Derivation of the asymptotic distribution builds on the unconditional probabilities are correct and serially uncorre- approximation of the discontinuous...

Ludzie pragną czasami się rozstawać, żeby móc tęsknić, czekać i cieszyć się z powrotem.

This example is an extreme case of quantile measure-
the usual Taylor expansion can be performed. Assumptions
ment error. Any noise introduced into the quantile estimate will
AN1 and AN2 impose sufficient conditions on the func-
change the conditional probability of a hit given the estimate
tion ft(β) and on the conditional density function of the error
itself.
terms to ensure that this smooth approximation will be suffi-
Therefore, none of these tests has power against this form of
ciently well behaved. The device for obtaining such an approx-
misspecification and none can be simply extended to examine
imation is provided by an extension of theorem 3 of Huber
other explanatory variables. We propose a new test that can be
(1967). This technique is standard in the regression quantile
easily extended to incorporate a variety of alternatives. Define
and LAD literature (Powell 1984, 1991; Weiss 1991). Alterna-


tive strategies for deriving the asymptotic distribution are the
Hitt(β 0 ) ≡ I yt < ft(β 0 ) − θ.
(6)
Engle and Manganelli: CAViaR
371
The Hitt(β 0 ) function assumes value ( 1 − θ) every time yt is TR and NR on R as specified in assumption DQ8). Make ex-less than the quantile and − θ otherwise. Clearly, the expected
plicit the dependence of the relevant variables on the num-
value of Hitt(β 0 ) is 0. Furthermore, from the definition of the
ber of observations, using appropriate subscripts. Define the
quantile function, the conditional expectation of Hitt(β 0 ) given q-vector measurable n X n( ˆ βT ), n = T
R
R + 1 , . . . , TR + NR,
any information known at t − 1 must also be 0. In particu-
as the typical row of X ( ˆ
βT ), possibly depending on ˆ β , and
R
TR
lar, Hitt(β 0 ) must be uncorrelated with its own lagged values
Hit ( ˆ
β
) ≡ [ HitT
), . . . , HitT
( ˆ
β
)].
and with f
TR
R +1 ( ˆ
βTR
R + NR
TR
t(β 0 ), and must have expected value equal to 0. If
Hit
Theorem 5 (Out-of-sample dynamic quantile test) . Under the
t(β 0 ) satisfies these moment conditions, then there will be
no autocorrelation in the hits, no measurement error as in (5),
assumptions of Theorems 1 and 2 and assumptions DQ1–DQ3,
and the correct fraction of exceptions. Whether there is the right
DQ8, and DQ9,
proportion of hits in each calendar year can be determined by




−1
DQ
Hit ˆ
β
X ˆ
β
X ˆ
β
· X ˆ β
checking the correlation of Hit
OOS ≡ N−1
R
TR
TR
TR
TR
t(β 0 ) with annual dummy vari-
ables. If other functions of the past information set, such as



×
d
X ˆ
β
∼
T
Hit ˆ
βT / θ( 1 − θ)
χ 2
rolling standard deviations or a GARCH volatility estimate, are
R
R
q
as R → ∞ .
suspected of being informative, then these can be incorporated.
Proof. See Appendix B.
A natural way to set up a test is to check whether the test
The in-sample DQ test is a specification test for the partic-
statistic T−1 / 2X ( ˆ
β)Hit ( ˆ
β) is significantly different from 0,
ular CAViaR process under study and it can be very useful for
where X t( ˆ β), t = 1 , . . . , T, the typical row of X ( ˆ β) (possibly model selection purposes. The simpler version of the out-of-depending on ˆ
β), is a q-vector measurable t and Hit ( ˆ β) ≡
sample DQ test, instead, can be used by regulators to check
[ Hit 1 ( ˆ β), . . ., HitT( ˆ β)].
whether the VaR estimates submitted by a financial institution
Let
M T ≡ (X (β 0 ) − E[ T−1X (β 0 )H∇ f (β 0 )]D−1 ×
T
satisfy some basic requirements that every good quantile esti-
∇ f (β 0 )), where H is a diagonal matrix with typical entry mate must satisfy, such as unbiasedeness, independent hits, and ht( 0| t). Theorem 4 derives the in-sample distribution of the
independence of the quantile estimates. The nicest features of
DQ test. The out-of-sample case is considered in Theorem 5.
the out-of-sample DQ test are its simplicity and the fact that
it does not depend on the estimation procedure: to implement
Theorem 4 (In-sample dynamic quantile test) . Under the as-
it, the evaluator (either the regulator or the risk manager) just
sumptions of Theorems 1 and 2 and assumptions DQ1–DQ6,
needs a sequence of VaRs and the corresponding values of the

−1 / 2
d
portfolio.