Package 'fEGarch'

Estimation of a Broad Family of EGARCH Models

Description

A library of quasi maximum-likelihood estimation (QMLE) methods for fitting various short- and long-memory models from a broad family of exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models. For the purpose of comparison, a FIAPARCH (fractionally integrated asymmetric power ARCH), a FIGARCH (fractionally integrated GARCH), a FITGARCH, a FIGJR-GARCH and their short-memory variants can be implemented as well.

Details

fEGarch is an R package for estimating a broad family of EGARCH models (Feng et al., 2025; Ayensu et al., 2025) including both short- and long-memory as well as a selection of varying transformations for the asymmetry and the magnitude term in such a model, for example in form of the FIMLog-GARCH (Feng et al., 2023). Log-GARCH specifications can be implemented as well as a special case of the broad EGARCH family. The six most common conditional distributions are supported, namely a normal distribution, a $t$- distribution, a generalized error distribution, as well as the skewed variants of these three distributions. Furthermore, as a novelty, an average Laplace (AL) distribution (see for example Feng et al., 2025) and its skewed version are provided as well. The main functions to implement these models are fEGarch_spec in combination with fEGarch. Further details on these models can also be found in the documentation of these two functions. For convenience, further specification functions for particular submodels are available as well: egarch_spec, loggarch_spec, megarch_spec, mloggarch_spec, fiegarch_spec, filoggarch_spec, fimegarch_spec and fimloggarch_spec.

As a popular alternative for the sake of comparison, a FIAPARCH model can be fitted as well using fiaparch. The corresponding documentation page also includes further information on the model. Similarly, figarch can be utilized for fitting FIGARCH models, fitgarch for fitting FITGARCH models and figjrgarch for fitting FIGJR-GARCH models. A general function for the estimation of additional GARCH-type models, including the aforementioned additional models as well as their short-memory variants, is garchm_estim.

In addition, the package provides functionalities in order to simultaneously model the conditional mean (using either autoregressive moving-average (ARMA) models or fractionally integrated ARMA (FARIMA) models) alongside the conditional variance. For this purpose, the function mean_spec can be utilized and its result needs to be passed to fEGarch alongside the result of either fEGarch_spec or one of its wrappers.

Further options include the specification of semiparametric volatility models (see also Ayensu et al., 2025), where a smooth, nonparametric scale function is at first estimated and removed from an observed series, before estimating a parametric model. The scale estimation is currently done through automated local polynomial regression with designated bandwidth selection algorithms under short memory and long memory.

Main Functions

The main functions of the package are:

fEGarch_spec:

setting the model specifications for a model from the broader EGARCH family,

mean_spec:

setting the model specifications for the conditional mean,

fEGarch:

fitting a model from the broad family of EGARCH models given a model specification and an observation series,

garchm_estim:

fitting a GARCH-type model selectable from a standard GARCH, a GJR-GARCH, a TGARCH, an APARCH, a FIGARCH, a FIGJR-GARCH, a FITGARCH and a FIAPARCH,

fEGarch_sim:

simulating from an EGARCH family model,

fiaparch_sim:

simulating from a FIAPARCH model,

figarch_sim:

simulating from a FIGARCH model,

figjrgarch_sim:

simulating from a FIGJR-GARCH model,

fitgarch_sim:

simulating from a FITGARCH model,

aparch_sim:

simulating from an APARCH model,

garch_sim:

simulating from a GARCH model,

gjrgarch_sim:

simulating from a GJR-GARCH model,

tgarch_sim:

simulating from a TGARCH model,

predict,fEGarch_fit-method:

multistep point forecasts of the conditional mean and the conditional standard deviation,

predict_roll,fEGarch_fit-method:

rolling point forecasts of the conditional mean and the conditional standard deviation over a test set.

measure_risk:

value at risk and expected shortfall computation for various model specifications.

find_dist:

fits all eight distributions considered in this package to a supposed iid series and selects the best fitted distribution following either BIC (the default) or AIC.

backtest_suite,fEGarch_risk-method:

runs a selection of functions for backtesting VaR and ES.

Datasets

The package includes a few datasets. Follow the corresponding links to the documentation of the datasets to find additional information including the sources.

UKinflation:

monthly inflation rate of the UK.

SP500:

daily log-returns of the S&P 500 index.

License

The package is distributed under the General Public License v3 ([GPL-3](https://tldrlegal.com/license/gnu-general-public-license-v3-(gpl-3))).

Author(s)

• Dominik Schulz (Department of Economics, Paderborn University),
  Author and Package Creator

• Yuanhua Feng (Department of Economics, Paderborn University),
  Author

• Christian Peitz (Financial Intelligence Unit, German Government),
  Author

• Oliver Kojo Ayensu (Department of Economics, Paderborn University),
  Author

References

• Ayensu, O. K., Feng, Y., & Schulz, D. (2025). Recent Extensions of Exponential GARCH Models: Theory and Application. Forthcoming preprint, Paderborn University.

• Baillie, R., Bollerslev, T., & Mikkelsen, H. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 74(1), 3-30. DOI: 10.1016/S0304-4076(95)01749-6.

• Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3): 307-327. DOI: 10.1016/0304-4076(86)90063-1.

• Bollerslev, T., & Mikkelsen, H. O. (1996). Modeling and pricing long memory in stock market volatility. Journal of Econometrics, 73(1), 151–184. DOI: 10.1016/0304-4076(95)01749-6.

• Conrad, C., & Haag, B. R. (2006). Inequality constraints in the fractionally integrated GARCH model. Journal of Financial Econometrics, 4(3): 413-449. DOI: 10.1093/jjfinec/nbj015.

• Conrad, C., & Karanasos, M. (2006). The impulse response function of the long memory GARCH process. Economics Letters, 90(1): 34-41. DOI: 10.1016/j.econlet.2005.07.001.

• Ding, Z., Granger, C. W. J., & Engle, R. F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1(1): 83-106. DOI: 10.1016/0927-5398(93)90006-D.

• Engle, R. F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, 50(4): 987-1007. DOI: 10.2307/1912773.

• Feng, Y., Beran, J., Ghosh, S., & Letmathe, S. (2020). Fractionally integrated Log-GARCH with application to value at risk and expected shortfall. Working Papers CIE No. 137, Paderborn University, Center for International Economics. URL: http://groups.uni-paderborn.de/wp-wiwi/RePEc/pdf/ciepap/WP137.pdf.

• Feng, Y., Gries, T., Letmathe, S., & Schulz, D. (2022). The smoots Package in R for Semiparametric Modeling of Trend Stationary Time Series. The R Journal, 14(1), 182-195. URL: https://journal.r-project.org/articles/RJ-2022-017/.

• Feng, Y., Gries, T., & Letmathe, S. (2023). FIEGARCH, modulus asymmetric FILog-GARCH and trend-stationary dual long memory time series. Working Papers CIE No. 156, Paderborn University. URL: https://econpapers.repec.org/paper/pdnciepap/156.htm.

• Feng, Y., Peitz, C., & Siddiqui, S. (2025). A few useful members of the EGARCH-family with short- or long-memory in volatility. Unpublished working paper at Paderborn University.

• Geweke, J. (1986). Modeling the persistence of conditional variances: A comment. Econometric Reviews, 5(1), 57-61. DOI: 10.1080/07474938608800088.

• Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On The Relation between The Expected Value and The Volatility of Nominal Excess Return on stocks. Journal of Finance 48(5), 1779-1801. DOI: 10.1111/j.1540-6261.1993.tb05128.x.

• Karanasos, M., Psaradakis, Z., & Sola, M. (2004). On the autocorrelation properties of long-memory GARCH processes. Journal of Time Series Analysis, 25(2): 265-281. DOI: 10.1046/j.0143-9782.2003.00349.x.

• Letmathe, S., Beran, J., & Feng, Y. (2023). An extended exponential SEMIFAR model with application in R. Communications in Statistics - Theory and Methods, 53(22), 7914–7926. DOI: 10.1080/03610926.2023.2276049.

• Milhoj, A. (1987). A Multiplicative Parameterization of ARCH Models. University of Copenhagen, Denmark.

• Missiakoulis, S. (1983). Sargan Densities: Which One?. Journal of Econometrics, 23(2): 223-233. DOI: 10.1016/0304-4076(93)90078-J

• Nelson, D. B. (1991). Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica, 59(2), 347–370. DOI: 10.2307/2938260.

• Nielsen, M. O., & Noel, A. L. (2021). To infinity and beyond: Efficient computation of ARCH($\infty$) models. Journal of Time Series Analysis, 42(3), 338–354. DOI: 10.1111/jtsa.12570.

• Pantula, S. G. (1986). Modeling the persistence of conditional variances: A comment. Econometric Reviews, 5(1), 71-74. DOI: 10.1080/07474938608800089.

• Tse, Y. K. (1987). A Note On Sargan Densities. Journal of Econometrics, 34(3): 349-354. DOI: 10.1016/0304-4076(87)90017-0

• Tse, Y. K. (1998). The conditional heteroskedasticity of the yen-dollar exchange rate. Journal of Applied Econometrics, 13(1): 49-55. DOI: 10.1002/(SICI)1099-1255(199801/02)13:1<49::AID-JAE459>3.0.CO;2-O.

• Zakoian, J.-M. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and Control, 18(5): 931-955. DOI: 10.1016/0165-1889(94)90039-6.

---

Methods for Accessing Model Estimation and Forecasting Output Elements

Description

Accessors to access the elements of the same name in output objects returned by either fEGarch, garchm_estim, predict or predict_roll.

Usage

## S4 method for signature 'fEGarch_fit'
sigt(x)

## S4 method for signature 'fEGarch_fit'
cmeans(x)

## S4 method for signature 'fEGarch_fit'
etat(x)

## S4 method for signature 'fEGarch_fit'
inf_criteria(x)

## S4 method for signature 'fEGarch_fit'
llhood(x)

## S4 method for signature 'fEGarch_fit'
pars(x)

## S4 method for signature 'fEGarch_fit'
se(x)

## S4 method for signature 'fEGarch_fit'
vcov_mat(x)

## S4 method for signature 'fEGarch_forecast'
sigt(x)

## S4 method for signature 'fEGarch_forecast'
cmeans(x)

Arguments

x	an object returned by either fEGarch, garchm_estim, predict (for sigt and cmeans) or predict_roll (for sigt and cmeans).

