varcast  R Documentation 
Onestep ahead forecasts of Value at Risk and Expected Shortfall for a selection of shortmemory and longmemory parametric as well as semiparametric GARCHtype models are computed.
varcast(
x,
a.v = 0.99,
a.e = 0.975,
model = c("sGARCH", "lGARCH", "eGARCH", "apARCH", "fiGARCH", "filGARCH"),
garchOrder = c(1, 1),
distr = c("norm", "std"),
n.out = 250,
smooth = c("none", "lpr"),
...
)
x 
a vector containing the price series. 
a.v 
confidence level for calculating VaR; is set to 
a.e 
confidence level for calculating ES; is set to 
model 
model to be estimated. Options are 'sGARCH', 'eGARCH', 'apARCH',
'lGARCH', 'fiGARCH' and 'filGARCH'; is set to 
garchOrder 
orders to be estimated; c(1, 1), i.e. p = q = 1, is the default. 
distr 
distribution to use for the innovations of the respective GARCH model;
is set to 
n.out 
size of outsample; is set to 
smooth 
a character object; defines the datadriven smoothing approach
for the estimation of the nonparametric scale function; for

... 
depending on the choice of 
Let Y_t
be a (demeaned) return series. The semiparametric extension of
the GARCH(p,q) model (Bollerslev, 1986) is called a SemiGARCH model
(Feng, 2004) and is defined by
Y_t = s(x_t)\sigma_t \eta_t,
with \eta_t \sim IID(0,1)
and
\sigma^2_t = \alpha_0 + \sum_{i=1}^p \alpha_i Y^2_{ti}
+ \sum_{j=1}^q \beta_j \sigma^2_{tj},
where \sigma_t > 0
are the conditional standard deviations, s(x_t) > 0
is
a nonparametric scale function with x_t
being the rescaled observation
time points on the interval [0,1] and \alpha_i
and \beta_j
are nonnegative real valued coefficients, except for
\alpha_0
, which must satisfy \alpha_0 > 0
. Furthermore, it is assumed that
Var(\sigma_t \eta_t) = 1
. In this functions, different
shortmemory and longmemory GARCHtype models are selectable for the parametric part of the
model. More specifically, the standard GARCH (Bollerslev, 1986), the
LogGARCH (Pantula, 1986; Geweke, 1986; Milhoj, 1988), the eGARCH
(Nelson, 1991), the APARCH (Ding et al., 1993), the FIGARCH (Baillie et al., 1996)
and the FILogGARCH (Feng et al., 2020) model are implemented. For more
information on the representations of the last three models mentioned, we
refer the reader to the corresponding references listed in the references
section.
While the innovations \eta_t
must be i.i.d. (independent and identically
distributed) with zeromean and unitvariance and while any
distribution that satisfies these conditions is suitable, the standardized
tdistribution is selected for the estimation of the models and computation
of the Value at Risk (VaR) as well as the Expected Shortfall (ES) within
this function.
For a given level \alpha \in (0, 1)
,
VaR(\alpha) = inf \{z \in R: F_L(z) \geq \alpha\}
defines the VaR at level alpha. In this definition, L
is the
loss variable (making a loss is denoted as a positive value, whereas gains
are negative values) and F_L
is its cumulative distribution function.
Explained differently, VaR(\alpha)
is the \alpha
quantile of the loss
distribution.
The ES for a level \alpha
, however, is given by
ES(\alpha) = (1 / (1  \alpha)) \int_{\alpha}^1 VaR(u)du,
i.e. it is the expected loss in case VaR(\alpha)
is exceeded. More
information on these risk measures can be found on pp. 6472 in McNeil et
al. (2015).
To apply the function, a numeric vector that contains the price series that
is to be analyzed ordered from past to present must be passed to the
argument x
. Furthermore, the user can set different levels of alpha
for the VaR and the ES via the arguments a.v
and a.e
,
respectively. A parametric shortmemory or longmemory GARCHtype model can be
selected by means of model
, which only accepts a singleelement character vector
as input. At the time of the release of package version 1.0.0, a standard
GARCH ('sGARCH'), a LogGARCH ('lGARCH'), an eGARCH ('eGARCH'), an APARCH
('apARCH'), a FIGARCH ('fiGARCH') and a FILogGARCH ('filGARCH') model can be selected,
each with conditional tdistribution. By default, a standard GARCH model is applied.
