8 | 43.1 | 64.7 | 85.7 | 93.6 | 90.1 | 72.2 | 88.8 |
MST-GARCH-K | 42.9 | 47.1 | 29.6 | 12.2 | 5.9 | 9.8 | 19.7 | 3.1 |
LST-GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
LST-GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GJR | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
EGARCH | 0 | 0 | 0 | 0.1 | 0 | 0 | 0 | 0 |
EGARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|
GARCH | 8.4 | 5.7 | 5.7 | 0 | 0 | 0 | 21.9 | 69.4 |
GARCH-T | 7.3 | 5.5 | 5.9 | 0.1 | 0 | 0.1 | 43.8 | 28.8 |
MSG-GARCH-H | 13.8 | 15.5 | 14.5 | 34.7 | 25.8 | 27.2 | 5.3 | 0 |
MST-GARCH-H | 10.0 | 10.3 | 6.0 | 27.6 | 16.8 | 20.2 | 0.2 | 0 |
MSG-GARCH-K | 22.2 | 34.2 | 41.7 | 30.0 | 51.9 | 45.3 | 5.0 | 0 |
MST-GARCH-K | 14.9 | 19.7 | 16.7 | 7.4 | 5.2 | 7.0 | 0.5 | 0 |
LST-GARCH | 3.8 | 1.7 | 1.2 | 0 | 0 | 0 | 0.2 | 0 |
LST-GARCH-T | 4.8 | 3.3 | 1.8 | 0.2 | 0.2 | 0 | 1.0 | 0 |
GJR | 2.3 | 1.1 | 1.5 | 0 | 0.1 | 0 | 6.9 | 1.4 |
GJR-T | 2.2 | 2.3 | 2.1 | 0 | 0 | 0 | 9.9 | 0.1 |
EGARCH | 5.1 | 0.5 | 1.1 | 0 | 0 | 0 | 1.9 | 0.2 |
EGARCH-T | 5.2 | 0.2 | 1.8 | 0 | 0 | 0.1 | 3.4 | 0.1 |
4.1.2. Experiment 2
MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|
GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
MSG-GARCH-H | 52.3 | 46.9 | 67.8 | 12.2 | 49.8 | 7.8 | 93.6 | 99.4 |
MST-GARCH-H | 45.7 | 45.9 | 30.2 | 7.6 | 7.6 | 2.3 | 6.3 | 5.0 |
MSG-GARCH-K | 0.9 | 3.9 | 1.1 | 65.8 | 30.6 | 70.8 | 0.1 | 0.1 |
MST-GARCH-K | 1 | 3.3 | 0.9 | 12.0 | 12.0 | 19.1 | 0 | 0 |
LST-GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
LST-GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GJR | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
EGARCH | 0.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
EGARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|
GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
MSG-GARCH-H | 57.9 | 54.6 | 76.8 | 34.2 | 57.2 | 39.0 | 94.9 | 99.2 |
MST-GARCH-H | 41.1 | 39.5 | 19.1 | 5.1 | 3.2 | 2.8 | 5.0 | 0.6 |
MSG-GARCH-K | 0.7 | 3.2 | 3.7 | 53.4 | 36.8 | 54.1 | 0.1 | 0.2 |
MST-GARCH-K | 0.2 | 2.7 | 0.4 | 7.3 | 3.3 | 4.1 | 0 | 0 |
LST-GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
LST-GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GJR | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
EGARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
EGARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|
GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
MSG-GARCH-H | 59.9 | 64.6 | 79.2 | 48.7 | 19.1 | 35.9 | 97.9 | 99.8 |
MST-GARCH-H | 39.5 | 32.8 | 19.1 | 13.1 | 2.8 | 7.5 | 2.0 | 0 |
MSG-GARCH-K | 0.6 | 1.8 | 1.7 | 31.2 | 73.0 | 50.0 | 0.1 | 0.2 |
MST-GARCH-K | 0 | 0.1 | 0 | 6.9 | 5.1 | 6.6 | 0 | 0 |
LST-GARCH | 0 | 0 | 0 | 0.1 | 0 | 0 | 0 | 0 |
