An insurance company is backtesting its VaR methodology using data from March 6th 2017 to March 24th 2017 (i.e. a total of 15 daily observations). The estimated daily 5% VaRs during this period are correspondingly 0.0316, 0.0304, 0.0292, 0.0366, 0.0354, 0.0371, 0.0441, 0.0424, 0.0419, 0.0402, 0.0538, 0.0531, 0.0643, 0.0713, 0.0758. The daily equity returns of the insurance company over this period are correspondingly -0.00406, -0.00153, -0.03036, 0.00446, 0.02047, -0.03473, 0.00618, 0.01403, 0.00213, - 0.04828, 0.01737, -0.04826, 0.04274, 0.03946, -0.03894.

(a) What is the hit sequence for this 15 observations?

(b) You would like to conduct the unconditional coverage test. Calculate the likeli- hood for the null hypothesis and the alternative hypothesis.

(c) Calculate the likelihood ratio statistic for the unconditional coverage test. Would you reject the null hypothesis? (Hint: The 95% quantile of 21 is 3.84)