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Appendix B: Backtesting analysis
I) Unconditional test, LR UC :
The most basic assumption is that the market risk model provides a correct
unconditional coverage, namely, H 0;U : E [H ;t ] = . The null hypothesis is rejected
for large values of the likelihood-ratio test de…ned as
LR UC = 2(N N ) log(1
N
N ) log(1 ) + 2N log N
N
log 2 (1)
P
t=1;N
(B.1)
where 2 (1) stands for a Chi-squared distribution with one degree of freedom, N
H t; is the number of exceptions, and N is the total number of out-ofsample
observations. Note that N =N = b H is simply the sample mean of H ;t ,
i:e, the sample equivalent of E [H ;t ] :
II) Independence test, LR IND :
If exceptions are serially correlated, the property of reliability conditional coverage
will be defective even if the unconditional coverage is correct, because the
risk of bankruptcy is higher. Christo¤ersen (1998) proposes the analysis of the
…rst-order serial correlation in H ;t through a binary …rst-order Markov chain with
transition probability matrix
=
1 01 01
; with
1 11 ij = Pr(H ;t = j j H ;t 1 = i); i; j 2 f0; 1g (B.2)
01
The approximate joint likelihood conditional on the …rst observation is
L(; H ;t j H ;1 ) = (1 01 ) n 00
n 01
01 (1 11 ) n 10
n 11
11 ; (B.3)
where n ij represents the number of transitions from state i to state j. The
maximum-likelihood estimators under the alternative hypothesis are b 01 = n 01 = (n 00 + n 01 ) ;
and b 11 = n 11 = (n 10 + n 11 ) : Under the null hypothesis of independence, we have
01 = 11 = 0 ; with 0 = ; from which the conditional binomial joint likelihood
is
L( 0 ; H ;t j H ;1 ) = (1 01 ) n 00+n 10
n 01+n 11
01 : (B.4)
Note that 0 can be estimated as b 0 = N =N. The likelihood ratio test for the
hypothesis of independence is given by
h
i
LR IND = 2 log L(^; H ;t j H 1 ) log L(b 0 ; H ;t j H ;1 ) 2 (1)
(B.5)
III) Conditional test, LR CC :
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