The conditional DE test is a one-sided
test to check if the test statistic is much larger than zero.
The test statistic for the conditional DE test is derived in several steps.
First, define the autocovariance for lag j:
where
ɑ = 1- VaRLevel.
Ht is
the cumulative failures or violations process:
Ht
= (α -
Ut)I(Ut
< α) / α, where I(x) is the
indicator function.
Ut
are the ranks or mapped returns
Ut
=
Pt(Xt),
where
Pt(Xt)
=
P(Xt
| θt) is the cumulative
distribution of the portfolio outcomes or returns
Xt
over a given test window t =
1,...N and
θt are the parameters
of the distribution. For simplicity, the subindex
t is both the return and the parameters,
understanding that the parameters are those used on date
t, even though those parameters are estimated
on the previous date t-1, or even prior to
that.
The exact theoretical mean α/2, as opposed to the sample mean, is used in the
autocovariance formula, as suggested in the paper by Du and Escanciano
[1].
The autocorrelation for lag j is then
The test statistic for m lags is
Significance of the Test
The test statistic
CES is a random
variable and a function of random return sequences or portfolio outcomes
X1,…,XN:
For returns observed in the test window 1,…,N, the test
statistic attains a fixed value:
In general, for unknown returns that follow a distribution of
Pt, the value
of CES is uncertain
and it follows a cumulative distribution function:
This distribution function computes a confidence interval and a
p-value. To determine the distribution
PC, the esbacktestbyde
class
supports the large-sample approximation and simulation methods. You can specify
one of these methods by using the optional name-value pair argument
CriticalValueMethod
.
For the large sample approximation method, the distribution
PC is derived
from an asymptotic analysis. If the number of observations N
is large, the test statistic is approximately distributed as a chi-square
distribution with m degrees of freedom:
Note that the limiting distribution is independent of α.
If αtest = 1 - test
confidence level, then the critical value CV is
the value that satisfies the equation
The p-value is determined as
The test rejects if
pvalue <
αtest.
For the simulation method, the distribution
PCis
estimated as follows
Simulate M scenarios of returns as
Compute the corresponding test statistic as
Define
PC
as the empirical distribution of the simulated test statistic values as
where I(.) is the indicator function.
In practice, simulating ranks is more efficient than simulating returns and
then transforming the returns into ranks. simulate
.
For the empirical distribution, the value of
1-PC(x)
may be different than
P[CES
≥ x] because the distribution may have nontrivial jumps
(simulated tied values). Use the latter probability for the estimation of
confidence levels and p-values.
If ɑtest = 1 - test
confidence level, then the critical value of levels
CV is the value that satisfies the equation
The reported critical value CV is one of the simulated test
statistic values
CsES
that approximately solves the preceding equation.
The p-value is determined as
The test rejects if
pvalue <
αtest.