Details

Convenience methods to access the elements of the same name that can otherwise be accessed via the operator @ within objects that inherit from class "fEGarch_fit", which covers objects returned by either fEGarch, garchm_estim, predict (for sigt and cmeans) or predict_roll (for sigt and cmeans).

As alternatives for sigt, cmeans and etat, see also sigma,fEGarch_fit-method, fitted,fEGarch_fit-method and residuals,fEGarch_fit-method.

Value

The element within the input object of the same name as the method is returned. Depending on the element that can be a numeric vector, an object of class "zoo" or a numeric matrix.

Examples

window.zoo <- get("window.zoo", envir = asNamespace("zoo"))
rt <- window.zoo(SP500, start = "2010-01-01", end = "2012-12-31")
est <- fEGarch(egarch_spec(), rt)

# Access estimated conditional standard deviations using
# the common operator "@" ...
sigt1 <- est@sigt

# ... or use the accessor method "sigt()"
sigt2 <- sigt(est)

zoo::plot.zoo(
 cbind("Approach 1" = sigt1, "Approach 2" = sigt2)
)

# Other methods
cmeans(est)
etat(est)
inf_criteria(est)
llhood(est)
pars(est)
se(est)
vcov_mat(est)

---

APARCH Model Fitting

Description

Fit an APARCH model under the six most common and further conditional distributions to observed data using quasi maximum-likelihood estimation.

Usage

aparch(
  rt,
  orders = c(1, 1),
  cond_dist = c("norm", "std", "ged", "ald", "snorm", "sstd", "sged", "sald"),
  meanspec = mean_spec(),
  Drange = c(0, 1),
  nonparspec = locpol_spec(),
  use_nonpar = FALSE,
  n_test = 0,
  start_pars = NULL,
  LB = NULL,
  UB = NULL,
  control = list(),
  control_nonpar = list(),
  mean_after_nonpar = FALSE,
  parallel = TRUE,
  ncores = max(1, future::availableCores() - 1),
  trunc = "none",
  presample = 50,
  Prange = c(1, 5),
  fix_delta = c(NA, 1, 2),
  skip_vcov = FALSE
)

Arguments

rt	the observed series ordered from past to present; can be a numeric vector, a "zoo" class time series object, or a "ts" class time series object.
orders	a two-element numeric vector containing the two model orders $p$ and $q$ (see Details for more information); currently, only the default orders = c(1, 1) is supported; other specifications of a two-element numeric vector will lead to orders = c(1, 1) being run and a warning message being returned.
cond_dist	the conditional distribution to consider as a character object; the default is a conditional normal distribution "norm"; available are also, however, a $t$-distribution ("std"), a generalized error distribution ("ged"), an average Laplace distribution ("ald"), and their four skewed variants ("snorm", "sstd", "sged", "sald").
meanspec	an object of class "mean_spec"; indicates the specifications for the model in the conditional mean.
Drange	a two-element numeric vector that indicates the boundaries of the interval over which to search for the fractional differencing parameter $D$ in a long-memory ARMA-type model in the conditional mean model part; by default, $D$ being searched for on the interval from 0 to $0.5 - 1\times 10^{-6}$; note that specific settings in the arguments LB and UB overwrite this argument.
nonparspec	an object of class "locpol_spec" returned by locpol_spec; defines the settings of the nonparametric smoothing technique for use_nonpar = TRUE.
use_nonpar	a logical indicating whether or not to implement a semiparametric extension of the volatility model defined through spec; see "Details" for more information.
n_test	a single numerical value indicating, how many observations at the end of rt not to include in the fitting process and to reserve for backtesting.
start_pars	the starting parameters for the numerical optimization routine; should be of the same length as the parameter output vector within the output object (also keeping the same order); for NULL, an internally saved default set of values is used; see "Details" for the order of elements; elements should be set with respect to a series rescaled to have sample variance one.
LB	the lower boundaries of the parameters in the numerical optimization routine; should be of the same length as the parameter output vector within the output object (also keeping the same order); for NULL, an internally saved default set of values is used; see "Details" for the order of elements; elements should be set with respect to a series rescaled to have sample variance one.
UB	the upper boundaries of the parameters in the numerical optimization routine; should be of the same length as the parameter output vector within the output object (also keeping the same order); for NULL, an internally saved default set of values is used; see "Details" for the order of elements; elements should be set with respect to a series rescaled to have sample variance one.
control	a list that is passed to control of the function solnp of the package Rsolnp.
control_nonpar	a list containing changes to the arguments for the hyperparameter estimation algorithm in the nonparametric scale function estimation for use_nonpar = TRUE; see "Details" for more information.
mean_after_nonpar	only for use_nonpar = TRUE; considers the unconditional mean of the parametric model part in the QMLE step in a semiparametric model; by default, a zero-mean model is considered for the parametric part in a semiparametric model.
parallel	only relevant for a (skewed) average Laplace (AL) distribution, i.e. if cond_dist in spec is set to cond_dist = "ald" or cond_dist = "sald"; parallel is a logical value indicating whether or not the slices for the positive integer-valued parameter of the SM distribution should be fitted in parallel for a speed boost.
ncores	only relevant for a (skewed) average Laplace (AL) distribution, i.e. if cond_dist in spec is set to cond_dist = "ald" or cond_dist = "sald", and if simultaneously parallel = TRUE; ncores is a single numeric value indicating the number of cores to use for parallel computations.
trunc	a positive integer indicating the finite truncation length of the infinite-order polynomials of the infinite-order representations of the long-memory model parts; the character "none" is an optional input that specifies that truncation should always be applied back to the first (presample) observation time point, i.e. that maximum length filters should be applied at all times.
presample	the presample length for initialization (for extended EGARCH- / Log-GARCH-type models only relevant for the FARIMA-part, as series in log-transformed conditional variance are initialized by zero).
Prange	a two-element vector that indicates the search boundaries for the parameter $P$ in a (skewed) average Laplace distribution.
fix_delta	let the parameter $\delta$ be either free (fix_delta = NA), fix it to 1 (fix_delta = 1) or to 2 (fix_delta = 2); the latter two are specific submodels of a FIAPARCH.
skip_vcov	a logical indicating whether or not to skip the computation of the variance-covariance matrix of the parameter estimators and therefore also standard error computation.

Details

Let $\left\{r_t\right\}$, with $t \in \mathbb{Z}$ as the time index, be a theoretical time series that follows

$r_t=\mu+\varepsilon_t \text{ with } \varepsilon_t=\sigma_t \eta_t \text{ and } \eta_t \sim \text{IID}(0,1), \text{ where}$

$\sigma_t^{\delta}=\omega+\sum_{i=1}^{p}\phi_i\left(\left|\varepsilon_{t-i}\right|-\gamma_{i}\varepsilon_{t-i}\right)^{\delta} + \sum_{j=1}^{j} \beta_{j}\sigma_{t-j}^{\delta}.$

Here, $\eta_t\sim\text{IID}(0,1)$ means that the innovations $\eta_t$ are independent and identically distributed (iid) with mean zero and variance one, whereas $\sigma_t > 0$ are the conditional standard deviations in $r_t$. Moreover, $\beta_j$, $j=1,2,\dots, q$, $\phi_i$, $i=1,2,\dots, p$, are real-valued coefficients. $p$ and $q$ are the model orders definable through the argument orders, where $p$ is the first element and $q$ is the second element in the argument. In addition, we have $\mu = E\left(r_t\right)$ as a real-valued parameter, and $\gamma \in (-1,1)$. $\omega > 0$ is the intercept. It is assumed that all $\beta_j$ and $\phi_i$ are non-negative.

See also the reference section for sources on the APARCH (Ding et al., 1993) and the FIAPARCH (Tse, 1998) models.

In the current package version, standard errors of parameter estimates are computed from the Hessian at the optimum of the log-likelihood using hessian. To ensure numerical stability and applicability to a huge variety of differently scaled data, parametric models are first fitted to data that is scaled to have sample variance 1. Parameter estimates and other quantities are then either retransformed or recalculated afterwards for the original data.

For a conditional average Laplace distribution, an optimal model for each distribution parameter $P$ from 1 to 5 is estimated (assuming that $P$ is then fixed to the corresponding value). Afterwards, $P$ is then estimated by selecting the estimated model among the five fitted models that has the largest log-likelihood. The five models are, by default, fitted simultaneously using parallel programming techniques (see also the arguments parallel and ncores, which are only relevant for a conditional average Laplace distribution). After the optimal model (including the estimate of $P$ called $\hat{P}$) has been determined, $P=\hat{P}$ is seen as fixed to obtain the standard errors via the Hessian matrix for the estimates of the continuous parameters. A standard error for $\hat{P}$ is therefore not obtained and the ones obtained for the remaining estimates do not account for $\hat{P}$.

An ARMA-APARCH or a FARIMA-APARCH can be fitted by adjusting the argument meanspec correspondingly.

As an alternative, a semiparametric extension of the pure models in the conditional variance can be implemented. If use_nonpar = TRUE, meanspec is omitted and before fitting a model in the conditional volatility following the remaining function arguments, a smooth scale function, i.e. a function representing the unconditional standard deviation over time, is being estimated following the specifications in nonparspec and control_nonpar. This preliminary step stabilizes the input series rt, as long-term changes in the unconditional variance are being estimated and removed before the parametric step using tsmooth. control_nonpar can be adjusted following to make changes to the arguments of tsmooth for short-memory specifications. These arguments specify settings for the automated bandwidth selection algorithms implemented by this function. By default, we use the settings InfR = "Nai", bStart = 0.15, cb = 0.05, and method = "lpr" for tsmooth. locpol_spec passed to nonparspec handles more direct settings of the local polynomial smoother itself. See the documentation for these functions to get a detailed overview of these settings. Assume $\{r_t\}$ to be the observed series, where $t = 1, 2, \dots, n$, then $r_t^{*} = r_t - \bar{r}$, with $\bar{r}$ being the arithmetic mean over the observed $r_t$, is computed and subsequently $y_t = \ln\left[\left(r_t^{*}\right)^2\right]$. The subtraction of $\bar{r}$ is necessary so that $r_t^{*}$ are all different from zero almost surely. Once $y_t$ are available, its trend $m(x_t)$, with $x_t$ as the rescaled time on the interval $[0, 1]$, is being estimated using tsmooth and denoted here by $\hat{m}(x_t)$. Then from $\hat{\xi}_t = y_t - \hat{m}(x_t)$ obtain $\hat{C} = -\ln\left\{\sum_{t=1}^{n}\exp\left(\hat{\xi}_t\right)\right\}$, and obtain the estimated scale function as $\hat{s}(x_t)=\exp\left[\left(\hat{\mu}(x_t) - \hat{C}\right) / 2\right]$. The stabilized / standardized version of the series $\left\{r_t\right\}$ is then $\tilde{r}_t = r_t^{*} / \hat{s}(x_t)$, to which a purely parametric volatility model following the remaining function arguments is then fitted. The estimated volatility at a given time point is then the product of the estimate of the corresponding scale function value and of the estimated conditional standard deviation (following the parametric model part) for that same time point. See for example Feng et al. (2022) or Letmathe et al. (2023) for more information on the semiparametric extension of volatility models.

