| covtest {ufRisk} | R Documentation |
Unconditional and Conditional Coverage Tests, Independence Test
Description
The conditional (Kupiec, 1995), the unconditional coverage test (Christoffersen, 1998) and the independence test (Christoffersen, 1998) of the Value-at-Risk (VaR) can be applied.
Usage
covtest(obj = list(Loss = NULL, VaR = NULL, p = NULL), conflvl = 0.95)
Arguments
obj |
a list that contains the following elements:
|
conflvl |
a numeric vector with one element; the significance
level at which the null hypotheses are evaluated; is set to |
Details
With this function, the conditional and the unconditional coverage
tests introduced by Kupiec (1995) and Christoffersen (1998) can be applied.
Given a return series r_t with n observations, divide the
series into n-K in-sample and K out-of-sample observations,
fit a model to the in-sample data and obtain rolling one-step forecasts of
the VaR for the out-of-sample time points.
Define
I_t = 1,
if -r_t > \widehat{VaR}_t (\alpha) or
I_t = 0,
otherwise,
for t = n + 1, n + 2, ..., n + K as the hit sequence, where \alpha is
the confidence level for the VaR (often \alpha = 0.95 or \alpha = 0.99).
Furthermore, denote p = \alpha and let w be the actual covered
proportion of losses in the data.
1. Unconditional coverage test:
H_{0, uc}: p = w
Let K_1 be the number of ones in I_t and analogously K_0 the number of
zeros (all conditional on the first observation).
Also calculate \hat{w} = K_0 / (K - 1). Obtain
L(I_t, p) = p^{K_0}(1 - p)^{K_1}
and
L(I_t, \hat{w}) = \hat{w}^{K_0}(1 - \hat{w})^{K_1}
and subsequently the test statistic
LR_{uc} = -2 * \ln \{L(I_t, p) / L(I_t, \hat{w})\}.
LR_{uc} now asymptotically follows a chi-square-distribution with one degree
of freedom.
2. Conditional coverage test:
The conditional coverage test combines the unconditional coverage test
with a test on independence. Denote by w_{ij} the probability of an i on day
t-1 being followed by a j on day t, where i and j correspond to the value of
I_t on the respective day.
H_{0, cc}: w_{00} = w{10} = p
with i = 0, 1 and j = 0, 1.
Let K_{ij} be the number of observations, where the values on two following days
follow the pattern ij. Calculate
L(I_t, \hat{w}_{00}, \hat{w}_{10})
= \hat{w}_{00}^{K_{00}}(1 - \hat{w}_{00})^{K_{01}} * \hat{w}_{10})^{K_{10}}(1 - \hat{w}_{10})^{K_{11}},
where \hat{w}_{00} = K_{00} / K_0 and \hat{w}_{10} = K_{10} / K_1. The test
statistic is then given by
LR_{cc} = -2 * \ln \{ L(I_t, p) / L(I_t, \hat{w}_{00}, \hat{w}_{10}) \},
which asymptotically follows a chi-square-distribution with two degrees of freedom.
3. Independence test:
H_{0,ind}: w_{00} = w_{10}
The asymptotically chi-square-distributed test statistic (one degree of freedom) is given by
LR_{ind} = -2 * \ln \{L(I_t, \hat{w}_{00}, \hat{w}_{10}) / L(I_t, \hat{w})\}.
—————————————————————————–
The function needs four inputs: the out-of-sample loss series obj$Loss, the
corresponding estimated VaR series obj$VaR, the coverage level obj$p,
for which the VaR has been calculated and the significance level conflvl,
at which the null hypotheses are evaluated. If an object returned by this function
is entered into the R console, a detailed overview of the test
results is printed.
Value
A list of class ufRisk with the following four elements:
- p
probability p stated in the null hypotheses of the coverage tests.
- p.uc
the p-value of the unconditional coverage test.
- p.cc
the p-value of the conditional coverage test.
- p.ind
the p-value of the independence test.
- conflvl
the significance level at which the null hypotheses are evaluated.
Author(s)
Sebastian Letmathe (Scientific Employee) (Department of Economics, Paderborn University)
Dominik Schulz (Scientific Employee) (Department of Economics, Paderborn University),
References
Christoffersen, P. F. (1998). Evaluating interval forecasts. International economic review, pp. 841-862.
Kupiec, P. (1995). Techniques for verifying the accuracy of risk measurement models. The J. of Derivatives, 3(2).
Examples
# Example for Walmart Inc. (WMT)
prices <- WMT$price.close
output <- varcast(prices)
Loss <- -output$ret.out
VaR <- output$VaR.v
covtest.data <- list(Loss = Loss, VaR = VaR, p = 0.99)
covtest(covtest.data)
# directly passing an output object of 'varcast()' to 'covtest()'
output <- varcast(prices)
covtest(output)