Econometric modeling of risk measures: A selective review of the recent literature

  • Ding-shi Tian
  • Zong-wu Cai
  • Ying FangEmail author


Since the financial crisis in 2008, the risk measures which are the core of risk management, have received increasing attention among economists and practitioners. In this review, the concentration is on recent developments in the estimation of the most popular risk measures, namely, value at risk (VaR), expected shortfall (ES), and expectile. After introducing the concept of risk measures, the focus is on discussion and comparison of their econometric modeling. Then, parametric and nonparametric estimations of tail dependence are investigated. Finally, we conclude with insights into future research directions.


Expectile Expected Shortfall Network Risk Nonparametric Estimation Tail Dependence Value at Risk 

MR Subject Classification

62-02 62M10 62G08 


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  1. [1]
    V V Acharya, R Engle, M Richardson. Capital shortfall: a new approach to ranking and regulating systemic risks. American Economic Review, 2012, 102(3): 59–64.Google Scholar
  2. [2]
    V V Acharya, L H Pedersen, T Philippon, M Richardson. Measuring systemic risk. Review of Financial Studies, 2017, 30(1): 2–47.Google Scholar
  3. [3]
    T Adrian, M K Brunnermeier. CoVaR. American Economic Review, 2016, 106(7): 1705–1741.Google Scholar
  4. [4]
    P Artzner, F Delbaen, J M Eber, D Heath. Coherent measures of risk. Mathematical Finance, 1999, 9(3): 203–228.MathSciNetzbMATHGoogle Scholar
  5. [5]
    S Basak, A Shapiro. Value-at-Risk-based risk management: optimal policies and asset prices. Review of Financial Studies, 2001, 14(2): 371–405.Google Scholar
  6. [6]
    F Bellini, B Klar, A Müller, E R Gianin. Generalized quantiles as risk measures. Insurance: Mathematics and Economics, 2014, 54: 41–48.MathSciNetzbMATHGoogle Scholar
  7. [7]
    J Butler, B Schachter. Estimating Value-at-Risk with a precision measure by combining kernel estimation with historical simulation. Review of Derivatives Research, 1998, 1: 371–390.Google Scholar
  8. [8]
    Z Cai. Regression quantiles for time series. Econometric theory, 2002, 18(1): 169–192.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Z Cai, Y Fang, D Tian. Assessing tail risk using expectile regressions with partially varying coefficients. Journal of Management Science and Engineering, 2018, 3(4): 183–213.Google Scholar
  10. [10]
    Z Cai, J Su, Sufianti. A regression analysis of expected shortfall. Statistics and Its Interface, 2015, 8(3): 295–303.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Z Cai, X Wang. Nonparametric estimation of conditional VaR and expected shortfall. Journal of Econometrics, 2008, 147(1): 120–130.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Z Cai, Z Xiao. Semiparametric quantile regression estimation in dynamic models with partially varying coefficients. Journal of Econometrics, 2012, 167(2): 413–425.MathSciNetzbMATHGoogle Scholar
  13. [13]
    Z Cai, X Xu. Nonparametric quantile estimations for dynamic smooth coefficient models. Journal of the American Statistical Association, 2008, 103(484): 1595–1608.MathSciNetzbMATHGoogle Scholar
  14. [14]
    V Chavez-Demoulin, P Embrechts, S Sardy. Extreme-quantile tracking for financial time series. Journal of Econometrics, 2014, 181(1): 44–52.MathSciNetzbMATHGoogle Scholar
  15. [15]
    S X Chen, C Y Tang. Nonparametric inference of Value-at-Risk for dependent financial returns. Journal of Financial Econometrics, 2005, 3(2): 227–255.Google Scholar
  16. [16]