The order for manual settings of start_pars, LB and UB is crucial. The correct order is: $\mu$, $\text{ar}_1,\dots,\text{ar}_{p^{*}}$, $\text{ma}_1,\dots,\text{ma}_{q^{*}}$,$D$,$\omega$, $\phi_1,\dots,\phi_p$, $\beta_1, \dots, \beta_{q}$, $\gamma_1,\dots,\gamma_p$, $\delta$, shape parameter, skewness parameter. Depending on the exact model specification, parameters irrelevant for the specification at hand should be dropped in start_pars, LB and UB.

Value

An object of S4-class "fEGarch_fit_aparch" is returned. It contains the following elements.

pars:

a named numeric vector with the parameter estimates.

se:

a named numeric vector with the obtained standard errors in accordance with the parameter estimates.

vcov_mat:

the variance-covariance matrix of the parameter estimates with named columns and rows.

rt:

the input object rt (or at least the training data, if n_test is greater than zero); if rt was a "zoo" or "ts" object, the formatting is kept.

cmeans:

the estimated conditional means; if rt was a "zoo" or "ts" object, the formatting is also applied to cmeans.

sigt:

the estimated conditional standard deviations (or for use_nonpar = TRUE the estimated total volatilities, i.e. scale function value times conditional standard deviation); if rt was a "zoo" or "ts" object, the formatting is also applied to sigt.

etat:

the obtained residuals; if rt was a "zoo" or "ts" object, the formatting is also applied to etat.

orders:

a two-element numeric vector stating the considered model orders.

cond_dist:

a character value stating the conditional distribution considered in the model fitting.

long_memo:

a logical value stating whether or not long memory was considered in the model fitting.

llhood:

the log-likelihood value obtained at the optimal parameter combination.

inf_criteria:

a named two-element numeric vector with the corresponding AIC (first element) and BIC (second element) of the fitted parametric model part; for purely parametric models, these criteria are valid for the entire model; for semiparametric models, they are only valid for the parametric step and are not valid for the entire model.

meanspec:

the settings for the model in the conditional mean; is an object of class "mean_spec" that is identical to the object passed to the input argument meanspec.

test_obs:

the observations at the end up the input rt reserved for testing following n_test.

scale_fun:

the estimated scale function values, if use_nonpar = TRUE, otherwise NULL; formatting of rt is reused.

nonpar_model:

the estimation object returned by tsmooth for use_nonpar = TRUE.

trunc:

the input argument trunc.

References

• Ding, Z., Granger, C. W. J., & Engle, R. F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1(1): 83-106. DOI: 10.1016/0927-5398(93)90006-D.

• Feng, Y., Gries, T., Letmathe, S., & Schulz, D. (2022). The smoots Package in R for Semiparametric Modeling of Trend Stationary Time Series. The R Journal, 14(1), 182-195. URL: https://journal.r-project.org/articles/RJ-2022-017/.

• Letmathe, S., Beran, J., & Feng, Y. (2023). An extended exponential SEMIFAR model with application in R. Communications in Statistics - Theory and Methods, 53(22), 7914–7926. DOI: 10.1080/03610926.2023.2276049.

• Tse, Y. K. (1998). The conditional heteroskedasticity of the yen-dollar exchange rate. Journal of Applied Econometrics, 13(1): 49-55. DOI: 10.1002/(SICI)1099-1255(199801/02)13:1<49::AID-JAE459>3.0.CO;2-O.

Examples

window.zoo <- get("window.zoo", envir = asNamespace("zoo"))
rt <- window.zoo(SP500, end = "2002-12-31")
model <- aparch(rt)
model

---

Simulate From APARCH Models

Description

A streamlined simulation function to simulate from asymmetric power autoregressive conditional heteroskedasticity (APARCH) models.

Usage

aparch_sim(
  pars = list(mu = 0, ar = numeric(0), ma = numeric(0), D = 0, omega = 4e-04, phi = 0.05,
    beta = 0.8, gamma = 0.1, delta = 2, df = 10, shape = 2, P = 3, skew = 1),
  cond_dist = c("norm", "std", "ged", "ald", "snorm", "sstd", "sged", "sald"),
  n = 1000,
  nstart = 5000,
  trunc = "none"
)

Arguments

pars	a named list with the parameter specifications; the user can provide a named list with only the settings they would like to adjust relative to the default settings.
cond_dist	a one-element character vector specifying the conditional distribution to consider.
n	the number of observations to return.
nstart	the number of burn-in observations to simulate before the final n values to keep; the first nstart values are not returned; if a dual model, i.e. with model in the conditional mean and in the conditional variance, is considered, two times nstart is considered in the first simulation step in the conditional variance, so that n + nstart values can be fed into the second simulation step for the conditional mean.
trunc	a truncation for the finite-order coefficient series in long-memory models; can either be the character "none" for truncation back to the very first observation at each time point, or to any positive integer for setting the corresponding truncation length of the infinite-order representation polynomial.

Details

See the documentation on aparch for information on the APARCH model. This function provides an easy way to simulate from these models.

Value

A list with four elements is returned: rt are the simulated observations, etat are the underlying innovations, sigt are the correspondingly simulated conditional standard deviations, and cmeans are the simulated conditional means. These four elements are formatted as "ts" class time series objects.

Examples

sim <- aparch_sim(n = 1000)
mat <- do.call(cbind, sim)
plot(mat, main = "")

---

Plot Method for Fitting Step Results in the Style of ggplot2

Description

This is method for producing various plots of the decomposition results returned by this package.

Usage

## S4 method for signature 'fEGarch_fit'
autoplot(object, which = NULL, ...)

Arguments

object	an object returned by the fitting functions of this package, for example by fEGarch.
which	various plots can be selected either via a keyword or a number; enter "returns" or 1 to show a plot of the input training series; enter "means" or 2 to show the fitted conditional means; enter "stand_deviations" or 3 to show the fitted conditional standard deviations; use "residuals" or 4 to show the standardized residuals following the fitted model; the default is which = NULL which then lets you select a plot interactively in the R console.
...	no purpose and only implemented for compatibility.

Details

Create predefined standard plots of the estimation objects returned by the fEGarch package. Plots are created in the ggplot2 plot style. The type of plot can be chosen either interactively from the console, or the argument which can be used to directly select the kind of plot to create (see also the description of the argument which) within the function call.

Value

A ggplot2-graphic object is returned, i.e. an object of classes "gg" and "ggplot".

Author(s)

• Dominik Schulz (Research Assistant) (Department of Economics, Paderborn University),
  Author and Package Creator

Examples

window.zoo <- get("window.zoo", envir = asNamespace("zoo"))
rt <- window.zoo(SP500, end = "2002-12-31")
# Pure conditional volatility model
spec <- fEGarch_spec()
model <- fEGarch(spec, rt)
autoplot(model, which = 3)

---

Plotting of Risk Measure Results (ggplot2)

Description

Plot risk measure results returned by measure_risk as a points-over-threshold plot in style of ggplot2.

Usage

## S4 method for signature 'fEGarch_risk'
autoplot(object, which = NULL, ...)

Arguments

object	an object returned by measure_risk.
which	one of the levels of VaR and ES saved in object, usually either 0.975 or 0.99 by default.
...	without use.

Value

Returns a ggplot2 plot object.

Examples

window.zoo <- get("window.zoo", envir = asNamespace("zoo"))
rt <- window.zoo(SP500, end = "2003-12-31")

egarch_spec() %>%
 fEGarch(rt = rt, n_test = 250) %>%
 predict_roll() %>%
 measure_risk() %>%
 autoplot(which = 0.99)

---

Log-Return Calculation From Closing Prices

Description

Makes log-returns available from an input closing price series.

Usage

close_to_lreturn(close)

Arguments

close

a closing price series as a numeric vector or some time series object like "ts" or "zoo" ordered chronologically.

Details

Let $P_t$, $t=1^,\dots,n$, be an observed closing price series. The function returns

$r_t = \ln{P_t}-\ln{P_{t-1}}$

, t = 2,...,n.

Value

Returns the log-return series following the input close. The output object has one observation less than close, but keeps potential time series formatting.

Examples

# Assume SP500 + 100 was a closing price series,
# which it is not
close <- SP500 + 100
close_to_lreturn(close)

---

MLE for Distribution Fitting

Description

Given a vector of values assumed to stem from independent and identically distributed (iid) random variables, fit a selection of distributions, from the normal distribution, the $t$-distribution, the generalized error distribution (GED), the average Laplace distribution (ALD), and their skewed variants, to the data using maximum-likelihood estimation (MLE).

