# On the Elicitability and Risk Model Comparison of Emerging Markets Equities

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Models and Empirical Methodology

#### 2.1. Univariate GAS Model Specification

#### 2.2. The FZL Function

#### 2.3. The MCS Procedure

## 3. Data and Preliminary Analysis

#### Descriptive Statistics

## 4. Conclusions and Recommendations

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Burzoni, M.; Peri, I.; Ruffo, C.M. On the properties of the Lambda value at risk: Robustness, elicitability and consistency. Quant. Financ.
**2017**, 17, 1735–1743. [Google Scholar] [CrossRef] [Green Version] - Cont, R.; Deguest, R.; He, X.D. Loss-based risk measures. Stat. Risk Model. Appl. Financ. Insur.
**2013**, 30, 133–167. [Google Scholar] [CrossRef] [Green Version] - Fissler, T.; Ziegel, J.F. Higher order elicitability and Osband’s principle. Ann. Stat.
**2016**, 44, 1680–1707. [Google Scholar] [CrossRef] - Fissler, T.; Ziegel, J.F.; Gneiting, T. Expected Shortfall is jointly elicitable with Value at Risk-Implications for backtesting. arXiv
**2015**, arXiv:1507.00244. [Google Scholar] - Nolde, N.; Ziegel, J.F. Elicitability and backtesting: Perspectives for banking regulation. Ann. Appl. Stat.
**2017**, 11, 1833–1874. [Google Scholar] [CrossRef] - Chang, C.-L.; Jimenez-Martin, J.-A.; Maasoumi, E.; McAleer, M.; Pérez-Amaral, T. Choosing expected shortfall over VaR in Basel III using stochastic dominance. Int. Rev. Econ. Financ.
**2019**, 60, 95–113. [Google Scholar] [CrossRef] [Green Version] - Kellner, R.; Rösch, D. Quantifying market risk with Value-at-Risk or Expected Shortfall?—Consequences for capital requirements and model risk. J. Econ. Dyn. Control
**2016**, 68, 45–63. [Google Scholar] [CrossRef] - Bernardi, M.; Catania, L. Comparison of Value-at-Risk models using the MCS approach. Comput. Stat.
**2016**, 31, 579–608. [Google Scholar] [CrossRef] - Hansen, P.R.; Lunde, A.; Nason, J.M. The Model Confidence Set. Econometrica
**2011**, 79, 453–497. [Google Scholar] [CrossRef] [Green Version] - Patton, A.J.; Ziegel, J.F.; Chen, R. Dynamic semiparametric models for expected shortfall (and Value-at-Risk). J. Econ.
**2019**. [Google Scholar] [CrossRef] [Green Version] - Bao, Y.; Lee, T.-H.; Saltoglu, B. Evaluating predictive performance of value-at-risk models in emerging markets: A reality check. J. Forecast.
**2006**, 25, 101–128. [Google Scholar] [CrossRef] - Del Brio, E.B.; Mora-Valencia, A.; Perote, J. VaR performance during the subprime and sovereign debt crises: An application to emerging markets. Emerg. Mark. Rev.
**2014**, 20, 23–41. [Google Scholar] [CrossRef] - Hwang, S.; Satchell, S.E. Modelling emerging market risk premia using higher moments. Int. J. Financ. Econ.
**1999**, 4, 271–296. [Google Scholar] [CrossRef] - Ji, Q.; Liu, B.-Y.; Zhao, W.-L.; Fan, Y. Modelling dynamic dependence and risk spillover between all oil price shocks and stock market returns in the BRICS. Int. Rev. Financ. Anal.
**2020**, 68, 101238. [Google Scholar] [CrossRef] - Miletic, M.; Miletic, S. Performance of Value at Risk models in the midst of the global financial crisis in selected CEE emerging capital markets. Econ. Res.-Ekon. Istraž.
**2015**, 28, 132–166. [Google Scholar] [CrossRef] [Green Version] - Kharas, H. The Emerging Middle Class in Developing Countries. Available online: https://www.oecd-ilibrary.org/development/the-emerging-middle-class-in-developing-countries_5kmmp8lncrns-en (accessed on 30 August 2021).
- Aizenman, J.; Binici, M.; Hutchison, M.M. The Transmission of Federal Reserve Tapering News to Emerging Financial Markets (Working Paper No. 19980); National Bureau of Economic Research: Cambridge, MA, USA, 2014. [Google Scholar] [CrossRef]
- Enginar, O.; Karan, M.B.; Büyükkara, G. Performances of Emerging Stock Exchanges During the Fed’s Tapering Announcements. In Global Approaches in Financial Economics, Banking, and Finance; Dincer, H., Hacioglu, Ü., Yüksel, S., Eds.; Springer: Berlin, Germany, 2018; pp. 415–443. [Google Scholar] [CrossRef]
- Ghosh, S.; Saggar, M. Volatility spillovers to the emerging financial markets during taper talk and actual tapering. Appl. Econ. Lett.