LST-GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GJR | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
EGARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
EGARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|
GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 1.3 | 21.8 |
MSG-GARCH-H | 88.2 | 82.0 | 87.2 | 73.0 | 83.6 | 64.2 | 89.9 | 74.9 |
MST-GARCH-H | 8.8 | 8.7 | 5.3 | 13.5 | 3.8 | 5.9 | 1.8 | 0.1 |
MSG-GARCH-K | 3.0 | 9.3 | 7.5 | 13.2 | 12.5 | 29.7 | 6.3 | 3.0 |
MST-GARCH-K | 0.1 | 0 | 0 | 0.2 | 0.1 | 0.2 | 0 | 0 |
LST-GARCH | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
LST-GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GJR | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
GJR-T | 0 | 0 | 0 | 0 | 0 | 0 | 0.6 | 0.1 |
EGARCH | 0 | 0 | 0 | 0.1 | 0 | 0 | 0 | 0 |
EGARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0.1 | 0.1 |
4.1.3. Experiment 3
MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|
GARCH | 4.3 | 9.9 | 6.9 | 0.1 | 0 | 0 | 56.2 | 96.3 |
GARCH-T | 4.0 | 8.8 | 6.4 | 0.3 | 0 | 0.2 | 2.6 | 0.2 |
MSG-GARCH-H | 1.8 | 2.2 | 4.0 | 29.6 | 21.9 | 13.7 | 1.3 | 0 |
MST-GARCH-H | 2.8 | 2.5 | 5.2 | 1.5 | 3.5 | 25.0 | 0 | 0 |
MSG-GARCH-K | 1.1 | 1.9 | 2.3 | 41.8 | 61.3 | 45.2 | 4 | 0 |
MST-GARCH-K | 4.3 | 7.5 | 6.5 | 3.9 | 3.4 | 5.1 | 0 | 0 |
LST-GARCH | 30.9 | 32.3 | 27.3 | 7.3 | 2.5 | 1.9 | 3.5 | 0 |
LST-GARCH-T | 16.7 | 11.1 | 18.8 | 6.0 | 4.6 | 1.9 | 0.1 | 0 |
GJR | 14.5 | 10.0 | 9.9 | 1.4 | 1.9 | 0.9 | 26.5 | 3.3 |
GJR-T | 14.8 | 13.5 | 11.4 | 1.1 | 0 | 0.3 | 1.0 | 0 |
EGARCH | 2.2 | 0.2 | 0.8 | 3.9 | 0.8 | 3.0 | 8.0 | 0.2 |
EGARCH-T | 2.6 | 0.1 | 0.5 | 3.1 | 0.1 | 2.8 | 0.4 | 0 |
MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|
GARCH | 0 | 0 | 0 | 0.1 | 0 | 0 | 7.4 | 34.8 |
GARCH-T | 0.1 | 0.4 | 0.1 | 0.2 | 0 | 0 | 0.6 | 0 |
MSG-GARCH-H | 0 | 0.2 | 0.1 | 28.0 | 25.1 | 14.1 | 0.1 | 0 |
MST-GARCH-H | 0 | 0.1 | 0.1 | 0.6 | 1.9 | 23.9 | 0 | 0 |
MSG-GARCH-K | 0.1 | 0.2 | 0 | 31.9 | 58.9 | 41.8 | 0.1 | 0 |
MST-GARCH-K | 0.1 | 0.4 | 0 | 3.2 | 4.0 | 4.2 | 0 | 0 |
LST-GARCH | 43.4 | 63.5 | 43.1 | 10.4 | 5.3 | 3.2 | 23.4 | 1.3 |
LST-GARCH-T | 33.0 | 19.8 | 34.5 | 8.0 | 2.3 | 1.8 | 2.0 | 0 |
GJR | 9.7 | 6.0 | 9.2 | 1.5 | 1.2 | 1.2 | 46 | 41.3 |
GJR-T | 11.1 | 9.3 | 10.8 | 2.7 | 0 | 0.9 | 3.0 | 0.1 |
EGARCH | 1.4 | 0.7 | 1.3 | 5.9 | 1.2 | 4.7 | 16.3 | 22.5 |
EGARCH-T | 1.1 | 0.4 | 0.8 | 7.5 | 2 | 4.2 | 1.1 | 0 |
MSE() | QLIKE() | MAE() | MSE() | QLIKE() | MAE() | AIC | BIC | |
---|---|---|---|---|---|---|---|---|
GARCH | 0 | 0 | 0 | 0.1 | 0 | 0 | 0 | 0.1 |
GARCH-T | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
MSG-GARCH-H | 0 | 0 | 0 | 33.4 | 33.2 | 22.8 | 0 | 0 |
MST-GARCH-H | 0 | 0 | 0 | 0.8 | 1.0 | 19.7 | 0 | 0 |
MSG-GARCH-K | 0 | 0 | 0 | 30.0 | 47.3 | 37.0 | 0 | 0 |