Usage

distr_est(
  x,
  dist = c("norm", "std", "ged", "ald", "snorm", "sstd", "sged", "sald"),
  fix_mean = NULL,
  fix_sdev = NULL,
  Prange = c(1, 5),
  skip_vcov = FALSE
)

norm_est(x, fix_mean = NULL, fix_sdev = NULL, skip_vcov = FALSE)

std_est(x, fix_mean = NULL, fix_sdev = NULL, skip_vcov = FALSE)

ged_est(x, fix_mean = NULL, fix_sdev = NULL, skip_vcov = FALSE)

ald_est(
  x,
  fix_mean = NULL,
  fix_sdev = NULL,
  Prange = c(1, 5),
  skip_vcov = FALSE
)

snorm_est(x, fix_mean = NULL, fix_sdev = NULL, skip_vcov = FALSE)

sstd_est(x, fix_mean = NULL, fix_sdev = NULL, skip_vcov = FALSE)

sged_est(x, fix_mean = NULL, fix_sdev = NULL, skip_vcov = FALSE)

sald_est(
  x,
  fix_mean = NULL,
  fix_sdev = NULL,
  Prange = c(1, 5),
  skip_vcov = FALSE
)

Arguments

x	a numeric vector with the data.
dist	a character value that specifies the distribution to consider; available are a normal distribution ("norm"), a $t$-distribution ("std"), a GED ("ged"), an ALD ("ald"), and their skewed variants ("snorm", "sstd", "sged", "sald").
fix_mean	optional; for the default NULL, a location parameter representing the (unconditional) mean of the distribution is also being estimated; for any numerical value, however, the mean will be fixed to the corresponding value and therefore excluded from the estimation itself.
fix_sdev	optional; for the default NULL, a scale parameter representing the (unconditional) standard deviation of the distribution is also being estimated; for any numerical value, however, the standard deviation will be fixed to the corresponding value and therefore excluded from the estimation itself.
Prange	a two-element numeric vector, giving the boundaries of the search space for the shape parameter $P$ in an ALD or its skewed variant.
skip_vcov	a logical indicating whether or not the computation of the variance-covariance matrix should be skipped.

Details

Let $x$ be an individual observation. Let $\mu$ a real-valued location parameter, representing the unconditional mean of the distribution, and $\sigma$ a real-valued scale parameter, representing the unconditional standard deviation. Generally, let $\theta$ be a vector with all parameters of the underlying distribution. The likelihood of $x$ is given through

$L_x^{\text{norm}}(\theta)=\frac{\sigma^{-1}}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)$

for a normal distribution,

$L_x^{\text{std}}(\theta)=\frac{\sigma^{-1}\Gamma\left(\frac{\nu + 1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)\sqrt{\pi(\nu-2)}}\left[1+\frac{1}{\nu - 2}\left(\frac{x-\mu}{\sigma}\right)^2\right]^{-\frac{\nu + 1}{2}}$

for a $t$-distribution with $\nu$ as the degrees of freedom and $\Gamma$ as the gamma function,

$L_x^{\text{ged}}(\theta)=\frac{\sigma^{-1}\beta}{2}\sqrt{\frac{C_{\Gamma,3}}{C_{\Gamma,1}^3}} \exp\left\{-\left|\frac{x-\mu}{\sigma}\right|^{\beta}\left(\frac{C_{\Gamma,3}}{C_{\Gamma,1}}\right)^{\frac{\beta}{2}}\right\}$

for a GED with $\beta$ as its real-valued shape and with $C_{\Gamma,i}=\Gamma\left(\frac{i}{\beta}\right)$, $i\in\left\{1,3\right\}$, and in

$L_x^{\text{ald}}(\theta)=\frac{\sigma^{-1}sB}{2}\exp\left(-s\left|\frac{x-\mu}{\sigma}\right|\right)\sum_{j=0}^{P}c_j\left(s\left|\frac{x-\mu}{\sigma}\right|\right)^j$

for an ALD with $P$ as its discrete shape, where $s = \sqrt{2(P+1)}$,

$B=2^{-2P} {{2P}\choose{P}}, \hspace{4mm} P \geq 0,$

and

$c_{j}=\frac{2(P-j + 1)}{j(2P-j+1)}c_{j-1}, \hspace{4mm} j =2,3,\dots,P,$

with $c_0 = c_1 = 1$. The individual-observation likelihoods for the skewed variants are derived analogously from the idea by Fernandez and Steel (1998). The log-likelihoods to maximize over are then just the sum of the log-transformed likelihoods for each observation.

distr_est is a general purpose distribution fitting function, where the distribution can be selected through the argument dist. norm_est, std_est, ged_est, ald_est, snorm_est, sstd_est, sged_est, and sald_est are wrappers around distr_est in order to directly provide fitting functions for the different distributions available in this package.

Value

Returns a list with the following elements.

References

• Fernandez, C., & Steel, M. F. J. (1998). Bayesian Modeling of Fat Tails and Skewness. Journal of the American Statistical Association, 93(441), 359–371. DOI: 10.1080/01621459.1998.10474117.

Examples

# Draw obs. from GED and implement standard deviation 1.2
# and mean 3.1
x <- rged_s(4000, shape = 1.5) * 1.2 + 3.1
# Fit GED
ged_est(x)
# Fit GED differently using distr_est()
distr_est(x, dist = "ged")
# Fit GED while fixing mean and standard deviation
ged_est(x, fix_mean = 3.1, fix_sdev = 1.2)
# Fit another distribution
sstd_est(x)

---

Subspecification of EGARCH Family Models

Description

Two common subspecifications of the broad EGARCH family, namely for a EGARCH-type and a Log-GARCH-type model.

Usage

egarch_type_spec(
  orders = c(1, 1),
  long_memo = TRUE,
  cond_dist = c("norm", "std", "ged", "ald", "snorm", "sstd", "sged", "sald"),
  powers = c(1, 1),
  modulus = c(FALSE, FALSE)
)

loggarch_type_spec(
  orders = c(1, 1),
  long_memo = TRUE,
  cond_dist = c("norm", "std", "ged", "ald", "snorm", "sstd", "sged", "sald")
)

Arguments

orders	a two-element numeric vector with the model orders; the first element is the order $p$ for the term based on $\ln\left(\sigma_t^2\right)$, i.e. the log-transformed conditional variance, while the second element is the order $q$ for the innovation-based term (see Details below for more information).
long_memo	a logical value that indicates whether the long-memory version of the model should be considered or not.
cond_dist	a character value stating the underlying conditional distribution to consider; available are a normal distribution ("norm"), a $t$-distribution ("std"), a generalized error distribution ("ged"), an average Laplace distribution ("ald") and the skewed versions of them ("snorm", "sstd", "sged", "sald").
powers	a two-element numeric vector that states the exponents in the power-transformations of the asymmetry and the magnitude terms in that order (see Details for more information).
modulus	a two-element logical vector indicating if the innovations in the asymmetry and the magnitude terms (in that order) should use a modulus transformation (see Details for more information).

Details

These are wrappers for fEGarch_spec. egarch_type_spec() is when setting model_type in fEGarch_spec to eGARCH. loggarch_type_spec() is a shortcut when setting model_type = "loggarch". See fEGarch_spec for further details.

Value

An object of class "egarch_type_spec" or "loggarch_type_spec" is returned.

---

Fitting Function for Models of the Broader EGARCH Family

Description

Use quasi-maximum-likelihood estimation to fit a model from the broader EGARCH family to some observed time series.

Usage

fEGarch(
  spec = egarch_spec(),
  rt,
  drange = c(0, 1),
  meanspec = mean_spec(),
  Drange = c(0, 1),
  nonparspec = locpol_spec(),
  use_nonpar = FALSE,
  n_test = 0,
  start_pars = NULL,
  LB = NULL,
  UB = NULL,
  control = list(),
  control_nonpar = list(),
  mean_after_nonpar = FALSE,
  parallel = TRUE,
  ncores = max(1, future::availableCores() - 1),
  trunc = "none",
  presample = 50,
  Prange = c(1, 5),
  skip_vcov = FALSE
)

Arguments

spec	an S4 object of class "egarch_type_spec" or "loggarch_type_spec" as returned by the various spec functions of this package, for example fEGarch_spec or its wrappers like megarch_spec or mloggarch_spec, among others.
rt	the input time series to fit the model to ordered from past to present; can also be a "zoo" class object or a "ts" class object.
drange	a two-element numeric vector that indicates the boundaries of the interval over which to search for the fractional differencing parameter $d$ in a long-memory GARCH-type model in the conditional volatility model part; by default, $d$ being searched for on the interval from 0 to 1; note that specific settings in the arguments LB and UB overwrite this argument.
meanspec	an object of class "mean_spec"; indicates the specifications for the model in the conditional mean.
Drange	a two-element numeric vector that indicates the boundaries of the interval over which to search for the fractional differencing parameter $d$ in a long-memory ARMA-type model in the conditional mean model part; by default, $D$ being searched for on the interval from 0 to 1; note that specific settings in the arguments LB and UB overwrite this argument.
nonparspec	an object of class "locpol_spec" returned by locpol_spec; defines the settings of the nonparametric smoothing technique for use_nonpar = TRUE.
use_nonpar	a logical indicating whether or not to implement a semiparametric extension of the volatility model defined through spec; see "Details" for more information.
n_test	a single numerical value indicating, how many observations at the end of rt not to include in the fitting process and to reserve for backtesting.
start_pars	a vector with starting parameters for the optimization; must be of the same length as the output vector of parameters; the default NULL uses internally saved default sets of starting values; see "Details" for the order of elements.
LB	a vector with lower boundaries for parameters; must be of the same length as the output vector of parameters; the default NULL uses internally saved default sets of lower boundary values; see "Details" for the order of elements.
UB	a vector with upper boundaries for parameters; must be of the same length as the output vector of parameters; the default NULL uses internally saved default sets of upper boundary values; see "Details" for the order of elements.
control	a list that is passed to control of the function solnp of the package Rsolnp.
control_nonpar	a list containing changes to the arguments for the hyperparameter estimation algorithm in the nonparametric scale function estimation for use_nonpar = TRUE; see "Details" for more information.
mean_after_nonpar	only for use_nonpar = TRUE; considers the unconditional mean of the parametric model part in the QMLE step in a semiparametric model; by default, a zero-mean model is considered for the parametric part in a semiparametric model.
parallel	only relevant for a (skewed) average Laplace (AL) distribution, i.e. if cond_dist in spec is set to cond_dist = "ald" or cond_dist = "sald"; parallel is a logical value indicating whether or not the slices for the positive integer-valued parameter of the SM distribution should be fitted in parallel for a speed boost.
ncores	only relevant for a (skewed) average Laplace (AL) distribution, i.e. if cond_dist in spec is set to cond_dist = "ald" or cond_dist = "sald", and if simultaneously parallel = TRUE; ncores is a single numeric value indicating the number of cores to use for parallel computations.
trunc	a positive integer indicating the finite truncation length of the infinite-order polynomials of the infinite-order representations of the long-memory model parts; the character "none" is an optional input that specifies that truncation should always be applied back to the first (presample) observation time point, i.e. that maximum length filters should be applied at all times.
presample	the presample length for initialization (for extended EGARCH- / Log-GARCH-type models only relevant for the FARIMA-part, as series in log-transformed conditional variance are initialized by zero).
Prange	a two-element vector that indicates the search boundaries for the parameter $P$ in a (skewed) average Laplace distribution.
skip_vcov	a logical indicating whether or not to skip the computation of the variance-covariance matrix of the parameter estimators and therefore also standard error computation.