**2017**, 24, 122–127. [Google Scholar] [CrossRef] - Mishra, P.; Moriyama, K.; N’Diaye, P.M.P.; Nguyen, L. Impact of Fed Tapering Announcements on Emerging Markets; International Monetary Fund: Washington, DC, USA, 2014. [Google Scholar]
- Blazsek, S.; Hernández, H. Analysis of electricity prices for Central American countries using dynamic conditional score models. Empir. Econ.
**2018**, 55, 1807–1848. [Google Scholar] [CrossRef] - Gong, X.-L.; Liu, X.-H.; Xiong, X. Measuring tail risk with GAS time varying copula, fat tailed GARCH model and hedging for crude oil futures. Pac. Basin Financ. J.
**2019**, 55, 95–109. [Google Scholar] [CrossRef] - Troster, V.; Tiwari, A.K.; Shahbaz, M.; Macedo, D.N. Bitcoin returns and risk: A general GARCH and GAS analysis. Financ. Res. Lett.
**2018**, 30, 187–193. [Google Scholar] [CrossRef] - Owusu Junior, P.; Alagidede, I. Risks in emerging markets equities: Time-varying versus spatial risk analysis. Phys. A Stat. Mech. Appl.
**2020**, 542, 123474. [Google Scholar] [CrossRef] - Khalaf, L.; Leccadito, A.; Urga, G. Multilevel and Tail Risk Management. J. Financ. Econom.
**2021**. [Google Scholar] [CrossRef] - Kratz, M.; Lok, Y.H.; McNeil, A.J. Multinomial VaR backtests: A simple implicit approach to backtesting expected shortfall. J. Bank. Financ.
**2018**, 88, 393–407. [Google Scholar] [CrossRef] [Green Version] - Pradhan, A.K.; Tiwari, A.K. Estimating the market risk of clean energy technologies companies using the expected shortfall approach. Renew. Energy
**2021**, 177, 95–100. [Google Scholar] [CrossRef] - Barendse, S. Interquantile Expectation Regression (SSRN Scholarly Paper No. ID 2937665). Available online: https://papers.ssrn.com/abstract=2937665 (accessed on 30 August 2021).
- Dimitriadis, T.; Bayer, S. A Joint Quantile and Expected Shortfall Regression Framework. arXiv
**2017**, arXiv:1704.02213. [Google Scholar] [CrossRef] - Couperier, O.; Leymarie, J. Backtesting Expected Shortfall via Multi-Quantile Regression. Available online: https://halshs.archives-ouvertes.fr/halshs-01909375/document (accessed on 30 August 2021).
- Han, A.; Hausman, J.A. Flexible parametric estimation of duration and competing risk models. J. Appl. Econom.
**1990**, 5, 1–28. [Google Scholar] [CrossRef] - White, H. A Reality Check for Data Snooping. Econometrica
**2000**, 68, 1097–1126. [Google Scholar] [CrossRef] - Romano, J.P.; Wolf, M. Stepwise Multiple Testing as Formalized Data Snooping. Econometrica
**2005**, 73, 1237–1282. [Google Scholar] [CrossRef] [Green Version] - Hansen, P.R.; Lunde, A. A forecast comparison of volatility models: Does anything beat a GARCH (1, 1)? J. Appl. Econom.
**2005**, 20, 873–889. [Google Scholar] [CrossRef] [Green Version] - Giacomini, R.; White, H. Tests of Conditional Predictive Ability. Econometrica
**2006**, 74, 1545–1578. [Google Scholar] [CrossRef] [Green Version] - Taylor, J.W. Forecasting Value at Risk and Expected Shortfall Using a Semiparametric Approach Based on the Asymmetric Laplace Distribution. J. Bus. Econ. Stat.
**2019**, 37, 121–133. [Google Scholar] [CrossRef] - Diebold, F.X.; Mariano, R.S. Comparing Predictive Accuracy. J. Bus. Econ. Stat.
**1995**, 13, 253–263. [Google Scholar] [CrossRef] [Green Version] - Dimitrakopoulos, D.N.; Kavussanos, M.G.; Spyrou, S.I. Value at risk models for volatile emerging markets equity portfolios. Q. Rev. Econ. Financ.
**2010**, 50, 515–526. [Google Scholar] [CrossRef] - Mollah, S.; Mobarek, A. Global Stock Market Integration: Co-Movement, Crises, and Efficiency in Developed and Emerging Markets; Springer: Berlin, Germany, 2016. [Google Scholar]
- Gneiting, T. Making and Evaluating Point Forecasts. J. Am. Stat. Assoc.
**2011**, 106, 746–762. [Google Scholar] [CrossRef] [Green Version] - Weber, S. Distribution-Invariant Risk Measures, Information, and Dynamic Consistency. Math. Financ.
**2006**, 16, 419–441. [Google Scholar] [CrossRef] - Creal, D.; Koopman, S.J.; Lucas, A. Generalized autoregressive score models with applications. J. Appl. Econom.
**2013**, 28, 777–795. [Google Scholar] [CrossRef] [Green Version] - Harvey, A.C. Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series (Volume 52); Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Ardia, D.; Boudt, K.; Catania, L. Downside Risk Evaluation with the R Package GAS (SSRN Scholarly Paper No. ID 2871444). Available online: https://papers.ssrn.com/abstract=2871444 (accessed on 30 August 2021).
- Mariano, R.S.; Preve, D. Statistical tests for multiple forecast comparison. J. Econom.
