Bootstrapping Time-Varying Uncertainty Intervals for Extreme Daily Return Periods
Abstract
:1. Introduction
1.1. Literature Review
2. Methodology
2.1. Bayesian Inferences to Parameter Estimates
The Likelihood Function
2.2. Value-at-Risk and Expected Shortfall
2.3. Cross-Validation on Prediction and Uncertainty Problems
2.4. Fixed-Design Residual Bootstrap
2.5. Backtesting Value-at-Risk and Expected Shortfall Forecasts
Algorithm 1: Fixed-design Residual Bootstrap |
2.6. Density Forecasts Using Threshold and Quantile-Weighted Scoring Rules for VaR and ES
2.7. Data and Software
3. Empirical Results
3.1. Exploratory Data Analysis
3.2. SARIMA–GAS–GEVD Framework
3.3. Comparative Analysis
3.4. Evaluation of the Prediction Experiment
3.5. Forecasting and Backtesting Procedure
3.6. Return Level Periods and Bootstrapping Uncertainty Intervals
3.7. Discussion of Results
4. Conclusions and Recommendations
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
AE | Absolute Error |
ADF | Augmented Dickey Fuller |
AIC | Akaike Information Criterion |
ARMA-GARCH | Autoregressive-Moving Average-Generalized Autoregressive-Conditional–Heteroscedasticity |
BCR | Balanced Classification Rate |
BM | Block Minima |
CC | Conditional Coverage |
CDF | Cumulative Distribution Function |
CRPS | Continuous Ranked Probability Score |
DQ | Dynamic Quantile |
ES | Expected Shortfall |
EVT | Extreme Value Theory |
FN | False Negative |
FP | False Positive |
FTSE/JSE-ALSI | Financial Time Series exchange/Johannesburg Stock Exchange–All Share Index |
GARCH | Generalized Autoregressive Conditional Heteroscedasticity |
GAS | Generalised Autoregressive Score |
GAS-GEVD | Generalised Autoregressive Score–Generalised Extreme Value Distribution |
GEVD | Generalised Extreme Value Distribution |
i.i.d | Independent and Identically Distributed |
JSE-ALSI | Johannesburg Stock Exchange–All Share Index |
MAE | Mean Absolute Error |
MAPE | Mean Absolute Percentage Error |
MCMC | Markov-chain-Monte-Carlo |
MCC | Matthews Correlation Coefficient |
MLE | Maximum Likelihood Estimation |
MIDAS | Mixing Data Sampling |
MC | Monte Carlo |
Q-Q | Quantile-Quantile |
SARIMA | Seasonal Autoregressive Integrated Moving Average |
SARIMA-GAS-GEVD | Seasonal Autoregressive Integrated Moving Average–Generalised Autoregressive Score–Generalised Extreme Value Distribution |
TN | True Negative |
TR | Tail Risk |
TP | True Positive |
UC | Unconditional Coverage |
VaR | Value-at-Risk |
wCRPS | Weighted Continuous Ranked Probability Score |
1 | This assumption states that a score of the empirical distribution when computing the conditional Value at Risk measures is constant over time. |
2 | se herein referenced standard error of ξ. |
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Threshold Weights | Emphasis | Quantile Weights |
---|---|---|
Center | ||
Tails | ||
Left Tail | ||
Right Tail |
Skewness | Kurtosis | |
---|---|---|
Returns on FTSE/JSE-ALSI | −18.83 | 688.84 |
Parameter | Estimate | Std Error | t-Value | |
---|---|---|---|---|
65.5515 | 0.8782 | 7.4040 | 0.0000 | |
0.0702 | 0.0895 | 8.7172 | 0.0000 | |
3.7575 | 3.7575 | 1.4740 | 0.0702 | |
9.9995 | 0.0005 | 2.0578 | 0.0000 | |
−1.7119 | 0.1209 | −1.4114 | 0.0000 | |
−0.9666 | 0.0005 | −2.0670 | 0.0000 | |
−0.9356 | 0.0065 | −1.5098 | 0.0000 | |
Unconditional Parameters | ||||
−0.0046 | 0.00245 | −7.999 | 1.091 |
Test | SARIMA | GAS | GEVD | SARIMA-GAS-GEVD |
---|---|---|---|---|
LL | 108 | 110 | 102 | 99 |
Rank | 3 | 4 | 2 | 1 |
AIC | −1710 | −1712 | −1719 | −1725 |
Rank | 4 | 3 | 2 | 1 |
BIC | −1794 | −1799 | −1802 | −1800 |
Rank | 4 | 3 | 1 | 2 |
MAE | 0.9023 | 0.988 | 0.8774 | 0.9930 |
Rank | 2 | 3 | 1 | 4 |
MPAE | 1.247 | 1.004 | 1.002 | 0.798 |
Rank | 4 | 3 | 2 | 1 |
Metric | Mean | Std Error | Training Set | Validation Set |
---|---|---|---|---|
Precision | 0.9514 | 0.0108 | 0.9510 | 0.9701 |
Recall | 0.9664 | 0.0107 | 0.9645 | 0.9489 |
F1Score | 0.9580 | 0.0063 | 0.9577 | 0.9594 |
MCC | 0.8468 | 0.0015 | 0.8489 | 0.8513 |
BCR | 0.8953 | 0.0076 | 0.8909 | 0.9102 |
Wilcoxon Signed-Rank Test | Model Power Test | |||
---|---|---|---|---|
Parameters | SARIMA–GAS–GEVD | Data | Mean Difference | Actual Power |
z-value | 8.67 | Training | −0.0194 | 0.8241 |
p-value | 0.001 | Validation | 0.0298 | 0.8556 |
Threshold Weights | Emphasis | p-Value | Quantile Weights | p-Value |
---|---|---|---|---|
Center | 0.1675 | 0.1892 | ||
Tails | 0.0587 | 0.0774 | ||
Left Tail | 0.1157 | 0.1422 | ||
Right Tail | 0.1005 | 0.9911 |
Risk Measure | TR Test | DQ Test | ADmax | ADMean | AE |
---|---|---|---|---|---|
VaR | 0.4816 | 7.3246 | 13.4596 | 0.5664197 | 1.1342 |
ES | 0.5364 | 0.3958 | 0.48743 | 0.4873 | 1.0982 |
Period | Returns | Bootstrap Replicates | 99% CI |
---|---|---|---|
3 Years | 4.3760 | 15,000 | (4.2996; 4.3977) |
5 Years | 5.2416 | 15,000 | (5.166; 5.374) |
10 Yeas | 6.3842 | 15,000 | (6.287; 6.439) |
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Makatjane, K.; Tsoku, T. Bootstrapping Time-Varying Uncertainty Intervals for Extreme Daily Return Periods. Int. J. Financial Stud. 2022, 10, 10. https://doi.org/10.3390/ijfs10010010
Makatjane K, Tsoku T. Bootstrapping Time-Varying Uncertainty Intervals for Extreme Daily Return Periods. International Journal of Financial Studies. 2022; 10(1):10. https://doi.org/10.3390/ijfs10010010
Chicago/Turabian StyleMakatjane, Katleho, and Tshepiso Tsoku. 2022. "Bootstrapping Time-Varying Uncertainty Intervals for Extreme Daily Return Periods" International Journal of Financial Studies 10, no. 1: 10. https://doi.org/10.3390/ijfs10010010