Details

For details on the models in the conditional variance, see fEGarch_spec. For details on the models in the conditional mean, see mean_spec. The combined model defined through mean_spec and fEGarch_spec is the specified model. It can be thought of as the model described in mean_spec with $\left\{r_t\right\}$ therein being governed by a model from the EGARCH family (see for example Feng et al., 2025) as described in fEGarch_spec, however with mean of $\left\{r_t\right\}$ fixed to zero.

The specified model is then fitted using quasi maximum likelihood estimation, where pre-sample values of $g\left(\eta_{t-1}\right)$ are filled in through its theoretical expectation of zero, which is analogous to not setting pre-sample values in the long-memory case. In addition, in short-memory models, pre-sample values of $ln(\sigma_t^2)$ are roughly approximated through $\ln\left(\hat{\sigma}_{t}^{2}\right)$, where $\hat{\sigma}_{t}^{2}$ is the sample variance of the observations.

See the references section for sources on the EGARCH (Nelson, 1991), FIEGARCH (Bollerslev and Mikkelsen, 1996), Log-GARCH (Geweke, 1986; Pantula, 1986; Milhoj, 1987) and FILog-GARCH (Feng et al., 2020) models. For information on the FIMLog-GARCH, see Feng et al. (2023).

In the current package version, standard errors of parameter estimates are computed from the Hessian at the optimum of the log-likelihood using hessian. To ensure numerical stability and applicability to a huge variety of differently scaled data, parametric models are first fitted to data that is scaled to have sample variance 1. Parameter estimates and other quantities are then either retransformed or recalculated afterwards for the original data.

For a conditional average Laplace distribution, an optimal model for each distribution parameter $P$ from 1 to 5 is estimated (assuming that $P$ is then fixed to the corresponding value). Afterwards, $P$ is then estimated by selecting the estimated model among the five fitted models that has the largest log-likelihood. The five models are, by default, fitted simultaneously using parallel programming techniques (see also the arguments parallel and ncores, which are only relevant for a conditional average Laplace distribution). After the optimal model (including the estimate of $P$ called $\hat{P}$) has been determined, $P=\hat{P}$ is seen as fixed to obtain the standard errors via the Hessian matrix for the estimates of the continuous parameters. A standard error for $\hat{P}$ is therefore not obtained and the ones obtained for the remaining estimates do not account for $\hat{P}$.

As an alternative, a semiparametric extension of the pure models in the conditional variance can be implemented. If use_nonpar = TRUE, meanspec is omitted and before fitting a zero-mean model in the conditional volatility following spec, a smooth scale function, i.e. a function representing the unconditional standard deviation over time, is being estimated following the specifications in nonparspec and control_nonpar. This preliminary step stabilizes the input series rt, as long-term changes in the unconditional variance are being estimated and removed before the parametric step using either tsmooth or tsmoothlm depending on whether spec specifies a model with short memory or with long memory. control_nonpar can be adjusted following the arguments of tsmooth for short-memory specifications of spec, on the other hand changes to arguments of tsmoothlm can be passed to it for long-memory specifications. These arguments specify settings for the automated bandwidth selection algorithms implemented by these two functions. By default, we use the settings Mcf = "NP", InfR = "Opt", bStart = 0.15, bvc = "Y", cb = 0.05, and method = "lpr" within the call to tsmooth, as well as pmin = 0, pmax = 1, qmin = 0, qmax = 1, InfR = "Opt", bStart = 0.15, cb = 0.05, and method = "lpr" for tsmoothlm. locpol_spec passed to nonparspec handles more direct settings of the local polynomial smoother itself. See the documentation for these functions to get a detailed overview of these settings. Assume $\{r_t\}$ to be the observed series, where $t = 1, 2, \dots, n$, then $r_t^{*} = r_t - \bar{r}$, with $\bar{r}$ being the arithmetic mean over the observed $r_t$, is computed and subsequently $y_t = \ln\left[\left(r_t^{*}\right)^2\right]$. The subtraction of $\bar{r}$ is necessary so that $r_t^{*}$ are all different from zero almost surely. Once $y_t$ are available, its trend $m(x_t)$, with $x_t$ as the rescaled time on the interval $[0, 1]$, is being estimated using either tsmooth or tsmoothlm and denoted here by $\hat{m}(x_t)$. Then from $\hat{\xi}_t = y_t - \hat{m}(x_t)$ obtain $\hat{C} = -\ln\left\{\sum_{t=1}^{n}\exp\left(\hat{\xi}_t\right)\right\}$, and obtain the estimated scale function as $\hat{s}(x_t)=\exp\left[\left(\hat{\mu}(x_t) - \hat{C}\right) / 2\right]$. The stabilized / standardized version of the series $\left\{r_t\right\}$ is then $\tilde{r}_t = r_t^{*} / \hat{s}(x_t)$, to which a purely parametric volatility model following spec is then fitted. The estimated volatility at a given time point is then the product of the estimate of the corresponding scale function value and of the estimated conditional standard deviation (following the parametric model part) for that same time point. See for example Feng et al. (2022) or Letmathe et al. (2023) for more information on the semiparametric extension of volatility models. Moreover, if bwidth in the object passed to nonparspec is not at its default NULL but instead a numeric value between 0 and 0.5, the automated bandwidth selection is skipped and the provided bandwidth in bwidth is used.

The order for manual settings of start_pars, LB and UB is crucial. The correct order is: $\mu$, $\text{ar}_1,\dots,\text{ar}_{p^{*}}$, $\text{ma}_1,\dots,\text{ma}_{q^{*}}$,$D$,$\omega_{\sigma}$, $\phi_1,\dots,\phi_p$, $\psi_1, \dots, \psi_{q-1}$, $\kappa$, $\gamma$, $d$, $\text{shape parameter}$, $\text{skewness parameter}$ for Type I models (see fEGarch_spec). For Type II models, we have $\mu$, $\text{ar}_1,\dots,\text{ar}_{p^{*}}$, $\text{ma}_1,\dots,\text{ma}_{q^{*}}$,$D$,$\omega_{\sigma}$, $\phi_1,\dots,\phi_p$, $\psi_1, \dots, \psi_{q}$, $d$, shape parameter, skewness parameter. Depending on the exact model specification, parameters irrelevant for the specification at hand should be dropped in start_pars, LB and UB.

Value

An object of S4-class "fEGarch_fit_egarch" "fEGarch_fit_loggarch" is returned depending on the selected model type in the model specification. It contains the following elements.

pars:

a named numeric vector with the parameter estimates.

se:

a named numeric vector with the obtained standard errors in accordance with the parameter estimates.

vcov_mat:

the variance-covariance matrix of the parameter estimates with named columns and rows.

rt:

the input object rt (or at least the training data, if n_test is greater than zero); if rt was a "zoo" or "ts" object, the formatting is kept.

sigt:

the estimated conditional standard deviations (or for use_nonpar = TRUE the estimated total volatilities, i.e. scale function value times conditional standard deviation); if rt was a "zoo" or "ts" object, the formatting is also applied to sigt.

cmeans:

the estimated conditional means; if rt was a "zoo" or "ts" object, the formatting is also applied to cmeans.

etat:

the obtained residuals; if rt was a "zoo" or "ts" object, the formatting is also applied to etat.

orders:

a two-element numeric vector stating the considered model orders.

cond_dist:

a character value stating the conditional distribution considered in the model fitting.

long_memo:

a logical value stating whether or not long memory was considered in the model fitting.

llhood:

the log-likelihood value obtained at the optimal parameter combination.

inf_criteria:

a named two-element numeric vector with the corresponding AIC (first element) and BIC (second element) of the fitted parametric model part; for purely parametric models, these criteria are valid for the entire model; for semiparametric models, they are only valid for the parametric step and are not valid for the entire model.

powers:

a two-element numeric vector stating the powers considered for the asymmetry term (first element) and the magnitude term (second element); only exists, if a type I model was fitted.

modulus:

a two-element logical vector stating the modulus transformations (TRUE, otherwise FALSE) considered for the asymmetry term (first element) and the magnitude term (second element); only exists, if a type I model was fitted.

meanspec:

the settings for the model in the conditional mean; is an object of class "mean_spec" that is identical to the object passed to the input argument meanspec.

test_obs:

the observations at the end up the input rt reserved for testing following n_test.

scale_fun:

the estimated scale function values, if use_nonpar = TRUE, otherwise NULL; formatting of rt is reused.

nonpar_model:

the estimation object returned by either tsmooth or tsmoothlm for use_nonpar = TRUE.

trunc:

the input argument trunc.

References

• Bollerslev, T., & Mikkelsen, H. O. (1996). Modeling and pricing long memory in stock market volatility. Journal of Econometrics, 73(1), 151–184. DOI: 10.1016/0304-4076(95)01749-6.

• Feng, Y., Beran, J., Ghosh, S., & Letmathe, S. (2020). Fractionally integrated Log-GARCH with application to value at risk and expected shortfall. Working Papers CIE No. 137, Paderborn University, Center for International Economics. URL: http://groups.uni-paderborn.de/wp-wiwi/RePEc/pdf/ciepap/WP137.pdf.

• Feng, Y., Gries, T., Letmathe, S., & Schulz, D. (2022). The smoots Package in R for Semiparametric Modeling of Trend Stationary Time Series. The R Journal, 14(1), 182-195. URL: https://journal.r-project.org/articles/RJ-2022-017/.

• Feng, Y., Gries, T., & Letmathe, S. (2023). FIEGARCH, modulus asymmetric FILog-GARCH and trend-stationary dual long memory time series. Working Papers CIE No. 156, Paderborn University. URL: https://econpapers.repec.org/paper/pdnciepap/156.htm.

• Feng, Y., Peitz, C., & Siddiqui, S. (2025). A few useful members of the EGARCH-family with short- or long-memory in volatility. Unpublished working paper at Paderborn University.

• Geweke, J. (1986). Modeling the persistence of conditional variances: A comment. Econometric Reviews, 5(1), 57-61. DOI: 10.1080/07474938608800088.

