arch.univariate.kupiec_pof

arch.univariate.kupiec_pof(returns: ndarray[tuple[int], dtype[float64]] | Series | list[float], var: ndarray[tuple[int], dtype[float64]] | Series | list[float], p: float = 0.05) → WaldTestStatistic[source]

Kupiec proportion-of-failures test of the coverage of VaR.

Parameters:
returns: ndarray[tuple[int], dtype[float64]] | Series | list[float]

Realized returns, a 1-d array of finite numbers.

var: ndarray[tuple[int], dtype[float64]] | Series | list[float]

VaR forecasts as positive losses, a 1-d array of finite numbers. See var_exceptions for the sign convention and for how the inputs are aligned.

p: float = 0.05

The probability of an exception that the VaR is designed for, which is the size of the tail, in the interval (0, 0.5]. It is 0.05 for a 95% VaR and 0.01 for a 99% VaR. The default is 0.05, a 95% VaR.

Returns:

The likelihood-ratio statistic, which is asymptotically \(\chi^2_1\) under the null that the probability of an exception is p, with its p-value.

Return type:

arch.utility.testing.WaldTestStatistic

Raises:

  • TypeError – If returns or var cannot be converted to numbers, or p is not a real number.

  • ValueError – If returns or var is not valid, as in var_exceptions, or p is not in (0, 0.5].

See also

christoffersen_independence

Test of the independence of the exceptions.

christoffersen_conditional_coverage

Joint test of coverage and independence.

traffic_light

The Basel zone of the number of exceptions.

Notes

With \(x\) exceptions in \(T\) observations and \(\hat{\pi}=x/T\), the statistic of Kupiec (1995) [1] is

\[LR_{uc} = -2\ln\left[\frac{(1-p)^{T-x}p^x} {(1-\hat{\pi})^{T-x}\hat{\pi}^x}\right].\]

The test is two-sided, so too few exceptions reject as well as too many. The terms \(0\ln 0\) of a series with no exception, or only exceptions, are 0, which is the limit of the likelihood, so that the statistic is \(-2T\ln(1-p)\) and \(-2T\ln p\).

References