The unconditional DE test is a
two-sided test to check if the test statistic is close to an expected value of
ɑ/2, where ɑ = 1- VaRLevel.
The test statistic for the unconditional DE test is
where
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.
Significance of the Test
The test statistic
UES is a random
variable and a function of random return sequences:
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 UES is uncertain
and follows a cumulative distribution function:
This distribution function computes a confidence interval and a
p-value. To determine the distribution
PU, 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
PU is derived
from an asymptotic analysis. If the number of observations N
is large, the test statistic
UES is
distributed as
where N(μ,σ2) is the normal
distribution with mean μ and variance σ2.
Because the test statistic cannot be smaller than 0 or greater than 1, the
analytical confidence interval limits are clipped to the interval [0,1].
Therefore, if the analytical value is negative, the test statistic is reset to
0, and if the analytical value is greater than 1, it is reset to 1.
The p-value is
The test rejects if
pvalue <
αtest.
For the simulation method, the distribution
PUis
estimated as follows
Simulate M scenarios of returns as
Compute the corresponding test statistic as
Define
PU
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. For more information, see simulate
.
For the empirical distribution, the value of
1-PU(x)
can differ from the value of
P[UES
≥ 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 confidence intervals levels
CIlower and
CIupper are the
values that satisfy equations:
The reported confidence interval limits
CIlower and
CIupper are
simulated test statistic values
UsES
that approximately solve the preceding equations.
The p-value is determined as
The test rejects if
pvalue <
αtest.