**2012**, 169, 123–130. [Google Scholar] [CrossRef] - West, K.D. Asymptotic inference about predictive ability. Econom. J. Econom. Soc.
**1996**, 64, 1067–1084. [Google Scholar] [CrossRef] - González-Rivera, G.; Lee, T.-H.; Mishra, S. Forecasting volatility: A reality check based on option pricing, utility function, value-at-risk, and predictive likelihood. Int. J. Forecast.
**2004**, 20, 629–645. [Google Scholar] [CrossRef] - Bernardi, M.; Catania, L.; Petrella, L. Are news important to predict the Value-at-Risk? Eur. J. Financ.
**2017**, 23, 535–572. [Google Scholar] [CrossRef] - Bernardi, M.; Catania, L. The Model Confidence Set Package for R. arXiv
**2014**, arXiv:1410.8504. [Google Scholar] - Cajueiro, D.O.; Tabak, B.M. Testing for time-varying long-range dependence in volatility for emerging markets. Phys. A Stat. Mech. Appl.
**2005**, 346, 577–588. [Google Scholar] [CrossRef] - McNeil, A.J.; Frey, R. Estimation of tail-related risk measures for heteroscedastic financial time series: An extreme value approach. J. Empir. Financ.
**2000**, 7, 271–300. [Google Scholar] [CrossRef] - McNeil, A.J.; Frey, R.; Embrechts, P. Quantitative Risk Management: Concepts, Techniques and Tools—Revised Edition; Princeton University Press: Princeton, NJ, USA, 2015. [Google Scholar]
- Fernández, C.; Steel, M.F.J. On Bayesian Modeling of Fat Tails and Skewness. J. Am. Stat. Assoc.
**1998**, 93, 359–371. [Google Scholar] [CrossRef] [Green Version] - Zhu, D.; Galbraith, J.W. A generalized asymmetric Student-t distribution with application to financial econometrics. J. Econom.
**2010**, 157, 297–305. [Google Scholar] [CrossRef] - Zhu, D.; Galbraith, J.W. Modeling and forecasting expected shortfall with the generalized asymmetric Student-t and asymmetric exponential power distributions. J. Empir. Financ.
**2011**, 18, 765–778. [Google Scholar] [CrossRef] - Kotz, S.; Kozubowski, T.; Podgorski, K. The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance; Springer Science & Business Media: Berlin, Germany, 2012. [Google Scholar]
- Cheung, Y.-W.; Fatum, R.; Yamamoto, Y. The exchange rate effects of macro news after the global Financial Crisis. J. Int. Money Financ.
**2019**, 95, 424–443. [Google Scholar] [CrossRef] - Crotty, J. Structural causes of the global financial crisis: A critical assessment of the ‘new financial architecture’. Camb. J. Econ.
**2009**, 33, 563–580. [Google Scholar] [CrossRef] [Green Version] - Martin, R. The local geographies of the financial crisis: From the housing bubble to economic recession and beyond. J. Econ. Geogr.
**2011**, 11, 587–618. [Google Scholar] [CrossRef] [Green Version] - Mollah, S.; Quoreshi, A.M.M.S.; Zafirov, G. Equity market contagion during global financial and Eurozone crises: Evidence from a dynamic correlation analysis. J. Int. Financ. Mark. Inst. Money
**2016**, 41, 151–167. [Google Scholar] [CrossRef] [Green Version] - Nguyen, T.H.; Pontell, H.N. Mortgage origination fraud and the global economic crisis. Criminol. Public Policy
**2010**, 9, 591–612. [Google Scholar] [CrossRef] - Marcellino, M.; Stock, J.H.; Watson, M.W. A comparison of direct and iterated multistep AR methods for forecasting macroeconomic time series. J. Econom.
**2006**, 135, 499–526. [Google Scholar] [CrossRef] [Green Version] - Fundamental Review of the Trading Book. Available online: https://www.bis.org/publ/bcbs265.htm (accessed on 30 August 2021).
- Emerging Markets. Available online: https://www.msci.com/documents/10199/c0db0a48-01f2-4ba9-ad01-226fd5678111 (accessed on 30 August 2021).
- Trucíos, C.; Hotta, L.K.; Ruiz, E. Robust bootstrap forecast densities for GARCH returns and volatilities. J. Stat. Comput. Simul.
**2017**, 87, 3152–3174. [Google Scholar] [CrossRef] - Jiménez, I.; Mora-Valencia, A.; Perote, J. Risk quantification and validation for Bitcoin. Oper. Res. Lett.
**2020**, 48, 534–541. [Google Scholar] [CrossRef]

