| varcast {ufRisk} | R Documentation |
Calculation of one-step ahead forecasts of Value at Risk and Expected Shortfall (parametric and semiparametric)
Description
One-step ahead forecasts of Value at Risk and Expected Shortfall for a selection of short-memory and long-memory parametric as well as semiparametric GARCH-type models are computed.
Usage
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"),
...
)
Arguments
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 out-sample; is set to |
smooth |
a character object; defines the data-driven smoothing approach
for the estimation of the nonparametric scale function; for
|
... |
depending on the choice of |
Details
Let Y_t be a (demeaned) return series. The semiparametric extension of
the GARCH(p,q) model (Bollerslev, 1986) is called a Semi-GARCH 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_{t-i}
+ \sum_{j=1}^q \beta_j \sigma^2_{t-j},
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 non-negative 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
short-memory and long-memory GARCH-type models are selectable for the parametric part of the
model. More specifically, the standard GARCH (Bollerslev, 1986), the
Log-GARCH (Pantula, 1986; Geweke, 1986; Milhoj, 1988), the eGARCH
(Nelson, 1991), the APARCH (Ding et al., 1993), the FIGARCH (Baillie et al., 1996)
and the FI-Log-GARCH (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 zero-mean and unit-variance and while any
distribution that satisfies these conditions is suitable, the standardized
t-distribution 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. 64-72 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 short-memory or long-memory GARCH-type model can be
selected by means of model, which only accepts a single-element character vector
as input. At the time of the release of package version 1.0.0, a standard
GARCH ('sGARCH'), a Log-GARCH ('lGARCH'), an eGARCH ('eGARCH'), an APARCH
('apARCH'), a FIGARCH ('fiGARCH') and a FI-Log-GARCH ('filGARCH') model can be selected,
each with conditional t-distribution. By default, a standard GARCH model is applied.
The orders of the GARCH-type 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 out-sample 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 (in-sample), while the VaR and the ES are obtained for the last
n.out observations (out-sample) based on the estimated model for the
in-sample. Moreover, the data-driven 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 plug-in 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, Log-GARCH and FI-Log-GARCH 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.
Value
This function returns a list with the following elements.
- model
selected model for estimation
- mean
the estimated mean of the in-sample returns
- model.fit
estimated model parameters for the parametric part of the in-sample
- np.est
the estimation results for the nonparametric part of the in-sample model
- ret.in
in-sample return series
- ret.out
out-sample return series
- sig.in
estimated in-sample total volatility
- sig.fc
out-sample forecasts of the total volatility
- scale
the estimated nonparametric scale function values for the in-sample
- scale.fc
the scale function forecast for the out-sample
- VaR.e
out-sample forecasts of the (1-
a.e)100% VaR- VaR.v
out-sample forecasts of the (1-
a.v)100% VaR- ES
out-sample forecasts of the (1-
a.e)100% ES- dfree
estimated degrees of freedom for the standardized returns
- a.v
coverage level for the 99 % VaR
- a.e
coverage level for 97.5 % VaR
- garchOrder
the orders p and q of the implemented GARCH-type model
Author(s)
Sebastian Letmathe (Scientific Employee) (Department of Economics, Paderborn University)
Dominik Schulz (Scientific Employee) (Department of Economics, Paderborn University),
References
Baillie, R. T., Bollerslev, T., & Mikkelsen, H. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. In: Journal of Econometrics, 74.1, pp. 3-30.
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. In: Journal of Econometrics 31.3, pp. 307-327.
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. 83-106.
Feng, Y. (2004). Simultaneously modeling conditional heteroskedasticity and scale change. In: Econometric Theory 20.3, pp. 563-596.
Feng, Y., Beran, J., Letmathe, S., & Ghosh, S. (2020). Fractionally integrated Log-GARCH 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. 79-97.
Geweke, J. (1986). Comment on: Modelling the persistence of conditional variances. In: Econometric Reviews 5, pp. 57-61.
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, 347-370.
Examples
# 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 out-of-sample 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 out-of-sample obs.', ylab = 'losses, VaR and ES',
main = '97.5% VaR (green) and ES (red) for the WMT return series')