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Journal of the Korean Data & Information Science Society = 한국데이터정보과학회지, v.24 no.3, 2013년, pp.625 - 636
Shim, Jooyong (Department of Data Science, Inje University) , Hwang, Changha (Department of Statistics, Dankook University)
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In this paper we study four kernel machines for estimating expected shortfall, which are constructed through combinations of support vector quantile regression (SVQR), restricted SVQR (RSVQR), least squares support vector machine (LS-SVM) and support vector expectile regression (SVER). These kernel ...
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Artzner, P., Delbaen, F., Eber, J. M. and Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203-28.
Bae, J., Hwang, C. and Shim, J. (2012). Two-step LS-SVR for censored regression. Journal of the Korean Data & Information Science Society, 23, 393-401.
Bollerslev, T. (1987). A conditional heteroskedastic time series model for speculative prices and rates of returns. Review of Economics and Statistics, 69, 542-547.
Efron, B. (1991). Regression percentiles using asymmetric squared error loss. Statistica Sinica, 1, 93-125.
Efron (1991) showed that the expectile could be used to estimate VaR and ES because there existed a one-to-one mapping from expectiles to quantiles.
From Efron (1991) the expectile regression function can be used to estimate VaR and ES because there exists a one-to-one mapping from expectile regression functions to quantile regression functions.
He, X. (1997). Quantile curves without crossing. The American Statistician, 51, 86-192.
where s(xi) is assumed to be positive, ∊i is assumed to have median 0 and |∊i | is assumed to have median 1. For noncrossing quantile regression, we employ the basic ideas of restricted regression quantile of He (1997) with support vector median regression as follows:
Hwang, C. and Shim, J. (2011). Cox proportional hazard model with L1 penalty. Journal of the Korean Data & Information Science Society, 22, 613-618.
Hwang, C. and Shim, J. (2012). Mixed effects least squares support vector machine for survival data analysis. Journal of the Korean Data & Information Science Society, 23, 739-748.
Jorion, P. (2007). Value at risk: The new benchmark for managing financial risk, McGraw-Hill, New York.
Value at risk (VaR) and expected shortfall (ES) are two closely related and widely used risk measures (Jorion, 2007).
Kuhn, H. W. and Tucker, A. W. (1951). Nonlinear programming. In Proceedings of 2nd Berkeley Symposium, University of California Press, Berkeley, 481-492.
where # is obtained via Kuhn-Tucker conditions (Kuhn and Tucker, 1951) such as
Leorato, S., Peracchi, F. and Tanase, A. (2012). Asymptotically ecient estimation of the conditional expected shortfall. Computational Statistics and Data Analysis, 56, 768-784.
Li, Y., Liu, Y. and Ji, Z. (2007). Quantile regression in reproducing kernel Hilbert spaces. Journal of the American Statistical Association, 102, 255-268.
Mercer, J. (1909). Functions of positive and negative type and their connection with theory of integral equations. Philosophical Transactions of Royal Society A, 415-446.
is a kernel function obtained from the Mercer’s condition (Mercer, 1909).
Shim, J. and Lee, J. (2010). Restricted support vector quantile regression without crossing. Journal of the Korean Data & Information Science Society, 21, 1319-1325.
In this paper, we consider four kernel machines for estimating ES, which are based on IQRF, SVQR by Takeuchi et al. (2006), RSVQR by Shim and Lee (2010), LS-SVM by Suykens and Vanderwalle (1999) and SVER by Wang et al. (2011).
We now introduce RSVQR proposed by Shim and Lee (2010). RSVQR is based on the following nonlinear heteroscedastic model,
Shim, J., Kim, Y., Lee, J. and Hwang, C. (2012). Estimating value at risk with semiparametric support vector quantile regression. Computational Statistics, 27, 685-700.
Shim, J. and Seok, K. (2012). Semiparametric kernel logistic regression with longitudinal data. Journal of the Korean Data & Information Science Society, 23, 385-392.
Suykens, J. A. K. and Vanderwalle, J. (1999). Least square support vector machine classifier. Neural Processing Letters, 9, 293-300.
In this paper, we consider four kernel machines for estimating ES, which are based on IQRF, SVQR by Takeuchi et al. (2006), RSVQR by Shim and Lee (2010), LS-SVM by Suykens and Vanderwalle (1999) and SVER by Wang et al. (2011).
The LS-SVM, a modified version of SVM in a least squares sense, has been proposed for the classification and the regression by Suykens and Vanderwalle (1999).
Takeuchi, I., Le, Q. V., Sears, T. D. and Smola, A. J. (2006). Nonparametric quantile estimation. Journal of Machine Learning Research, 7, 1231-1264.
In this paper, we consider four kernel machines for estimating ES, which are based on IQRF, SVQR by Takeuchi et al. (2006), RSVQR by Shim and Lee (2010), LS-SVM by Suykens and Vanderwalle (1999) and SVER by Wang et al. (2011).
Takeuchi et al. (2006) first considered quantile regression by SVM formulation. Li et al. (2007) derived a simple formula for the effective dimension of the SVQR model, which allows convenient selection of the hyperparameters.
Taylor, J. W. (2000). A quantile regression neural network approach to estimating the conditional density of multiperiod returns. Journal of Forecasting, 19, 299-311.
Taylor, J. W. (2008a). Estimating value at risk and expected shortfall using expectiles. Journal of Financial Econometrics, 6, 231-252.
ES is defined as the conditional expectation of the return given that it is less than the VaR (Artzner et al., 1999; Taylor, 2008a).
Taylor (2008a) proposed to estimate VaR and ES via linear asymmetric least squares regression.
The estimation of ES from expectile regression function can be obtained by solving the following equation (Taylor, 2008a),
Vapnik, V. N. (1995). The nature of statistical learning theory, Springer, New York.
The support vector machine (SVM), first developed by Vapnik (1995) and his group at AT&T Bell Laboratories, solves the weak point of neural network such as the existence of local minima in the area of statistical learning theory and structural risk minimization.
Wang, Y., Wang, S. and Lai, K. (2011). Measuring financial risk with generalized asymmetric least squares regression. Applied Soft Computing, 11, 5793-5800.
Yuan, M. (2006). GACV for quantile smoothing splines. Computational Statistics & Data Analysis, 50, 813-829.
For the model selection Yuan (2006) proposed the generalized approximate cross validation function as follows,
where λ is the set of hyperparameters, def f is a measure of the effective dimensionality of the fitted model and Yuan (2006) used def f = # with a differentiable modified check function.
Zhu, D. and Galbraith, J. W. (2011). Modeling and forecasting expected shortfall with the generalized asymmetric Student-t and asymmetric exponential power distributions. Journal of Empirical Finance, 18, 765-778.
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