China | S. Korea | Taiwan | India | Brazil | S. Africa | Russia | Mexico | Thailand | |
---|---|---|---|---|---|---|---|---|---|

EC-GFC periods | |||||||||

In-sample: 5 January 2007 to 18 April 2011 | |||||||||

Mean | 0.0003 | 0.0003 | 0.0001 | 0.0003 | 0.0005 | 0.0002 | −0.0002 | 0.0001 | 0.0006 |

Variance | 0.0005 | 0.0006 | 0.0003 | 0.0005 | 0.0008 | 0.0005 | 0.0009 | 0.0004 | 0.0004 |

Skewness | 0.03 | −0.13 | –0.20 | 0.19 | −0.34 | −0.26 | −0.40 | 0.02 | −0.59 |

Excess kurtosis | 5.02 | 17.11 | 2.40 | 6.89 | 7.39 | 4.31 | 13.79 | 6.46 | 6.17 |

Normtest.W* | 0.94 | 0.86 | 0.96 | 0.94 | 0.90 | 0.95 | 0.85 | 0.91 | 0.94 |

Observations | 1117 | 1117 | 1117 | 1117 | 1117 | 1117 | 1117 | 1117 | 1117 |

Out-of-sample: 19 April 2011 to 7 June 2013 | |||||||||

Mean | −0.0003 | −0.0002 | −0.0001 | −0.0004 | −0.0008 | −0.0003 | −0.0007 | 0.000 | 0.0003 |

Variance | 0.0002 | 0.0003 | 0.0002 | 0.0002 | 0.0003 | 0.0003 | 0.0003 | 0.0002 | 0.0002 |

Skewness | −0.077 | −0.248 | −0.167 | 0.030 | −0.401 | −0.082 | −0.477 | −0.533 | 0.034 |

Excess kurtosis | 2.896 | 2.457 | 1.931 | 1.334 | 2.815 | 1.562 | 2.642 | 3.795 | 2.959 |

Normtest.W* | 0.959 | 0.965 | 0.970 | 0.985 | 0.971 | 0.982 | 0.963 | 0.961 | 0.967 |

Observations | 559 | 559 | 559 | 559 | 559 | 559 | 559 | 559 | 559 |

Post-crises period | |||||||||

In-sample: 10 June 2013 to 21 July 2017 | |||||||||

Mean | 0.0002 | 0.0001 | 0.0002 | 0.0002 | −0.0003 | 0.0001 | −0.0002 | −0.0002 | −0.0001 |

Variance | 0.0002 | 0.0001 | 0.0001 | 0.0001 | 0.0004 | 0.0003 | 0.0004 | 0.0002 | 0.0002 |

Skewness | −0.17 | −0.16 | −0.15 | −0.51 | 0.18 | −0.25 | −0.03 | −0.60 | −0.07 |

Excess kurtosis | 3.07 | 1.45 | 2.31 | 4.22 | 1.82 | 2.94 | 7.26 | 4.97 | 3.87 |

Normtest.W* | 0.96 | 0.98 | 0.97 | 0.95 | 0.98 | 0.97 | 0.93 | 0.96 | 0.95 |

Observations | 990 | 990 | 990 | 990 | 990 | 990 | 990 | 990 | 990 |

Out-of-sample: 22 July 2017 to 18 February 2019 | |||||||||

Mean | 0.0003 | 0.0001 | 0.0001 | 0.0001 | 0.0004 | −0.0002 | 0.0001 | −0.0003 | 0.0004 |

Variance | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0003 | 0.0003 | 0.0002 | 0.0002 | 0.0001 |

Skewness | −0.12 | −0.36 | −0.96 | −0.37 | −1.26 | −0.22 | −1.82 | −0.57 | −0.09 |

