conditionalDE
Conditional Du-Escanciano (DE) expected shortfall (ES) backtest
Syntax
Description
runs the conditional expected shortfall (ES) backtest by Du and Escanciano [1].
The conditional test supports critical values by large-scale approximation and
by finite-sample simulation.TestResults
= conditionalDE(ebtde
)
[
specifies options using one or more name-value pair arguments in addition to the
input argument in the previous syntax.TestResults
,SimTestStatistic
] = conditionalDE(___,Name,Value
)
Examples
Create an esbacktestbyde
Object and Run a ConditionalDE Test
Create an esbacktestbyde
object for a t model with 10 degrees of freedom and 2 lags, and then run a conditionalDE
test.
load ESBacktestDistributionData.mat rng('default'); % For reproducibility ebtde = esbacktestbyde(Returns,"t",... 'DegreesOfFreedom',T10DoF,... 'Location',T10Location,... 'Scale',T10Scale,... 'PortfolioID',"S&P",... 'VaRID',["t(10) 95%","t(10) 97.5%","t(10) 99%"],... 'VaRLevel',VaRLevel); conditionalDE(ebtde,'NumLags',2)
ans=3×13 table
PortfolioID VaRID VaRLevel ConditionalDE PValue TestStatistic CriticalValue AutoCorrelation Observations CriticalValueMethod NumLags Scenarios TestLevel
___________ _____________ ________ _____________ __________ _____________ _____________ _______________ ____________ ___________________ _______ _________ _________
"S&P" "t(10) 95%" 0.95 reject 3.2121e-09 39.113 5.9915 0.11009 1966 "large-sample" 2 NaN 0.95
"S&P" "t(10) 97.5%" 0.975 reject 1.6979e-07 31.177 5.9915 0.087348 1966 "large-sample" 2 NaN 0.95
"S&P" "t(10) 99%" 0.99 reject 9.1526e-05 18.598 5.9915 0.076814 1966 "large-sample" 2 NaN 0.95
Input Arguments
ebtde
— esbacktestbyde
object
object
esbacktestbyde
object, which contains a copy of the
data (the PortfolioData
, VarData
,
and ESData
properties) and all combinations of
portfolio ID, VaR ID, and VaR levels to be tested. For more information
on creating an esbacktestbyde
object, see esbacktestbyde
.
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: TestResults =
conditionalDE(ebtde,'CriticalValueMethod','simulation','NumLags',10,'TestLevel',0.99)
CriticalValueMethod
— Method to compute critical values, confidence intervals, and p-values
'large-sample'
(default) | character vector with values of 'large-sample'
or 'simulation'
| string with values of "large-sample"
or
"simulation"
Method to compute critical values, confidence intervals, and
p-values, specified as the comma-separated
pair consisting of 'CriticalValueMethod'
and a
character vector or string with a value of
'large-sample'
or
'simulation'
.
Data Types: char
| string
NumLags
— Number of lags in conditionalDE
test
1
(default) | positive integer
Number of lags in the conditionalDE
test,
specified as the comma-separated pair consisting of
'NumLags'
and a positive integer.
Data Types: double
TestLevel
— Test confidence level
0.95
(default) | numeric value between 0
and
1
Test confidence level, specified as the comma-separated pair
consisting of 'TestLevel'
and a numeric value
between 0
and 1
.
Data Types: double
Output Arguments
TestResults
— Results
table
Results, returned as a table where the rows correspond to all combinations of portfolio ID, VaR ID, and VaR levels to be tested. The columns correspond to the following:
'PortfolioID'
— Portfolio ID for the given data'VaRID'
— VaR ID for each of the VaR levels'VaRLevel'
— VaR level'ConditionalDE'
— Categorical array with the categories'accept'
and'reject'
, which indicate the result of the conditional DE test'PValue'
— P-value of the conditional DE test'TestStatistic'
— Conditional DE test statistic'CriticalValue'
— Critical value for the conditional DE test'AutoCorrelation'
— Autocorrelation for the reported number of lags'Observations'
— Number of observations'CriticalValueMethod'
— Method used to compute confidence intervals and p-values'NumLags'
— Number of lags'Scenarios'
— Number of scenarios simulated to get the p-values'TestLevel'
— Test confidence level
Note
If you specify CriticalValueMethod
as
'large-sample'
, the function reports the
number of 'Scenarios'
as
NaN
.
For the test results, the terms 'accept'
and 'reject'
are used for convenience.
Technically, a test does not accept a model; rather, a test
fails to reject it.
SimTestStatistic
— Simulated values of the test statistics
numeric array
Simulated values of the test statistics, returned as an
NumVaRs
-by-NumScenarios
numeric array.
More About
Conditional DE Test
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 = (ɑ - Ut)I(Ut < ɑ) / ɑ, 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 ɑ/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 ɑ.
If ɑtest = 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 < ɑtest.
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 < ɑtest.
References
[1] Du, Z., and J. C. Escanciano. "Backtesting Expected Shortfall: Accounting for Tail Risk." Management Science. Vol. 63, Issue 4, April 2017.
[2] Basel Committee on Banking Supervision. "Minimum Capital Requirements for Market Risk". January 2016 (https://www.bis.org/bcbs/publ/d352.pdf).
Version History
Introduced in R2019b
See Also
esbacktestbyde
| summary
| runtests
| unconditionalDE
| simulate
| esbacktestbysim
Topics
- Workflow for Expected Shortfall (ES) Backtesting by Du and Escanciano
- Rolling Windows and Multiple Models for Expected Shortfall (ES) Backtesting by Du and Escanciano
- Expected Shortfall Estimation and Backtesting
- Overview of Expected Shortfall Backtesting
- ES Backtest Using Du-Escanciano Method
- Comparison of ES Backtesting Methods
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