The orders of the GARCHtype models can be defined with garchOrder
,
which is a numeric vector with two elements. Its first element is the
ARCH order p, whereas the GARCH order q can be adjusted via the second
element. If no adjustments are made, the orders p = q = 1 are selected. The
number of outsample observations is set via the argument n.out
. If n
is the total number of observations of the
whole price series, the model is estimated for the first n  n.out
observations (insample), while the VaR and the ES are obtained for the last
n.out observations (outsample) based on the estimated model for the
insample. Moreover, the datadriven estimation method of the underlying
scale function can be adjusted via the argument smooth
. If
smooth = 'lpr'
is selected, the scale function is obtained by
applying an iterative plugin algorithm logarithm of the squared
centralized returns. Depending on the setting of model
an algorithm
proposed by Feng, Gries and Fritz (2020) or by Letmathe, Feng and Uhde
(2021) is employed. In the former case, the function msmooth()
of the
smoots
package is applied and for the latter the tsmoothlm()
function of the esemifar
package is used. An ellipsis ...
is
implemented to allow for additional arguments for msmooth()
and
tsmoothlm()
.
NOTE:
This function makes use of the arima()
function of the stats package,
of the fracdiff()
function of the fracdiff
package, of the
ugarchspec()
and ugarchfit()
functions of the rugarch
package, of the msmooth()
function of the smoots
package
and of the esemifar()
function of the esemifar
for estimation.
Moreover, LogGARCH and FILogGARCH models in the parametric part of the
complete models are estimated via their ARMA and FARIMA representations,
respectively, and must therefore satisfy p \geq q
.
This function returns a list with the following elements.
selected model for estimation
the estimated mean of the insample returns
estimated model parameters for the parametric part of the insample
the estimation results for the nonparametric part of the insample model
insample return series
outsample return series
estimated insample total volatility
outsample forecasts of the total volatility
the estimated nonparametric scale function values for the insample
the scale function forecast for the outsample
outsample forecasts of the (1a.e
)100% VaR
outsample forecasts of the (1a.v
)100% VaR
outsample forecasts of the (1a.e
)100% ES
estimated degrees of freedom for the standardized returns
coverage level for the 99 % VaR
coverage level for 97.5 % VaR
the orders p and q of the implemented GARCHtype model
Sebastian Letmathe (Scientific Employee) (Department of Economics,
Paderborn University)
Dominik Schulz (Scientific Employee) (Department of Economics,
Paderborn University),
Baillie, R. T., Bollerslev, T., & Mikkelsen, H. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. In: Journal of Econometrics, 74.1, pp. 330.
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. In: Journal of Econometrics 31.3, pp. 307327.
Ding, Z., Granger, C.W., and Engle, R.F. (1993). A long memory property of stock market returns and a new model. In: Journal of Empirical Finance 1.1, pp. 83106.
Feng, Y. (2004). Simultaneously modeling conditional heteroskedasticity and scale change. In: Econometric Theory 20.3, pp. 563596.
Feng, Y., Beran, J., Letmathe, S., & Ghosh, S. (2020). Fractionally integrated LogGARCH with application to value at risk and expected shortfall (No. 137). Paderborn University, CIE Center for International Economics.
Pantula, S.G. (1986). Modeling the persistence of conditional variances: a comment. In: Econometric Reviews 5, pp. 7997.
Geweke, J. (1986). Comment on: Modelling the persistence of conditional variances. In: Econometric Reviews 5, pp. 5761.
Letmathe, S., Feng, Y., & Uhde, A. (2021). Semiparametric GARCH models with long memory applied to Value at Risk and Expected Shortfall (No. 141). Paderborn University, CIE Center for International Economics.
McNeil, A.J., Frey, R., and Embrechts, P. (2015). Quantitative risk management: concepts, techniques and tools  revised edition. Princeton University Press.
Milhoj, A. (1988). A Multiplicative parametrization of ARCH models. Universitetets Statistiske Institut.
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. In: Econometrica: Journal of the Econometric Society, 347370.
# Example for Walmart Inc. (WMT)
prices = WMT$price.close
# forecasting VaR and ES
results = varcast(prices, model = 'sGARCH', n.out = 250)
ret.out = results$ret.out
n.out = length(ret.out)
VaR97.5 = results$VaR.e
VaR99 = results$VaR.v
ES = results$ES
# plotting VaR at 99% coverage
matplot(1:n.out, cbind(ret.out, VaR99),
type = 'hl',
xlab = 'number of outofsample obs.', ylab = 'losses, VaR and ES',
main = '99% VaR (red) for the WMT return series')
# plotting VaR at 97.5% coverage and corresponding ES
matplot(1:n.out, cbind(ret.out, ES, VaR97.5),
type = 'hll',
xlab = 'number of outofsample obs.', ylab = 'losses, VaR and ES',
main = '97.5% VaR (green) and ES (red) for the WMT return series')
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