    V Chernozhukov, L Umantsev. Conditional Value-at-Risk: aspects of modeling and estimation. Empirical Economics, 2001, 26(1): 271–292.Google Scholar
  17. [17]
    J Danielsson, C G de Vries. Tail index and quantile estimation with very high frequency data. Journal of Empirical Finance, 1997, 4(2): 241–257.Google Scholar
  18. [18]
    R Davis, S Resnick. Tail estimates motivated by extreme value theory. Annals of Statistics, 1984, 12(4): 1467–1487.MathSciNetzbMATHGoogle Scholar
  19. [19]
    F X Diebold, K Yımaz. On the network topology of variance decompositions: measuring the connectedness of financial firms. Journal of Econometrics, 2014, 182(1): 119–134.MathSciNetzbMATHGoogle Scholar
  20. [20]
    D Duffie, J Pan. An overview of value at risk. Journal of Derivatives, 1997, 4(3): 7–49.Google Scholar
  21. [21]
    P Embrechts, M Hofert. Statistics and quantitative risk management for banking and insurance. Annual Review of Statistics and Its Application, 2014, 1: 493–514.Google Scholar
  22. [22]
    S Emmer, M Kratz, D Tasche. What is the best risk measure in practice? A comparison of standard measures. Journal of Risk, 2015, 18(2): 31–60.Google Scholar
  23. [23]
    R F Engle, S Manganelli. CAViaR: conditional autoregressive value at risk by regression quantiles. Journal of Business & Economic Statistics, 2004, 22(4): 367–381.MathSciNetGoogle Scholar
  24. [24]
    J Fan, I Gijbels. Local Polynomial Modeling and Its Applications, 1996, New York: Chapman & Hall/CRC Press.zbMATHGoogle Scholar
  25. [25]
    J Fan, T C Hu, Y K Truong. Robust nonparametric function estimation. Scandinavian Journal of Statistics, 1994, 21(4): 433–446.MathSciNetzbMATHGoogle Scholar
  26. [26]
    J Fan, Q Yao. Efficient estimation of conditional variance functions in stochastic regression. Biometrika, 1998, 85(3): 645–660.MathSciNetzbMATHGoogle Scholar
  27. [27]
    J Fan, Q Yao, H Tong. Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika, 1996, 83(1): 189–206.MathSciNetzbMATHGoogle Scholar
  28. [28]
    Y Fan, W K Härdle, W Wang, L Zhu. Single-index-based CoVaR with very high-dimensional covariates. Journal of Business & Economic Statistics, 2017, 36(2): 212–226.MathSciNetGoogle Scholar
  29. [29]
    T Gneiting. Making and evaluating point forecasts. Journal of the American Statistical Association, 2011, 106(494): 746–762.MathSciNetzbMATHGoogle Scholar
  30. [30]
    C Gourieroux, J P Laurent, O Scaillet. Sensitivity analysis of values at risk. Journal of Empirical Finance, 2000, 7(3): 225–245.Google Scholar
  31. [31]
    P Hall, R C Wolff, Q Yao. Methods for estimating a conditional distribution function. Journal of the American Statistical Association, 1999, 94(445): 154–163.MathSciNetzbMATHGoogle Scholar
  32. [32]
    W K Härdle, W Wang, L Yu. TENET: tail-event driven network risk. Journal of Econometrics, 2016, 192(2): 499–513.MathSciNetzbMATHGoogle Scholar
  33. [33]
    N Hautsch, J Schaumburg, M Schienle. Financial network systemic risk contributions. Review of Finance, 2014, 19(2): 685–738.zbMATHGoogle Scholar
  34. [34]
    T Honda. Nonparametric estimation of a conditional quantile for α-mixing processes. Annals of the Institute of Statistical Mathematics, 2000, 52(3): 459–470.MathSciNetzbMATHGoogle Scholar
  35. [35]
    T Honda. Quantile regression in varying coefficient models. Journal of Statistical Planning and Inference, 2004, 121(1): 113–125.MathSciNetzbMATHGoogle Scholar
  36. [36]
    J C Hull, A D White. Value at risk when daily changes in market variables are not normally distributed. Journal of Derivatives, 1998, 5(3): 9–19.Google Scholar
  37. [37]
    E Jondeau, S P Huang, M Rockinger. Financial modeling under non-Gaussian distributions, 2007, Berlin: Spring-Verlag.zbMATHGoogle Scholar