Excess kurtosis | 0.57 | 1.71 | 6.98 | 0.87 | 12.68 | 1.21 | 17.03 | 3.08 | 2.23 |

Normtest.W* | 0.99 | 0.98 | 0.93 | 0.99 | 0.92 | 0.99 | 0.91 | 0.97 | 0.96 |

Observations | 497 | 497 | 497 | 497 | 497 | 497 | 497 | 497 | 497 |

Mean | Variance | Skewness | Excess Kurtosis | Normtest.W* | Observations |
---|---|---|---|---|---|

EC-GFC periods | |||||

In-sample: 5 January 2007 to 18 April 2011 | |||||

0.0002 | 0.0003 | −0.4159 | 6.3206 | 0.92 | 1117 |

Out-of-sample: 19 April 2011 to 7 June 2013 | |||||

−0.0003 | 0.0001 | −0.3898 | 3.4102 | 0.96 | 559 |

Post-crises period | |||||

In-sample: 10 June 2013 to 21 July 2017 | |||||

−0.0001 | 0.0001 | −0.2912 | 1.9301 | 0.98 | 990 |

Out-of-sample: 22 July 2017 to 18 February 2019 | |||||

0.0002 | 0.0001 | −0.5911 | 1.0884 | 0.98 | 497 |

Model | Rank_{R,M} | t_{i} | p-Value_{R,M} | Loss | Model | Rank_{R,M} | t_{i} | p-Value_{R,M} | Loss |
---|---|---|---|---|---|---|---|---|---|

Eurozone and Global Financial Crises (EC-GFC) Periods:19 April 2011 to 7 June 2013 | |||||||||

