This code computes the Likelihood Ratio (LR) tests proposed by Kupiec (1995) to backtest Value-at-Risk (VaR) forecasts. The first LR test statistic, denoted LRuc in the code, test the assumption of unconditional coverage (UC). Under the null, the probability to get a VaR failure or violation (a violation is said to occur if the ex-post losses are larger than the VaR) is equal to the coverage rate α (1% for instance). This statistic has a chi-squared distribution with one degree of freedom. The second LR statistic proposed by Kupiec (1995) test the Time Until the First Failure (TUFF). Under the null of UC, the TUFF has a geometric distribution with an expected value of 1/α. The LR TUFF statistic has also a chi-squared distribution with one degree of freedom.
White Halbert and Giacomini Raffaella