  38. [38]
    P Jorion. Value at Risk: A New Benchmark for Measuring Derivatives Risk, 1997, New York: Irwin Professional Publishers.Google Scholar
  39. [39]
    B Kai, R Li, H Zou. Local composite quantile regression smoothing: an efficient and safe alternative to local polynomial regression. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2010, 72(1): 49–69.MathSciNetzbMATHGoogle Scholar
  40. [40]
    B Kai, R Li, H Zou. New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. Annals of Statistics, 2011, 39(1): 305–332.MathSciNetzbMATHGoogle Scholar
  41. [41]
    M O Kim. Quantile regression with varying coefficients. Annals of Statistics, 2007, 35(1): 92–108.MathSciNetzbMATHGoogle Scholar
  42. [42]
    R Koenker. Quantile Regression, 2005, New York: Cambridge University Press.zbMATHGoogle Scholar
  43. [43]
    R Koenker, G Bassett. Regression quantiles. Econometrica, 1978, 46(1): 33–50.MathSciNetzbMATHGoogle Scholar
  44. [44]
    R Koenker, Q Zhao. Conditional quantile estimation and inference for ARCH models. Econometric Theory, 1996, 12(5): 793–813.MathSciNetGoogle Scholar
  45. [45]
    E Kong, Y Xia. A single-index quantile regression model and its estimation. Econometric Theory, 2012, 28(4): 730–768.MathSciNetzbMATHGoogle Scholar
  46. [46]
    C M Kuan, J H Yeh, Y C Hsu. Assessing value at risk with CARE, the conditional autoregressive expectile models. Journal of Econometrics, 2009, 150(2): 261–270.MathSciNetzbMATHGoogle Scholar
  47. [47]
    K Kuester, S Mittnik, M S Paolella. Value-at-Risk prediction: a comparison of alternative strategies. Journal of Financial Econometrics, 2006, 4(1): 53–89.Google Scholar
  48. [48]
    N S Lambert, D M Pennock, Y Shoham. Eliciting properties of probability distributions. In Proceedings of the 9th ACM Conference on Electronic Commerce, 2008, pp. 129–138.Google Scholar
  49. [49]
    G Maguluri, C H Zhang. Estimation in the mean residual life regression model. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 1994, 56(3): 477–489.MathSciNetzbMATHGoogle Scholar
  50. [50]
    H Markowitz. Portfolio selection. Journal of Finance, 1952, 7(1): 77–91.Google Scholar
  51. [51]
    C Martins-Filho, F Yao, M Torero. Nonparametric estimation of conditional Value-at-Risk and expected shortfall based on extreme value theory. Econometric Theory, 2018, 34(1): 23–67.MathSciNetzbMATHGoogle Scholar
  52. [52]
    A J McNeil. Estimating the tails of loss severity distributions using extreme value theory. ASTIN Bulletin, 1997, 27(1): 117–137.Google Scholar
  53. [53]
    A J McNeil, R Frey. Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance, 2000, 7(3): 271–300.Google Scholar
  54. [54]
    J P Morgan. Value at Risk, 1996, New York: RiskMetrics Technical Document.Google Scholar
  55. [55]
    W K Newey, J L Powell. Asymmetric least squares estimation and testing. Econometrica, 1987, 55(4): 819–847.MathSciNetzbMATHGoogle Scholar
  56. [56]
    A K Nikoloulopoulos, H Joe, H Li. Vine copulas with asymmetric tail dependence and applications to financial return data. Computational Statistics & Data Analysis, 2012, 56(11): 3659–3673.MathSciNetzbMATHGoogle Scholar
  57. [57]
    D Oakes, T Dasu. Inference for the proportional mean residual life model. Lecture Notes-Monograph Series, 2003, 43: 105–116.MathSciNetzbMATHGoogle Scholar
  58. [58]
    K Osband. Providing Incentives for Better Cost Forecasting, 1985, University of California, Berkeley: PhD thesis.Google Scholar
  59. [59]
    J Pickands. Statistical inference using extreme order statistics. Annals of Statistics, 1975, 3(1): 119–131.MathSciNetzbMATHGoogle Scholar