MST-GARCH-K | 0 | 0 | 0 | 4.4 | 5.4 | 3.9 | 0 | 0 |
LST-GARCH | 48.0 | 62.8 | 49.5 | 7.2 | 3.8 | 2.4 | 28.9 | 1.8 |
LST-GARCH-T | 19.6 | 26.0 | 26.3 | 8.8 | 8.4 | 6.3 | 1.3 | 0 |
GJR | 18.1 | 5.3 | 13.5 | 6.3 | 0.8 | 3.2 | 55.8 | 92.8 |
GJR-T | 14.3 | 5.9 | 10.7 | 8.4 | 0.1 | 4.5 | 3.1 | 0.3 |
EGARCH | 0 | 0 | 0 | 0.4 | 0 | 5.7 | 10.9 | 5.0 |
EGARCH-T | 0 | 0 | 0 | 0.2 | 0 | 0.2 | 0 | 0 |
4.2. Discussion
5. Conclusions
Acknowledgments
Conflicts of Interest
References
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- 3.See [11] for example.
- 4.There are a number of expansions of these two MS-GARCH processes. For example, Gallo and Otrento [27] introduce asymmetric effects in each regime variance.
- 5.Hu and Shin [32] introduced a test procedure which tests the null hypothesis of a GARCH process against an MS-GARCH process.
- 6.For each experiment, we estimate these models: GARCH, GARCH-T, LST-GARCH, LST-GARCH-T, GJR-GARCH, GJR-GARCH-T, EGARCH, AEGARCH-T, MSG(2)-GARCH-H, MSG(2)-GARCH-K, MST(2)-GARCH-H and MST(2)-GARCH-K.
- 7.Results for are available on demand, results remain the same. We do not consider smaller sample size since in financial application, we used to study daily data.
- 9.We set , and . We generate 2000 more observations than required to minimize any starting bias.
- 10.The probabilities of being in regime . The long-run probability of the first regime: is equal to .
- 11.δ is computed as follows: .
- 12.We set , and . We generate 2000 more observations than required, to minimize any starting bias.
- 13.We generate 2000 more observations than required to minimize any starting bias.
- 14.Estimation computed with Gaussian kernel and Silverman’s rule of thumb.
- 15.Figure 3(a) is related to the 40th replication of the first and the second experiments with matrix , BIC selects the right specification when data are simulated with MSG-GARCH-H but it selects the GARCH model for data simulated with MS-GARCHG-K. Figure 3(b) is related to the 66th replication of the first and second experiments with where there is no selection problem.
- 16.Estimation computed with Gaussian kernel and Silverman’s rule of thumb.
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Chuffart, T. Selection Criteria in Regime Switching Conditional Volatility Models. Econometrics 2015, 3, 289-316. https://doi.org/10.3390/econometrics3020289
Chuffart T. Selection Criteria in Regime Switching Conditional Volatility Models. Econometrics. 2015; 3(2):289-316. https://doi.org/10.3390/econometrics3020289
Chicago/Turabian StyleChuffart, Thomas. 2015. "Selection Criteria in Regime Switching Conditional Volatility Models" Econometrics 3, no. 2: 289-316. https://doi.org/10.3390/econometrics3020289