• Letmathe, S., Beran, J., & Feng, Y. (2023). An extended exponential SEMIFAR model with application in R. Communications in Statistics - Theory and Methods, 53(22), 7914–7926. DOI: 10.1080/03610926.2023.2276049.

• Milhoj, A. (1987). A Multiplicative Parameterization of ARCH Models. University of Copenhagen, Denmark.

• Nelson, D. B. (1991). Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica, 59(2), 347–370. DOI: 10.2307/2938260.

• Pantula, S. G. (1986). Modeling the persistence of conditional variances: A comment. Econometric Reviews, 5(1), 71-74. DOI: 10.1080/07474938608800089.

Examples

window.zoo <- get("window.zoo", envir = asNamespace("zoo"))
rt <- window.zoo(SP500, end = "2002-12-31")
# Pure conditional volatility model
spec <- fEGarch_spec()
model <- fEGarch(spec, rt)
model
# Simultaneously model conditional mean
spec <- egarch_spec()
model2 <- suppressWarnings(
 fEGarch(spec, rt, meanspec = mean_spec(orders = c(1, 1)))
)
model2

---

Simulate From Models of the Broader EGARCH Family

Description

A streamlined simulation function to simulate from models that are part of the broader EGARCH family specifiable through fEGarch_spec.

Usage

fEGarch_sim(
  spec = egarch_spec(),
  pars = list(mu = 0, ar = numeric(0), ma = numeric(0), D = 0, omega_sig = -9, phi = 0.8,
    psi = numeric(0), kappa = -0.2, gamma = 0.3, d = 0, df = 10, shape = 2, P = 3, skew =
    1),
  n = 1000,
  nstart = 5000,
  trunc = "none"
)

Arguments

spec	an object of class "egarch_type_spec" or "loggarch_type_spec" returned by fEGarch_spec or related wrapper functions; note that the model orders in the conditional mean and in the conditional variance as well as the long-memory settings are not controlled through the element orders and long_memo of spec but solely through the parameter settings in pars.
pars	a named list with the parameter specifications; the user can provide a named list with only the settings they would like to adjust relative to the default settings.
n	the number of observations to return.
nstart	the number of burn-in observations to simulate before the final n values to keep; the first nstart values are not returned; if a dual model, i.e. with model in the conditional mean and in the conditional variance, is considered, two times nstart is considered in the first simulation step in the conditional variance, so that n + nstart values can be fed into the second simulation step for the conditional mean.
trunc	a truncation for the finite-order coefficient series in long-memory models; can either be the character "none" for truncation back to the very first observation at each time point, or to any positive integer for setting the corresponding truncation length of the infinite-order representation polynomial.

Details

See the documentation on fEGarch_spec for information on the models of the broader EGARCH family. This function provides an easy way to simulate from these models.

Value

A list with four elements is returned: rt are the simulated observations, etat are the underlying innovations, sigt are the correspondingly simulated conditional standard deviations, and cmeans are the simulated conditional means. These four elements are formatted as "ts" class time series objects.

Examples

spec <- megarch_spec()
sim <- fEGarch_sim(spec = spec)
mat <- do.call(cbind, sim)
plot(mat, main = "")

---

General EGARCH Family Model Specification

Description

Create an object with specifications for a model from the broader EGARCH family.

Usage

fEGarch_spec(
  model_type = c("egarch", "loggarch"),
  orders = c(1, 1),
  long_memo = FALSE,
  cond_dist = c("norm", "std", "ged", "ald", "snorm", "sstd", "sged", "sald"),
  powers = c(1, 1),
  modulus = c(FALSE, FALSE)
)

Arguments

model_type	a character value (either "egarch" or "loggarch") that indicates the type of model to implement (see Details for more information).
orders	a two-element numeric vector with the model orders; the first element is the order $p$ for the term based on $\ln\left(\sigma_t^2\right)$, i.e. the log-transformed conditional variance, while the second element is the order $q$ for the innovation-based term (see Details below for more information).
long_memo	a logical value that indicates whether the long-memory version of the model should be considered or not.
cond_dist	a character value stating the underlying conditional distribution to consider; available are a normal distribution ("norm"), a $t$-distribution ("std"), a generalized error distribution ("ged"), an average Laplace distribution ("ald") and the skewed versions of them ("snorm", "sstd", "sged", "sald").
powers	a two-element numeric vector that states the exponents in the power-transformations of the asymmetry and the magnitude terms in that order (see Details for more information).
modulus	a two-element logical vector indicating if the innovations in the asymmetry and the magnitude terms (in that order) should use a modulus transformation (see Details for more information).

Details

Let $\left\{r_t\right\}$, with $t \in \mathbb{Z}$ as the time index, be a theoretical time series that follows

$r_t=\mu+\sigma_t \eta_t \text{ with } \eta_t \sim \text{IID}(0,1), \text{ where}$

$\ln\left(\sigma_t^2\right)=\omega_{\sigma}+\theta(B)g(\eta_{t-1}).$

Here, $\eta_t\sim\text{IID}(0,1)$ means that the innovations $\eta_t$ are independent and identically distributed (iid) with mean zero and variance one, whereas $\sigma_t > 0$ are the conditional standard deviations in $r_t$. Note that $\ln\left(\cdot\right)$ denotes the natural logarithm. Moreover, $B$ is the backshift operator and $\theta(B) = 1 +\sum_{i=1}^{\infty}\theta_i B^{i}$, where $\theta_i$, $i=1,2,\dots$, are real-valued coefficients. $g\left(\eta_{t-1}\right)$ is a suitable function in $\eta_{t-1}$. Generally, $\left\{g\left(\eta_{t}\right)\right\}$ should be an iid zero-mean sequence with finite variance. We have $\mu = E\left(r_t\right)$ as a real-valued parameter. The real-valued parameter $\omega_{\sigma}$ is in fact $\omega_{\sigma}=E\left[\ln\left(\sigma_t^2\right)\right]$. This previous set of equations defines the broader family of EGARCH models (Feng et al., 2025; Ayensu et al., 2025), from which subtypes are described in the following that depend on the choice of $g$.

$\textbf{Type I}:$

We have $\theta(B) = \phi^{-1}(B)(1-B)^{-d}\psi(B)$, where

$\phi(B) = 1-\sum_{i=1}^{p}\phi_{i}B^{i} \text{ and}$

$\psi(B) = 1+\sum_{j=1}^{q-1}\psi_{j}B^{j},$

are characteristic polynomials with real coefficients $\phi_{0},\dots,\phi_{p},\psi_{0},\dots,\psi_{q-1}$, by fixing $\phi_{0}=\psi_{0}=1$, and without common roots. Furthermore, the fractional differencing parameter is $d \in [0,1]$.

$g\left(\cdot\right)$ can be defined in different ways. Following a type I specification (model_type = "egarch"), we have

$g\left(\eta_t\right)=\kappa \left\{g_a\left(\eta_t\right) - E\left[g_a\left(\eta_t\right)\right] \right\} + \gamma\left\{g_m\left(\eta_t\right)-E\left[ g_m\left(\eta_t\right)\right]\right\}$

with $g_a\left(\eta_t\right)$ and $g_m\left(\eta_t\right)$ being suitable transformations of $\eta_t$ and where $\kappa$ and $\gamma$ are two additional real-valued parameters. In case of a simple (FI)EGARCH, we have $g_a\left(\eta_t\right) = \eta_t$ (and therefore with $E\left[g_a\left(\eta_t\right)\right] = 0$) and $g_m\left(\eta_t\right) = \left|\eta_t \right|$.

Generally, we consider two cases:

$g_{1,a}\left(\eta_t\right)=\text{sgn}\left(\eta_t\right)\left|\eta_t\right|^{p_a}/p_a \text{ and}$

$g_{2,a}\left(\eta_t\right)=\text{sgn}\left(\eta_t\right)\left[\left(\left|\eta_t\right| + 1\right)^{p_a} - 1\right] / p_{a}$

whereas

$g_{1,m}\left(\eta_t\right)=\left|\eta_t\right|^{p_m}/p_m \text{ and}$

$g_{2,m}\left(\eta_t\right)=\left[\left(\left|\eta_t\right| + 1\right)^{p_m} - 1\right] / p_{m}.$

Note that $\text{sgn}\left(\eta_t\right)$ denotes the sign of $\eta_t$. $g_{1,\cdot}$ incorporates a power transformation and $g_{2,\cdot}$ a modulus transformation together with a power transformation. The choices $g_{1,a}$ and $g_{2,a}$ correspond to setting the first element in modulus to FALSE or TRUE, respectively, under model_type = "egarch", where $p_a$ can be selected via the first element in powers. As a special case, for $p_a = 0$, a log-transformation is employed and the division through $p_a$ is dropped, i.e. $g_{1,a}\left(\eta_t\right)=\text{sgn}\left(\eta_t\right)\ln\left(\left|\eta_t\right|\right)$ and $g_{2,a}\left(\eta_t\right)=\text{sgn}\left(\eta_t\right)\ln\left(\left|\eta_t\right|+1\right)$ are employed for $p_a=0$. Completely analogous thoughts hold for $g_{1,m}$ and $g_{2,m}$ and the second elements in the arguments modulus and powers. The aforementioned model family is a type I model selectable through model_type = "egarch". Simple (FI)EGARCH models are given through selection of $g_{1,a}(\cdot)$ and $g_{1,m}(\cdot)$ in combination with $p_a = p_m = 1$. Another of such type I models is the FIMLog-GARCH (Feng et al., 2023), where instead $g_{2,a}(\cdot)$ and $g_{2,m}(\cdot)$ are selected with $p_a = p_m = 0$.