Brazil 1% | Brazil 2.5% | ||||||||

SNORM | 2 | −1.98 | 1.00 | −3.05 | SNORM | 3 | −1.43 | 1.00 | −3.16 |

STD | 3 | −1.50 | 1.00 | −2.93 | STD | 1 | −2.56 | 1.00 | −3.16 |

SSTD | 1 | −2.23 | 1.00 | −2.97 | SSTD | 2 | −2.49 | 1.00 | −3.13 |

AST | 6 | 1.81 | 0.11 | −2.51 | AST | 6 | 1.95 | 0.08 | −2.89 |

AST1 | 5 | 1.81 | 0.11 | −2.51 | AST1 | 5 | 1.95 | 0.08 | −2.89 |

ALD | 4 | −1.12 | 1.00 | −2.96 | ALD | 4 | −1.15 | 1.00 | −3.13 |

p-value | 0.105 | p-value | 0.084 | ||||||

Mexico 1% | Mexico 2.5% | ||||||||

SNORM | 2 | −1.32 | 1.00 | −3.17 | SNORM | 4 | −0.77 | 1.00 | −3.32 |

STD | 4 | 0.13 | 1.00 | −3.05 | STD | 2 | −1.17 | 1.00 | −3.33 |

SSTD | 1 | −2.96 | 1.00 | −3.13 | SSTD | 1 | −3.47 | 1.00 | −3.32 |

AST | 6 | 1.43 | 0.29 | −2.93 | AST | 6 | 1.68 | 0.22 | −3.21 |

AST1 | 5 | 1.43 | 0.29 | −2.93 | AST1 | 5 | 1.68 | 0.22 | −3.21 |

ALD | 3 | −0.93 | 1.00 | −3.15 | ALD | 3 | −0.87 | 1.00 | −3.32 |

p-value | 0.292 | p-value | 0.223 | ||||||

Russia 1% | Russia 2.5% | ||||||||

SNORM | 2 | −1.11 | 1.00 | −2.69 | SNORM | 2 | −0.21 | 1.00 | −2.88 |

STD | 3 | 1.08 | 0.44 | −2.60 | STD | 3 | 0.54 | 0.83 | −2.86 |

SSTD | 4 | 2.06 | 0.06 | −2.54 | SSTD | 4 | 1.89 | 0.10 | −2.82 |

ALD | 1 | −1.50 | 1.00 | −2.76 | ALD | 1 | −1.78 | 1.00 | −2.93 |

p-value | 0.059 | p-value | 0.095 | ||||||

South Africa 1% | South Africa 2.5% | ||||||||

SNORM | 1 | −0.96 | 1.00 | −2.92 | SNORM | 3 | 1.54 | 0.18 | −3.03 |

STD | 3 | 0.56 | 0.86 | −2.86 | STD | 2 | −0.16 | 1.00 | −3.07 |

SSTD | 4 | 1.19 | 0.43 | −2.85 | ALD | 1 | −1.10 | 1.00 | −3.10 |

ALD | 2 | −0.66 | 1.00 | −2.93 | |||||

p-value | 0.425 | p-value | 0.180 | ||||||

China 1% | China 2.5% | ||||||||

SNORM | 1 | −0.52 | 1.00 | −3.06 | SNORM | 3 | 1.03 | 0.44 | −3.21 |

STD | 3 | −0.27 | 1.00 | −3.04 | STD | 2 | −0.30 | 1.00 | −3.24 |

SSTD | 4 | 1.41 | 0.30 | −2.98 | ALD | 1 | −0.57 | 1.00 | −3.25 |

ALD | 2 | −0.34 | 1.00 | −3.05 | |||||

p-value | 0.300 | p-value | 0.443 | ||||||

India 1% | India 2.5% | ||||||||

SNORM | 4 | 0.23 | 0.97 | −2.90 | SNORM | 4 | 0.86 | 0.62 | −3.08 |

STD | 1 | −5.01 | 1.00 | −3.17 | STD | 1 | −5.55 | 1.00 | −3.33 |

SSTD | 2 | −3.92 | 1.00 | −3.08 | SSTD | 2 | −4.11 | 1.00 | −3.22 |

AST | 6 | 2.17 | 0.06 | −2.66 | AST | 6 | 2.62 | 0.02 | −2.95 |

AST1 | 5 | 2.17 | 0.06 | −2.66 | AST1 | 5 | 2.62 | 0.02 | −2.95 |

ALD | 3 | −1.31 | 1.00 | −3.06 | ALD | 3 | −2.66 | 1.00 | −3.28 |

p-value | 0.061 | p-value | 0.016 | ||||||

South Korea 1% | South Korea 2.5% | ||||||||

SNORM | 4 | −0.86 | 1.00 | −2.87 | SNORM | 4 | −0.98 | 1.00 | −3.05 |

STD | 1 | −2.95 | 1.00 | −2.99 | STD | 1 | −3.53 | 1.00 | −3.14 |

SSTD | 2 | −1.73 | 1.00 | −2.88 | SSTD | 3 | −1.53 | 1.00 | −3.04 |

AST | 6 | 1.98 | 0.09 | −2.52 | AST | 6 | 2.43 | 0.03 | −2.81 |

AST1 | 5 | 1.98 | 0.09 | −2.52 | AST1 | 5 | 2.43 | 0.03 | −2.81 |

ALD | 3 | −1.26 | 1.00 | −2.95 | ALD | 2 | −1.58 | 1.00 | −3.09 |

p-value | 0.087 | p-value | 0.031 | ||||||

Taiwan 1% | Taiwan 2.5% | ||||||||

SNORM | 3 | −1.36 | 1.00 | −3.12 | SNORM | 3 | −1.26 | 1.00 | −3.28 |

STD | 1 | −3.52 | 1.00 | −3.08 | STD | 1 | −4.68 | 1.00 | −3.31 |

SSTD | 4 | −1.31 | 1.00 | −2.89 | SSTD | 4 | −1.21 | 1.00 | −3.16 |

