rugarch (version 1.0-16)

ESTest: Expected Shortfall Test.

Description

Implements the Expected Shortfall Test of McNeil and Frey.

Usage

ESTest(alpha = 0.05, actual, ES, VaR, conf.level = 0.95, 
boot = FALSE, n.boot = 1000)

Arguments

alpha

The quantile (coverage) used for the VaR.

actual

A numeric vector of the actual (realized) values.

ES

The numeric vector of the Expected Shortfall (ES).

VaR

The numeric vector of VaR.

conf.level

The confidence level at which the Null Hypothesis is evaluated.

boot

Whether to bootstrap the test.

n.boot

Number of bootstrap replications to use.

Value

• A list with the following items:
• expected.exceed
  The expected number of exceedances (length actual x coverage).
• actual.exceed
  The actual number of exceedances.
• H1
  The Alternative Hypothesis of the one sided test (see details).
• boot.p.value
  The bootstrapped p-value (if used).
• p.value
  The p-value.
• Decision
  The one-sided test Decision on H0 given the confidence level and p-value (not the bootstrapped).

Details

The Null hypothesis is that the excess conditional shortfall (excess of the actual series when VaR is violated), is i.i.d. and has zero mean. The test is a one sided t-test against the alternative that the excess shortfall has mean greater than zero and thus that the conditional shortfall is systematically underestimated. Using the bootstrap to obtain the p-value should alleviate any bias with respect to assumptions about the underlying distribution of the excess shortfall.

References

McNeil, A.J. and Frey, R. and Embrechts, P. (2000), Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach, Journal of Empirical Finance,7, 271--300.

Examples

data(dji30ret)
spec = ugarchspec(mean.model = list(armaOrder = c(1,1), include.mean = TRUE),
variance.model = list(model = "gjrGARCH"), distribution.model = "sstd")
fit = ugarchfit(spec, data = dji30ret[1:1000, 1, drop = FALSE])
spec2 = spec
setfixed(spec2)<-as.list(coef(fit))
filt = ugarchfilter(spec2, dji30ret[1001:2500, 1, drop = FALSE], n.old = 1000)
actual = dji30ret[1001:2500,1]
# location+scale invariance allows to use [mu + sigma*q(p,0,1,skew,shape)]
VaR = fitted(filt) + sigma(filt)*qdist("sstd", p=0.05, mu = 0, sigma = 1, 
skew  = coef(fit)["skew"], shape=coef(fit)["shape"])
# calculate ES
f = function(x) qdist("sstd", p=x, mu = 0, sigma = 1, 
skew  = coef(fit)["skew"], shape=coef(fit)["shape"])
ES = fitted(filt) + sigma(filt)*integrate(f, 0, 0.05)$value/0.05
print(ESTest(0.05, actual, ES, VaR, boot = TRUE))