# VaR Duration Test

### Description

Implements the VaR Duration Test of Christoffersen and Pelletier.

### Usage

1 | ```
VaRDurTest(alpha, actual, VaR, conf.level = 0.95)
``` |

### Arguments

`alpha` |
The quantile (coverage) used for the VaR. |

`actual` |
A numeric vector of the actual (realized) values. |

`VaR` |
The numeric vector of VaR. |

`conf.level` |
The confidence level at which the Null Hypothesis is evaluated. |

### Details

The duration of time between VaR violations (no-hits) should ideally be independent and not cluster. Under the null hypothesis of a correctly specified risk model, the no-hit duration should have no memory. Since the only continuous distribution which is memory free is the exponential, the test can conducted on any distribution which embeds the exponential as a restricted case, and a likelihood ratio test then conducted to see whether the restriction holds. Following Christoffersen and Pelletier (2004), the Weibull distribution is used with parameter ‘b=1’ representing the case of the exponential. A future release will include the choice of using a bootstrap method to evaluate the p-value, and until then care should be taken when evaluating series of length less than 1000 as a rule of thumb.

### Value

A list with the following items:

`b` |
The estimated Weibull parameter which when restricted to the value of 1 results in the Exponential distribution. |

`uLL` |
The unrestricted Log-Likelihood value. |

`rLL` |
The restricted Log-Likelihood value. |

`LRp` |
The Likelihood Ratio Test Statistic. |

`H0` |
The Null Hypothesis. |

`Decision` |
The on H0 given the confidence level |

### Author(s)

Alexios Ghalanos

### References

Christoffersen, P. and Pelletier, D. 2004, Backtesting value-at-risk: A
duration-based approach, *Journal of Financial Econometrics*, **2(1)**,
84–108.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 | ```
## Not run:
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"])
print(VaRDurTest(0.05, actual, VaR))
# Try with the Normal Distribution (it fails)
spec = ugarchspec(mean.model = list(armaOrder = c(1,1), include.mean = TRUE),
variance.model = list(model = "gjrGARCH"), distribution.model = "norm")
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("norm", p=0.05, mu = 0, sigma = 1)
print(VaRDurTest(0.05, actual, VaR))
## End(Not run)
``` |