AST | 6 | 2.03 | 0.06 | −2.25 | AST | 6 | 2.50 | 0.02 | −2.78 |

AST1 | 5 | 2.03 | 0.06 | −2.25 | AST1 | 5 | 2.50 | 0.02 | −2.78 |

ALD | 2 | −2.02 | 1.00 | −3.09 | ALD | 2 | −3.08 | 1.00 | −3.32 |

p-value | 0.064 | p-value | 0.022 | ||||||

Thailand 1% | Thailand 2.5% | ||||||||

SNORM | 4 | 0.74 | 0.80 | −2.96 | SNORM | 6 | 1.32 | 0.37 | −3.17 |

STD | 1 | −4.27 | 1.00 | −3.20 | STD | 1 | −5.55 | 1.00 | −3.41 |

SSTD | 2 | −2.49 | 1.00 | −3.13 | SSTD | 2 | −2.87 | 1.00 | −3.34 |

AST | 6 | 1.08 | 0.58 | −2.93 | AST | 5 | 1.02 | 0.55 | −3.20 |

AST1 | 5 | 1.08 | 0.58 | −2.93 | AST1 | 4 | 1.02 | 0.55 | −3.20 |

ALD | 3 | −0.36 | 1.00 | −3.07 | ALD | 3 | −0.28 | 1.00 | −3.28 |

p-value | 0.584 | p-value | 0.367 | ||||||

Post−crises (PC) period:22 July 2017 to 18 February 2019 | |||||||||

Brazil 1% | Brazil 2.5% | ||||||||

SNORM | 2 | −1.62 | 1.00 | −2.59 | SNORM | 4 | −0.85 | 1.00 | −2.95 |

STD | 4 | −0.91 | 1.00 | −2.57 | STD | 1 | −3.30 | 1.00 | −3.02 |

SSTD | 1 | −1.81 | 1.00 | −2.60 | SSTD | 3 | −1.09 | 1.00 | −2.96 |

AST | 6 | 2.17 | 0.06 | −2.32 | AST | 6 | 2.05 | 0.06 | −2.75 |

AST1 | 5 | 2.17 | 0.06 | −2.32 | AST1 | 5 | 2.05 | 0.06 | −2.75 |

ALD | 3 | −0.93 | 1.00 | −2.64 | ALD | 2 | −1.14 | 1.00 | −2.98 |

p-value | 0.058 | p-value | 0.062 | ||||||

Mexico 1% | Mexico 2.5% | ||||||||

SNORM | 4 | 0.84 | 0.72 | −2.91 | SNORM | 4 | 0.84 | 0.74 | −3.26 |

STD | 1 | −3.26 | 1.00 | −3.18 | STD | 1 | −3.74 | 1.00 | −3.41 |

SSTD | 3 | 0.01 | 1.00 | −3.02 | SSTD | 3 | −0.71 | 1.00 | −3.34 |

AST | 6 | 1.11 | 0.54 | −2.92 | AST | 6 | 1.45 | 0.35 | −3.23 |

AST1 | 5 | 1.11 | 0.54 | −2.92 | AST1 | 5 | 1.45 | 0.35 | −3.23 |

ALD | 2 | −1.09 | 1.00 | −3.17 | ALD | 2 | −1.62 | 1.00 | −3.40 |

p-value | 0.536 | p-value | 0.349 | ||||||

Russia 1% | Russia 2.5% | ||||||||

SNORM | 4 | −0.32 | 1.00 | −2.75 | SNORM | 4 | 0.26 | 0.97 | −3.08 |

STD | 2 | −2.06 | 1.00 | −2.82 | STD | 1 | −3.94 | 1.00 | −3.22 |

SSTD | 1 | −2.99 | 1.00 | −2.82 | SSTD | 2 | −3.24 | 1.00 | −3.20 |

AST | 6 | 1.20 | 0.40 | −2.48 | AST | 6 | 1.03 | 0.54 | −3.01 |

AST1 | 5 | 1.20 | 0.40 | −2.48 | AST1 | 5 | 1.03 | 0.54 | −3.01 |

ALD | 3 | −0.56 | 1.00 | −2.79 | ALD | 3 | −0.61 | 1.00 | −3.16 |

p-value | 0.402 | p-value | 0.536 | ||||||

South Africa 1% | South Africa 2.5% | ||||||||

SNORM | 3 | −0.77 | 1.00 | −2.72 | SNORM | 3 | 0.01 | 1.00 | −2.97 |

STD | 1 | −3.82 | 1.00 | −2.81 | STD | 1 | −3.48 | 1.00 | −3.06 |

SSTD | 4 | −0.31 | 1.00 | −2.63 | SSTD | 4 | 2.19 | 0.06 | −2.87 |

AST | 6 | 2.16 | 0.06 | −2.37 | ALD | 2 | −0.46 | 1.00 | −2.99 |

AST1 | 5 | 2.16 | 0.06 | −2.37 | |||||

ALD | 2 | −1.49 | 1.00 | −2.78 | |||||

p-value | 0.063 | p-value | 0.059 | ||||||

China 1% | China 2.5% | ||||||||

SNORM | 2 | −0.91 | 1.00 | −3.40 | SNORM | 4 | −1.58 | 1.00 | −3.50 |

STD | 3 | −0.15 | 1.00 | −3.38 | STD | 1 | −4.80 | 1.00 | −3.55 |

SSTD | 1 | −1.12 | 1.00 | −3.41 | SSTD | 3 | −1.58 | 1.00 | −3.50 |

ALD | 4 | 1.38 | 0.26 | −3.31 | AST | 6 | 2.78 | 0.01 | −3.25 |

AST1 | 5 | 2.78 | 0.01 | −3.25 | |||||

ALD | 2 | −1.69 | 1.00 | −3.48 | |||||

p-value | 0.264 | p-value | 0.010 | ||||||

India 1% | India 2.5% | ||||||||

SNORM | 2 | −1.24 | 1.00 | −3.53 | SNORM | 4 | −0.72 | 1.00 | −3.67 |