$\textbf{Type II}:$

As additional specifications of a Log-GARCH and a FILog-GARCH, which belong to the broader EGARCH family, we redefine

$\psi(B) = 1+\sum_{j=1}^{q}\psi_{j}B^{j}$

now as a polynomial of order $q$ and

$\ln\left(\sigma_t^2\right)=\omega_{\sigma}+\left[\phi^{-1}(B)(1-B)^{-d}\psi(B) - 1\right]\xi_t,$

where

$\xi_t=\ln\left(\eta_t^2\right)-E\left[\ln\left(\eta_t^2\right)\right].$

Everything else is defined as before. Since

$\phi^{-1}(B)(1-B)^{-d}\psi(B) - 1=\sum_{i=1}^{\infty}\gamma_i B^{i}=\gamma_1 B\left[\sum_{i=1}^{\infty}(\gamma_i/\gamma_1)B^{i-1}\right]=\gamma_1 B\left[1+\sum_{i=1}^{\infty}\theta_i B^{i}\right]=\theta(B)\gamma_1 B,$

where $\theta_i = \gamma_{i+1}/\gamma_{1}$, $i=0,1,\dots$, and by defining

$g\left(\eta_{t-1}\right)=\gamma_1\left\{\ln\left(\eta_{t-1}^2\right)-E\left[\ln\left(\eta_{t-1}^2\right)\right]\right\} = 2\gamma_1\left\{\ln\left(\left|\eta_{t-1}\right|\right)-E\left[\ln\left(\left|\eta_{t-1}\right|\right)\right]\right\},$

the equation of $\ln\left(\sigma_t^2\right)$ can be stated to be

$\ln\left(\sigma_t^2\right)=\omega_{\sigma}+\theta(B)g(\eta_{t-1})$

as in the broad EGARCH family at the very beginning. Therefore, Log-GARCH and FI-Log-GARCH models are equivalent to the type I models, where $\kappa = 0$ with usage of $g_{1,m}$ with $p_m = 0$ and where $\gamma = 2\gamma_1$. Nonetheless, in this package, the type II models make use of the more common parameterization of $\ln\left(\sigma_t^2\right)$ stated at the beginning of the type II model description.

This describes the established Log-GARCH models as part of the broad EGARCH family (type II models; model_type = "loggarch").

$\textbf{General information}:$

While the arguments powers and modulus are only relevant under a type I model, i.e. for model_type = "egarch", the arguments orders, long_memo and cond_dist are meaningful for both model_type = "egarch" and model_type = "loggarch", i.e. both under type I and II models. The first element of the two-element vector orders is the order $p$, while the second element is the order $q$. Furthermore, for long_memo = TRUE, the mentioned models are kept as they are, while for long_memo = FALSE the parameter $d$ is set to zero. cond_dist controls the conditional distribution. The unconditional mean $\mu$ is controlled via the function mean_spec; for include_mean = FALSE therein, $\mu$ is not being estimated and fixed to zero; its default is however include_mean = TRUE.

See also the closely related spec-functions that immediately create specifications of specific submodels of the broad EGARCH family. These functions are egarch_spec(), fiegarch_spec(), loggarch_spec(),filoggarch_spec(), megarch_spec(), mloggarch_spec() and mafiloggarch_spec(), which are all wrappers for fEGarch_spec().

See the references section for sources on the EGARCH (Nelson, 1991), FIEGARCH (Bollerslev and Mikkelsen, 1996), Log-GARCH (Geweke, 1986; Pantula, 1986; Milhoj, 1987) and FILog-GARCH (Feng et al., 2020) models.

Value

An object of either class "egarch_type_spec" or "loggarch_type_spec" is returned, depending on the choice for the input argument model_type.

References

• Ayensu, O. K., Feng, Y., & Schulz, D. (2025). Recent Extensions of Exponential GARCH Models: Theory and Application. Forthcoming preprint, Paderborn University.

• Bollerslev, T., & Mikkelsen, H. O. (1996). Modeling and pricing long memory in stock market volatility. Journal of Econometrics, 73(1), 151–184. DOI: 10.1016/0304-4076(95)01749-6.

• Feng, Y., Beran, J., Ghosh, S., & Letmathe, S. (2020). Fractionally integrated Log-GARCH with application to value at risk and expected shortfall. Working Papers CIE No. 137, Paderborn University, Center for International Economics. URL: http://groups.uni-paderborn.de/wp-wiwi/RePEc/pdf/ciepap/WP137.pdf.

• Feng, Y., Gries, T., & Letmathe, S. (2023). FIEGARCH, modulus asymmetric FILog-GARCH and trend-stationary dual long memory time series. Working Papers CIE No. 156, Paderborn University. URL: https://econpapers.repec.org/paper/pdnciepap/156.htm.

• Feng, Y., Peitz, C., & Siddiqui, S. (2025). A few useful members of the EGARCH-family with short- or long-memory in volatility. Unpublished working paper at Paderborn University.

• Geweke, J. (1986). Modeling the persistence of conditional variances: A comment. Econometric Reviews, 5(1), 57-61. DOI: 10.1080/07474938608800088.

• Milhoj, A. (1987). A Multiplicative Parameterization of ARCH Models. University of Copenhagen, Denmark.

• Nelson, D. B. (1991). Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica, 59(2), 347–370. DOI: 10.2307/2938260.

• Pantula, S. G. (1986). Modeling the persistence of conditional variances: A comment. Econometric Reviews, 5(1), 71-74. DOI: 10.1080/07474938608800089.

Examples

# EGARCH(1, 1) with cond. normal distribution
spec1 <- fEGarch_spec()
# EGARCH(2, 1) with cond. t-distribution
spec2 <- fEGarch_spec(orders = c(2, 1), cond_dist = "std")
# FIEGARCH(1, 1) with cond. normal distribution
spec3 <- fEGarch_spec(long_memo = TRUE)
# MEGARCH(1, 1) with cond. generalized error distribution
spec4 <- fEGarch_spec(modulus = c(TRUE, FALSE), powers = c(0, 1))
# Some unnamed specification
spec5 <- fEGarch_spec(
 model_type = "egarch",
 orders = c(1, 1),
 long_memo = TRUE,
 cond_dist = "std",
 powers = c(0.25, 0.75),
 modulus = c(TRUE, FALSE)
)

---

FIAPARCH Model Fitting

Description

Fit a fractionally integrated APARCH model under the six most common and further conditional distributions to observed data using quasi maximum-likelihood estimation.

Usage

fiaparch(
  rt,
  orders = c(1, 1),
  cond_dist = c("norm", "std", "ged", "ald", "snorm", "sstd", "sged", "sald"),
  drange = c(0, 1),
  meanspec = mean_spec(),
  Drange = c(0, 1),
  nonparspec = locpol_spec(),
  use_nonpar = FALSE,
  n_test = 0,
  start_pars = NULL,
  LB = NULL,
  UB = NULL,
  control = list(),
  control_nonpar = list(),
  mean_after_nonpar = FALSE,
  parallel = TRUE,
  ncores = max(1, future::availableCores() - 1),
  trunc = "none",
  presample = 50,
  Prange = c(1, 5),
  fix_delta = c(NA, 1, 2),
  skip_vcov = FALSE
)

Arguments

rt	the observed series ordered from past to present; can be a numeric vector, a "zoo" class time series object, or a "ts" class time series object.
orders	a two-element numeric vector containing the two model orders $p$ and $q$ (see Details for more information); currently, only the default orders = c(1, 1) is supported; other specifications of a two-element numeric vector will lead to orders = c(1, 1) being run and a warning message being returned.
cond_dist	the conditional distribution to consider as a character object; the default is a conditional normal distribution "norm"; available are also, however, a $t$-distribution ("std"), a generalized error distribution ("ged"), an average Laplace distribution ("ald"), and their four skewed variants ("snorm", "sstd", "sged", "sald").
drange	a two-element numeric vector that gives the boundaries of the search interval for the fractional differencing parameter $d$ in the conditional volatility model part; is overwritten by the settings of the arguments LB and UB.
meanspec	an object of class "mean_spec"; indicates the specifications for the model in the conditional mean.
Drange	a two-element numeric vector that indicates the boundaries of the interval over which to search for the fractional differencing parameter $D$ in a long-memory ARMA-type model in the conditional mean model part; by default, $D$ being searched for on the interval from 0 to $0.5 - 1\times 10^{-6}$; note that specific settings in the arguments LB and UB overwrite this argument.
nonparspec	an object of class "locpol_spec" returned by locpol_spec; defines the settings of the nonparametric smoothing technique for use_nonpar = TRUE.
use_nonpar	a logical indicating whether or not to implement a semiparametric extension of the volatility model defined through spec; see "Details" for more information.
n_test	a single numerical value indicating, how many observations at the end of rt not to include in the fitting process and to reserve for backtesting.
start_pars	the starting parameters for the numerical optimization routine; should be of the same length as the parameter output vector within the output object (also keeping the same order); for NULL, an internally saved default set of values is used; see "Details" for the order of elements; elements should be set with respect to a series rescaled to have sample variance one.
LB	the lower boundaries of the parameters in the numerical optimization routine; should be of the same length as the parameter output vector within the output object (also keeping the same order); for NULL, an internally saved default set of values is used; see "Details" for the order of elements; elements should be set with respect to a series rescaled to have sample variance one.
UB	the upper boundaries of the parameters in the numerical optimization routine; should be of the same length as the parameter output vector within the output object (also keeping the same order); for NULL, an internally saved default set of values is used; see "Details" for the order of elements; elements should be set with respect to a series rescaled to have sample variance one.
control	a list that is passed to control of the function solnp of the package Rsolnp.
control_nonpar	a list containing changes to the arguments for the hyperparameter estimation algorithm in the nonparametric scale function estimation for use_nonpar = TRUE; see "Details" for more information.
mean_after_nonpar	only for use_nonpar = TRUE; considers the unconditional mean of the parametric model part in the QMLE step in a semiparametric model; by default, a zero-mean model is considered for the parametric part in a semiparametric model.
parallel	only relevant for a (skewed) average Laplace (AL) distribution, i.e. if cond_dist in spec is set to cond_dist = "ald" or cond_dist = "sald"; parallel is a logical value indicating whether or not the slices for the positive integer-valued parameter of the SM distribution should be fitted in parallel for a speed boost.
ncores	only relevant for a (skewed) average Laplace (AL) distribution, i.e. if cond_dist in spec is set to cond_dist = "ald" or cond_dist = "sald", and if simultaneously parallel = TRUE; ncores is a single numeric value indicating the number of cores to use for parallel computations.
trunc	a positive integer indicating the finite truncation length of the infinite-order polynomials of the infinite-order representations of the long-memory model parts; the character "none" is an optional input that specifies that truncation should always be applied back to the first (presample) observation time point, i.e. that maximum length filters should be applied at all times.
presample	the presample length for initialization (for extended EGARCH- / Log-GARCH-type models only relevant for the FARIMA-part, as series in log-transformed conditional variance are initialized by zero).
Prange	a two-element vector that indicates the search boundaries for the parameter $P$ in a (skewed) average Laplace distribution.
fix_delta	let the parameter $\delta$ be either free (fix_delta = NA), fix it to 1 (fix_delta = 1) or to 2 (fix_delta = 2); the latter two are specific submodels of a FIAPARCH.
skip_vcov	a logical indicating whether or not to skip the computation of the variance-covariance matrix of the parameter estimators and therefore also standard error computation.