  60. [60]
    M Rocco. Extreme value theory in finance: a survey. Journal of Economic Surveys, 2014, 28(1): 82–108.MathSciNetGoogle Scholar
  61. [61]
    R Schmidt, U Stadtmüller. Non-parametric estimation of tail dependence. Scandinavian Journal of Statistics, 2006, 33(2): 307–335.MathSciNetzbMATHGoogle Scholar
  62. [62]
    K F Siburg, P Stoimenov, G N Weis. Forecasting portfolio-Value-at-Risk with nonparametric lower tail dependence estimates. Journal of Banking & Finance, 2015, 54: 129–140.Google Scholar
  63. [63]
    R L Smith. Estimating tails of probability distributions. Annals of Statistics, 1987, 15(3): 1174–1207.MathSciNetzbMATHGoogle Scholar
  64. [64]
    J W Taylor. A quantile regression approach to estimating the distribution of multiperiod returns. Journal of Derivatives, 1999, 7(1): 64–78.Google Scholar
  65. [65]
    J W Taylor. Estimating value at risk and expected shortfall using expectiles. Journal of Financial Econometrics, 2008, 6(2): 231–252.Google Scholar
  66. [66]
    C S Wang, Z Zhao. Conditional Value-at-Risk: semiparametric estimation and inference. Journal of Econometrics, 2016, 195(1): 86–103.MathSciNetzbMATHGoogle Scholar
  67. [67]
    H White, T H Kim, S Manganelli. Modeling autoregressive conditional skewness and kurtosis with multi-quantile CAViaR, 2008, University of California at San Diego: ECB Working Paper.zbMATHGoogle Scholar
  68. [68]
    H White, T H Kim, S Manganelli. VAR for VaR: measuring tail dependence using multivariate regression quantiles. Journal of Econometrics, 2015, 187(1): 169–188.MathSciNetzbMATHGoogle Scholar
  69. [69]
    T Wu, K Yu, Y Yu. Single-index quantile regression. Journal of Multivariate Analysis, 2010, 101(7): 1607–1621.MathSciNetzbMATHGoogle Scholar
  70. [70]
    W Wu, K Yu, G Mitra. Kernel conditional quantile estimation for stationary processes with application to conditional Value-at-Risk. Journal of Financial Econometrics, 2008, 6(2): 253–270.Google Scholar
  71. [71]
    S Xie, Y Zhou, A T Wan. A varying-coefficient expectile model for estimating value at risk. Journal of Business & Economic Statistics, 2014, 32(4): 576–592.MathSciNetGoogle Scholar
  72. [72]
    X Xu, A Mihoci, W K Härdle. Lcare-localizing conditional autoregressive expectiles. Journal of Empirical Finance, 2018, 48: 198–220.Google Scholar
  73. [73]
    Q Yao, H Tong. Asymmetric least squares regression estimation: a nonparametric approach. Journal of Nonparametric Statistics, 1996, 6(2–3): 273–292.MathSciNetzbMATHGoogle Scholar
  74. [74]
    K Yu, M Jones. A comparison of local constant and local linear regression quantile estimators. Computational Statistics & Data Analysis, 1997, 25(2): 159–166.zbMATHGoogle Scholar
  75. [75]
    K Yu, M Jones. Local linear quantile regression. Journal of the American statistical Association, 1998, 93(441): 228–237.MathSciNetzbMATHGoogle Scholar
  76. [76]
    J F Ziegel. Coherence and elicitability. Mathematical Finance, 2016, 26(4): 901–918.MathSciNetzbMATHGoogle Scholar
  77. [77]
    H Zou, M Yuan. Composite quantile regression and the oracle model selection theory. Annals of Statistics, 2008, 36(3): 1108–1126.MathSciNetzbMATHGoogle Scholar

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© Editorial Committee of Applied Mathematics 2019

Authors and Affiliations

  1. 1.Wang Yanan Institute for Studies in Economics, Department of Statistics, School of Economics, Ministry of Education Key Laboratory of Econometrics and Fujian Key Laboratory of Statistical ScienceXiamen UniversityXiamen, FujianChina
  2. 2.Department of EconomicsUniversity of KansasLawrenceUSA

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