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The unconditional coverage test introduced by Kupiec (1995) [78] is based on the number of violations, i.e. the number of times returns exceed the predicted VaR (T1) over a period

of time T for a given significance level. If the VaR model is correctly specified, the failure rate (ˆπ = T1

T ) should be equal to the pre-specified VaR level (α). The null hypothesis

H0 : π = α is evaluated through a likelihood ratio test:

LRuc = −2 ln L(Πα) L( bΠ) ! = −2 ln (1 − α) T0_αT1 (1 − ˆπ)T0_πˆT1  T →∞ −→ χ2₁ where T0= T − T1.

Christoffersen (1998) [27] developed a conditional coverage test from the unconditional coverage test (LRuc) and the independence test (LRind).

The LRindstatistic: LRind= −2 ln(L( bΠ)/L( bΠ1)) is the likelihood ratio statistic for the

hypothesis of serial independence against first-order Markov dependence. It has an asymp- totic χ2₁ distribution. The likelihood function under the null hypothesis (π01= π11= π =

(T11+ T01)/T ) is L( bΠ) = (1 − ˆπ)T0πˆT1 where T0 = T00+ T10and T1= T11+ T01. The likeli-

hood function under the alternative hypothesis is L( bΠ1) = (1 − ˆπ01)T00ˆπ₀₁T01(1 − ˆπ11)T10ˆπ₁₁T11

previous period, ˆπ01= T01/(T00+ T01) and ˆπ11= T11/(T10+ T11).

He assumes that, under the alternative hypothesis of VaR inefficiency, the process of violations It(α), where It(α) = 1 if rt < V aR(α) and It(α) = 0 otherwise, can be

modeled as a Markov chain with πij = P r[It(α) = j|It−1(α) = i]. This leads to a test

of the null hypothesis of conditional coverage using a simple likelihood ratio statistic, LRcc= −2 ln(L(Πα)/L( bΠ1)) = LRuc+ LRind, which is asymptotically distributed χ22.

While this test is easy to use, it is rather limited for two main reasons, i) The inde- pendence is tested against a very particular form of alternative dependence structure that does not take into account a dependence of order higher than one, ii) The use of a Markov chain only considers the influence of past violations It(α) and not the influence of any

other exogenous variable.

The Dynamic Quantile Test proposed by Engle and Manganelli (2004) [39] overcomes these two drawbacks of the conditional coverage test. These authors suggest using a lin- ear regression model that links current violations to past violations. Let us define the auxiliary variable: Hitt(α) = It(α) − α so that Hitt(α) = 1 − α if rt< V aRt|t−1(α) and

Hitt(α) = −α otherwise. The null hypothesis of this test is that the sequence of hits

(Hitt) is uncorrelated with any variable that belongs to the information set Ωt−1 available

when the VaR was calculated and it has a mean value of zero, which implies that the hits are not autocorrelated.

The Dynamic Quantile test is a Wald test of the null hypothesis that all slopes in the regression model, Hitt(α) = δ0+ p X i=1 δiHitt−i+ q X j=p+1 δjXj + t

are zero, where Xj are explanatory variables contained in Ωt−1. The test statistic

has an asymptotic χ2_p+q+1 distribution. In our implementation of the test, we use p = 5 and q = 1 (where X1 = V aR(α)) as proposed by Engle and Manganelli (2004). By doing

so, we are testing whether the probability of an exception depends on the level of the VaR.

Lopez (1998, 1999) [85, 86] introduced loss functions in VaR evaluation to take into account the magnitude of the excesses that occur with respect to the VaR. Using that insight, Sarma et al. (2003) [113] introduced the Quadratic Loss Function (QLF) that uses squared distances between the observed returns and the V aR(α) predicted when a violation occurs, to ensure a greater penalty on large excesses:

lft+1=

(

(rt+1− V ar(α))2 if rt+1< V aR(α)

0 if rt+1≥ V aR(α)

A VaR model should be preferable to another if it has a lower average value of the loss function,  PT t=1 lft T  .

Later on, Giacomini and Komunjer (2005) [46] suggested the use of the Asymmetric Linear Tick Loss Function (AlTick) that takes into account the magnitude of the implicit cost associated with VaR forecasting errors. Hence, it takes into account not only the returns that exceed the VaR, but also the opportunity cost produced by an overestimation of VaR. When there are not exceptions, the loss function also penalizes due to the excess capital retained:

Lα(et+1) =

(

(α − 1)et+1 if et+1< 0

αet+1 if et+1≥ 0

where et+1 = rt+1− V aRt+1. Giacomini and Komunjer use the asymmetric linear

loss function of order α because the object of interest is the conditional α-quantile of the distribution of returns. If a quadratic loss function is used, the optimal forecast is the conditional mean of the distribution of returns and if, on the other hand, an absolute value loss function is used, the optimal forecast corresponds to the conditional median of the distribution of returns. For this reason, the AlTick is the implicit loss function whenever the object of interest is a forecast of a particular quantile of the conditional distribution of returns. A VaR model is preferable if it has a lower average value of the loss function.