STD | 1 | −3.21 | 1.00 | −3.59 | STD | 1 | −3.11 | 1.00 | −3.75 |

SSTD | 3 | −1.21 | 1.00 | −3.52 | SSTD | 3 | −0.80 | 1.00 | −3.67 |

AST | 5 | 1.92 | 0.09 | −3.18 | AST | 6 | 1.95 | 0.09 | −3.47 |

AST1 | 6 | 1.92 | 0.09 | −3.18 | AST1 | 5 | 1.95 | 0.09 | −3.47 |

ALD | 4 | −1.10 | 1.00 | −3.51 | ALD | 2 | −1.95 | 1.00 | −3.71 |

p-value | 0.08 | p-value | 0.094 | ||||||

South Korea 1% | South Korea 2.5% | ||||||||

SNORM | 3 | −0.40 | 1.00 | −3.18 | SNORM | 3 | −0.78 | 1.00 | −3.42 |

STD | 4 | 1.92 | 0.09 | −3.02 | STD | 4 | 2.08 | 0.06 | −3.30 |

SSTD | 1 | −1.77 | 1.00 | −3.21 | SSTD | 2 | −1.22 | 1.00 | −3.42 |

ALD | 2 | −1.18 | 1.00 | −3.26 | ALD | 1 | −1.97 | 1.00 | −3.48 |

p-value | 0.093 | p-value | 0.063 | ||||||

Taiwan 1% | Taiwan 2.5% | ||||||||

SNORM | 4 | −0.94 | 1.00 | −3.04 | SNORM | 4 | −1.42 | 1.00 | −3.44 |

STD | 1 | −2.10 | 1.00 | −3.13 | STD | 1 | −2.61 | 1.00 | −3.48 |

SSTD | 3 | −1.21 | 1.00 | −3.06 | SSTD | 3 | −1.49 | 1.00 | −3.45 |

AST | 5 | 1.73 | 0.14 | −2.56 | AST | 6 | 2.07 | 0.06 | −3.08 |

AST1 | 6 | 1.73 | 0.14 | −2.56 | AST1 | 5 | 2.07 | 0.06 | −3.08 |

ALD | 2 | −2.06 | 1.00 | −3.15 | ALD | 2 | −1.97 | 1.00 | −3.48 |

p-value | 0.140 | p-value | 0.065 | ||||||

Thailand 1% | Thailand 2.5% | ||||||||

SNORM | 4 | −0.24 | 1.00 | −3.55 | SNORM | 4 | 1.05 | 0.47 | −3.67 |

STD | 1 | −4.42 | 1.00 | −3.75 | STD | 1 | −5.34 | 1.00 | −3.95 |

SSTD | 2 | −2.80 | 1.00 | −3.64 | SSTD | 2 | −2.61 | 1.00 | −3.83 |

AST | 6 | 1.52 | 0.19 | −3.26 | AST | 6 | 1.44 | 0.24 | −3.63 |

AST1 | 5 | 1.52 | 0.19 | −3.26 | AST1 | 5 | 1.44 | 0.24 | −3.63 |

ALD | 3 | −0.86 | 1.00 | −3.60 | ALD | 3 | −1.06 | 1.00 | −3.81 |

p-value | 0.188 | p-value | 0.245 | ||||||

Eurozone and Global Financial Crises (EC-GFC) Periods:19 April 2011 to 7 June 2013 | |||||||||

EM index 1% | EM index 2.5% | ||||||||

ALD | 1 | −6.03 | 1.00 | −0.02 | ALD | 1 | −5.84 | 1.00 | −0.02 |

p-value | 0.000 | p-value | 0.000 | ||||||

Post−crises (PC) period:22 July 2017 to 18 February 2019 | |||||||||

EM index 1% | EM index 2.5% | ||||||||

ALD | 1 | −6.00 | 1.00 | −0.024 | ALD | 1 | −5.92 | 1.00 | −0.02 |

p-value | 0.000 | p-value | 0.000 |

_{i}denotes the test statistic derived from sample loss of the $ith$ model relative to the average across models in ${M}^{*}$ (Hansen et al. [9]). Values in bold print are overall p-value for the respective SSM (selected at the 95% confidence level).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Owusu Junior, P.; Alagidede, I.P.; Tiwari, A.K.
On the Elicitability and Risk Model Comparison of Emerging Markets Equities. *Math. Comput. Appl.* **2021**, *26*, 63.
https://doi.org/10.3390/mca26030063

**AMA Style**

Owusu Junior P, Alagidede IP, Tiwari AK.
On the Elicitability and Risk Model Comparison of Emerging Markets Equities. *Mathematical and Computational Applications*. 2021; 26(3):63.
https://doi.org/10.3390/mca26030063

**Chicago/Turabian Style**

Owusu Junior, Peterson, Imhotep Paul Alagidede, and Aviral Kumar Tiwari.
2021. "On the Elicitability and Risk Model Comparison of Emerging Markets Equities" *Mathematical and Computational Applications* 26, no. 3: 63.
https://doi.org/10.3390/mca26030063