Computes a point and interval estimate of the Marginal Expected Shortfall (MES) using an expectile based approach.
ExpectMES(data, tau, tau1, method="LAWS", var=FALSE, varType="asym-Dep", bias=FALSE,
bigBlock=NULL, smallBlock=NULL, k=NULL, alpha_n=NULL, alpha=0.05)A list with elements:
HatXMES: an estimate of the \(\tau'_n\)-th expectile based MES;
VarHatXMES: an estimate of the asymptotic variance of the expectile based MES estimator;
CIHatXMES: an estimate of the approximate \((1-\alpha)100\%\) confidence interval for \(\tau'_n\)-th MES.
A vector of \((1 \times n)\) observations.
A real in \((0,1)\) specifying the intermediate level \(\tau_n\). See Details.
A real in \((0,1)\) specifying the extreme level \(\tau'_n\). See Details.
A string specifying the method used to estimate the expecile. By default est="LAWS" specifies the use of the LAWS based estimator. See Details.
If var=TRUE then an estimate of the asymptotic variance of the MES estimator is computed.
A string specifying the type of asymptotic variance to compute. By default varType="asym-Dep" specifies the variance estimator for serial dependent observations. See Details.
A logical value. By default bias=FALSE specifies that no bias correction is computed. See Details.
An interger specifying the size of the big-block used to estimaste the asymptotic variance. See Details.
An interger specifying the size of the small-block used to estimaste the asymptotic variance. See Details.
An integer specifying the value of the intermediate sequence \(k_n\). See Details.
A real in \((0,1)\) specifying the quantile's extreme level to be use in order to estimate the expectile's extreme level.
A real in \((0,1)\) specifying the confidence level \((1-\alpha)100\%\) of the approximate confidence interval for the expecile at the intermedite level.
Simone Padoan, simone.padoan@unibocconi.it, http://mypage.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@ensai.fr, http://ensai.fr/en/equipe/stupfler-gilles/
For a dataset data of sample size \(n\), an estimate of the \(\tau'_n\)-th MES is computed. The estimation of the MES at the extreme level tau1 (\(\tau'_n\)) is indeed meant to be a prediction. Two estimators are available: the so-called Least Asymmetrically Weighted Squares (LAWS) based estimator and the Quantile-Based (QB) estimator. The definition of both estimators depends on the estimation of the tail index \(\gamma\). Here, \(\gamma\) is estimated using the Hill estimation (see HTailIndex for details).
The observations can be either independent or temporal dependent. See Section 4 in Padoan and Stupfler (2020) for details.
The so-called intermediate level tau or \(\tau_n\) is a sequence of positive reals such that \(\tau_n \to 1\) as \(n \to \infty\). See predExpectiles for details.
The so-called extreme level tau1 or \(\tau'_n\) is a sequence of positive reals such that \(\tau'_n \to 1\) as \(n \to \infty\). See predExpectiles for details.
When method='LAWS', then the \(\tau'_n\)-th MES is estimated using the LAWS based estimator. When method='QB', the expectile is instead estimated using the QB esimtator. See Sectino 4 in Padoan and Stupfler (2020) and in particular Corollary 4.3 and 4.4 for details. The definition of both estimators depend on the estimation of the tail index \(\gamma\). In particular, the tail index \(\gamma\) is estimated using the Hill estimator (see HTailIndex).
If var=TRUE then an esitmate of the asymptotic variance of the \(tau'_n\)-th MES is computed. Notice that the estimation of the asymptotic variance is only available when \(\gamma\) is estimated using the Hill estimator (see HTailIndex). With independent observations the asymptotic variance is estimated by \(\hat{\gamma}^2\), see Corollary 4.3 in Padoan and Stupfler (2020). This is achieved through varType="asym-Ind". With serial dependent observations the asymptotic variance is estimated by the formula in Corollary 4.3 of Padoan and Stupfler (2020). This is achieved through varType="asym-Dep". See Section 4 adn 5 in Padoan and Stupfler (2020) for details. In this latter case the computation of the serial dependence is based on the "big blocks seperated by small blocks" techinque which is a standard tools in time series, see e.g. Leadbetter et al. (1986).
