unconditionalDE
Unconditional Du-Escanciano (DE) expected shortfall (ES) backtest
Syntax
Description
runs the unconditional Du-Escanciano (DE) expected shortfall (ES) backtest [1]. The
unconditional test supports critical values by large-scale approximation and by
finite-sample simulation.TestResults
= unconditionalDE(ebtde
)
[
specifies options using one or more name-value pair arguments in addition to the
input argument in the previous syntax.TestResults
,SimTestStatistic
] = unconditionalDE(___,Name,Value
)
Examples
Create an esbacktestbyde
Object and Run an UnconditionalDE Test
Create an esbacktestbyde
object for a t model with 10 degrees of freedom, and then run an unconditionalDE
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); unconditionalDE(ebtde)
ans=3×14 table
PortfolioID VaRID VaRLevel UnconditionalDE PValue TestStatistic LowerCI UpperCI Observations CriticalValueMethod MeanLS StdLS Scenarios TestLevel
___________ _____________ ________ _______________ ________ _____________ _________ _________ ____________ ___________________ ______ _________ _________ _________
"S&P" "t(10) 95%" 0.95 accept 0.181 0.028821 0.019401 0.030599 1966 "large-sample" 0.025 0.0028565 NaN 0.95
"S&P" "t(10) 97.5%" 0.975 accept 0.086278 0.015998 0.0085028 0.016497 1966 "large-sample" 0.0125 0.0020394 NaN 0.95
"S&P" "t(10) 99%" 0.99 reject 0.016871 0.0080997 0.0024575 0.0075425 1966 "large-sample" 0.005 0.0012972 NaN 0.95
Input Arguments
ebtde
— esbacktestbyde
object
object
esbacktestbyde
(ebtde
) 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 =
unconditionalDE(ebtde,'CriticalValueMethod','large-sample','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
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'UnconditionalDE'
— Categorical array with the categories'accept'
and'reject'
, which indicate the result of the unconditional DE test'PValue'
— P-value of the unconditional DE test'TestStatistic'
— Unconditional DE test statistic'LowerCI'
— Confidence-interval lower limit for the unconditional DE test statistic'UpperCI'
— Confidence-interval upper limit for the unconditional DE test statistic'Observations'
— Number of observations'CriticalValueMethod'
— Method for computing confidence intervals and p-values'MeanLS'
— Mean of thelarge-sample
normal distribution; ifCriticalValueMethod
is'simulation'
,'MeanLS'
is reported asNaN
'StdLS'
— Standard deviation of thelarge-sample
normal distribution; ifCriticalValueMethod
is'simulation'
,'StdLS'
is reported asNaN
'Scenarios'
— Number of scenarios simulated to get the p-values; ifCriticalValueMethod
is'large-sample'
, the number of scenarios is reported asNaN
'TestLevel'
— Test confidence level
Note
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 a
NumVaRs
-by-NumScenarios
numeric array.
More About
Unconditional DE Test
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 = (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.
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 < ɑtest.
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 < 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
| conditionalDE
| 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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