The size of the big and small blocks are specified by the parameters bigBlock and smallBlock, respectively.
If bias=TRUE then \(\gamma\) is estimated using formula (4.2) of Haan et al. (2016). This is used by the LAWS and QB estimators. Furthermore, the \(\tau'_n\)--th quantile is estimated using the formula in page 330 of de Haan et al. (2016). This provides a bias corrected version of the Weissman estimator. This is used by the QB estimator.
However, in this case the asymptotic variance is not estimated using the formula in Haan et al. (2016) Theorem 4.2. Instead, for simplicity the asymptotic variance is estimated by the formula in Corollary 3.8, with serial dependent observations, and \(\hat{\gamma}^2\) with independent observation (see e.g. de Drees 2000, for the details).
k or \(k_n\) is the value of the so-called intermediate sequence \(k_n\), \(n=1,2,\ldots\). Its represents a sequence of positive integers such that \(k_n \to \infty\) and \(k_n/n \to 0\) as \(n \to \infty\). Practically, when tau=NULL and method='LAWS', then \(\tau_n=1-k_n/n\) is the intermediate level of the expectile to be stimated. \(k_n\) also specifies the number of k\(+1\) larger order statistics used in the definition of the Hill estimator (see HTailIndex for detail). Differently, When tau=NULL and method='QB', then \(\tau_n=1-k_n/n\) is the intermediate level of the quantile to be stimated.
If the quantile's extreme level is provided by alpha_n, then expectile's extreme level \(tau'_n\) is replaced by \(tau'_n(\alpha_n)\) which is estimated by the method described in Section 6 of Padoan and Stupfler (2020). See estExtLevel for details.
Given a small value \(\alpha\in (0,1)\) then an estimate of an asymptotic confidence interval for \(tau'_n\)-th expectile, with approximate nominal confidence level \((1-\alpha)100\%\), is computed. The confidence intervals are computed exploiting formula in Corollary 4.3, 4.4 and Theorem 6.2 of Padoan and Stupfler (2020) and (46) in Drees (2003). See Sections 4-6 in Padoan and Stupfler (2020) for details. When biast=TRUE confidence intervals are computed in the same way but after correcting the tail index estimate by an estimate of the bias term, see formula (4.2) in de Haan et al. (2016) for details.
Padoan A.S. and Stupfler, G. (2020). Extreme expectile estimation for heavy-tailed time series. arXiv e-prints arXiv:2004.04078, https://arxiv.org/abs/2004.04078.
Daouia, A., Girard, S. and Stupfler, G. (2018). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80, 263-292.
de Haan, L., Mercadier, C. and Zhou, C. (2016). Adapting extreme value statistics to nancial time series: dealing with bias and serial dependence. Finance and Stochastics, 20, 321-354.
Drees, H. (2003). Extreme quantile estimation for dependent data, with applications to finance. Bernoulli, 9, 617-657.
Drees, H. (2000). Weighted approximations of tail processes for \(\beta\)-mixing random variables. Annals of Applied Probability, 10, 1274-1301.
Leadbetter, M.R., Lindgren, G. and Rootzen, H. (1989). Extremes and related properties of random sequences and processes. Springer.
QuantMES, HTailIndex, predExpectiles, extQuantile
# Marginl Expected Shortfall expectile based estimation at the extreme level
# obtained with 2-dimensional data simulated from an AR(1) with bivariate
# Student-t distributed innovations
tsDist <- "AStudentT"
tsType <- "AR"
tsCopula <- "studentT"
# parameter setting
corr <- 0.8
dep <- 0.8
df <- 3
par <- list(corr=corr, dep=dep, df=df)
# Big- small-blocks setting
bigBlock <- 65
smallBlock <- 15
# quantile's extreme level
alpha_n <- 0.999
# sample size
ndata <- 2500
# Simulates a sample from an AR(1) model with Student-t innovations
data <- rbtimeseries(ndata, tsDist, tsType, tsCopula, par)
# Extreme MES expectile based estimation
MESHat <- ExpectMES(data, NULL, NULL, var=TRUE, k=150, bigBlock=bigBlock,
smallBlock=smallBlock, alpha_n=alpha_n)
MESHat
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