Details

Consider a FIAPARCH($p, d, q$) with constant asymmetry term $\gamma$ regardless of the lag. Let $\left\{r_t\right\}$, with $t \in \mathbb{Z}$ as the time index, be a theoretical time series that follows

$r_t=\mu+\varepsilon_t \text{ with } \varepsilon_t=\sigma_t \eta_t \text{ and } \eta_t \sim \text{IID}(0,1), \text{ where}$

$\sigma_t^{\delta}=\omega+\left[1-\beta^{-1}(B)\phi(B)(1-B)^{d}\right]\left(\left|\varepsilon_t\right|-\gamma\varepsilon_t\right)^{\delta}.$

Here, $\eta_t\sim\text{IID}(0,1)$ means that the innovations $\eta_t$ are independent and identically distributed (iid) with mean zero and variance one, whereas $\sigma_t > 0$ are the conditional standard deviations in $r_t$. Moreover, $B$ is the backshift operator and $\beta(B) = 1 - \sum_{j=1}^{q}\beta_j B^{j}$, where $\beta_j$, $j=1,2,\dots, q$, are real-valued coefficients. Furthermore, $\phi(B) = 1 - \sum_{i=1}^{p}\phi_i B^{i}$, where $\phi_i$, $i=1,2,\dots, p$, are real-valued coefficients. $p$ and $q$ are the model orders definable through the argument orders, where $p$ is the first element and $q$ is the second element in the argument. In addition, we have $\mu = E\left(r_t\right)$ as a real-valued parameter and $\gamma \in (-1,1)$ and $d \in [0,1]$ as the parameter for the level of integration. With $d = 0$ the model reduces to a short-memory APARCH, for $d=1$ we have a full integration, and for $d\in(0, 1)$, we have fractional integration, where $d\in(0, 0.5)$ is considered to describe a long-memory process. $\omega > 0$ is the intercept. It is assumed that all $\beta_j$ and $\phi_i$ are non-negative.

The pre-sample values of $\left(\left|\varepsilon_t\right|-\gamma\varepsilon_t\right)^{\delta}$ are replaced by the in-sample arithmetic mean of

$\left(\left|r_t^{*} - \bar{r}_t^{*}\right|- \gamma_0\left(r_t^{*}-\bar{r}_t^{*}\right)\right)^{\delta_0},$

where $r_t^{*}$ are considered as the observations and $\bar{r}_t^{*}$ is the sample mean of $r_t^{*}$. Initial values for $\gamma_0$ and $\delta_0$ as initial guesses for $\gamma$ and $\delta$ are obtained from a previous fitting of a short-memory APARCH.

Currently, only a model of orders $p=1$ with $q=1$ can be fitted; to ensure the non-negativity of all of the infinite-order coefficient series $\left[1-\beta^{-1}(B)\phi(B)(1-B)^{d}\right]$, which in combination with $\omega>0$ ensures that all the conditional volatilities are greater than zero, we employ inequality constraints ensuring that the first 50 coefficients of the infinite-order ARCH-representation are non-negative as an approximation to ensuring that all of the coefficients are non-negative. To ensure that they are non-negative, one may in theory consider the sufficient conditions mentioned in Bollerslev and Mikkelsen (1996) or Tse (1998), which are however sometimes restrictive, or the simultaneously necessary and sufficient conditions by Conrad and Haag (2006), which are however complex to implement properly.

See also the reference section for sources on the APARCH (Ding et al., 1993) and the FIAPARCH (Tse, 1998) models.

The truncated infinite order polynomial is computed following the idea by Nielsen and Noel (2021) as is the series of conditional variances for most computational efficiency. To ensure stability of the first fitted in-sample conditional standard deviations, we however use a small, but also adjustable (also to length zero) presample, which may introduce biases into the parameter estimators.

In the current package version, standard errors of parameter estimates are computed from the Hessian at the optimum of the log-likelihood using hessian. To ensure numerical stability and applicability to a huge variety of differently scaled data, parametric models are first fitted to data that is scaled to have sample variance 1. Parameter estimates and other quantities are then either retransformed or recalculated afterwards for the original data.

For a conditional average Laplace distribution, an optimal model for each distribution parameter $P$ from 1 to 5 is estimated (assuming that $P$ is then fixed to the corresponding value). Afterwards, $P$ is then estimated by selecting the estimated model among the five fitted models that has the largest log-likelihood. The five models are, by default, fitted simultaneously using parallel programming techniques (see also the arguments parallel and ncores, which are only relevant for a conditional average Laplace distribution). After the optimal model (including the estimate of $P$ called $\hat{P}$) has been determined, $P=\hat{P}$ is seen as fixed to obtain the standard errors via the Hessian matrix for the estimates of the continuous parameters. A standard error for $\hat{P}$ is therefore not obtained and the ones obtained for the remaining estimates do not account for $\hat{P}$.

An ARMA-FIAPARCH or a FARIMA-FIAPARCH can be fitted by adjusting the argument meanspec correspondingly.

As an alternative, a semiparametric extension of the pure models in the conditional variance can be implemented. If use_nonpar = TRUE, meanspec is omitted and before fitting a zero-mean model in the conditional volatility following the remaining function arguments, a smooth scale function, i.e. a function representing the unconditional standard deviation over time, is being estimated following the specifications in nonparspec and control_nonpar. This preliminary step stabilizes the input series rt, as long-term changes in the unconditional variance are being estimated and removed before the parametric step using tsmoothlm. control_nonpar can be adjusted following to make changes to the arguments of tsmoothlm for long-memory specifications. These arguments specify settings for the automated bandwidth selection algorithms implemented by this function. By default, we use the settings pmin = 0, pmax = 1, qmin = 0, qmax = 1, InfR = "Nai", bStart = 0.15, cb = 0.05, and method = "lpr" for tsmoothlm. locpol_spec passed to nonparspec handles more direct settings of the local polynomial smoother itself. See the documentation for these functions to get a detailed overview of these settings. Assume $\{r_t\}$ to be the observed series, where $t = 1, 2, \dots, n$, then $r_t^{*} = r_t - \bar{r}$, with $\bar{r}$ being the arithmetic mean over the observed $r_t$, is computed and subsequently $y_t = \ln\left[\left(r_t^{*}\right)^2\right]$. The subtraction of $\bar{r}$ is necessary so that $r_t^{*}$ are all different from zero almost surely. Once $y_t$ are available, its trend $m(x_t)$, with $x_t$ as the rescaled time on the interval $[0, 1]$, is being estimated using tsmoothlm and denoted here by $\hat{m}(x_t)$. Then from $\hat{\xi}_t = y_t - \hat{m}(x_t)$ obtain $\hat{C} = -\ln\left\{\sum_{t=1}^{n}\exp\left(\hat{\xi}_t\right)\right\}$, and obtain the estimated scale function as $\hat{s}(x_t)=\exp\left[\left(\hat{\mu}(x_t) - \hat{C}\right) / 2\right]$. The stabilized / standardized version of the series $\left\{r_t\right\}$ is then $\tilde{r}_t = r_t^{*} / \hat{s}(x_t)$, to which a purely parametric volatility model following the remaining function arguments is then fitted. The estimated volatility at a given time point is then the product of the estimate of the corresponding scale function value and of the estimated conditional standard deviation (following the parametric model part) for that same time point. See for example Feng et al. (2022) or Letmathe et al. (2023) for more information on the semiparametric extension of volatility models.

The order for manual settings of start_pars, LB and UB is crucial. The correct order is: $\mu$, $\text{ar}_1,\dots,\text{ar}_{p^{*}}$, $\text{ma}_1,\dots,\text{ma}_{q^{*}}$,$D$,$\omega$, $\phi$, $\beta$, $\gamma$, $\delta$, $d$, shape parameter, skewness parameter. Depending on the exact model specification, parameters irrelevant for the specification at hand should be dropped in start_pars, LB and UB.

Value

An object of S4-class "fEGarch_fit_fiaparch" is returned. It contains the following elements.

pars:

a named numeric vector with the parameter estimates.

se:

a named numeric vector with the obtained standard errors in accordance with the parameter estimates.

vcov_mat:

the variance-covariance matrix of the parameter estimates with named columns and rows.

rt:

the input object rt (or at least the training data, if n_test is greater than zero); if rt was a "zoo" or "ts" object, the formatting is kept.

cmeans:

the estimated conditional means; if rt was a "zoo" or "ts" object, the formatting is also applied to cmeans.

sigt:

the estimated conditional standard deviations (or for use_nonpar = TRUE the estimated total volatilities, i.e. scale function value times conditional standard deviation); if rt was a "zoo" or "ts" object, the formatting is also applied to sigt.

etat:

the obtained residuals; if rt was a "zoo" or "ts" object, the formatting is also applied to etat.

orders:

a two-element numeric vector stating the considered model orders.

cond_dist:

a character value stating the conditional distribution considered in the model fitting.

long_memo:

a logical value stating whether or not long memory was considered in the model fitting.

llhood:

the log-likelihood value obtained at the optimal parameter combination.

inf_criteria:

a named two-element numeric vector with the corresponding AIC (first element) and BIC (second element) of the fitted parametric model part; for purely parametric models, these criteria are valid for the entire model; for semiparametric models, they are only valid for the parametric step and are not valid for the entire model.

meanspec:

the settings for the model in the conditional mean; is an object of class "mean_spec" that is identical to the object passed to the input argument meanspec.

test_obs:

the observations at the end up the input rt reserved for testing following n_test.

scale_fun:

the estimated scale function values, if use_nonpar = TRUE, otherwise NULL; formatting of rt is reused.

nonpar_model:

the estimation object returned by tsmoothlm for use_nonpar = TRUE.

trunc:

the input argument trunc.

References

• Bollerslev, T., & Mikkelsen, H. O. (1996). Modeling and pricing long memory in stock market volatility. Journal of Econometrics, 73(1): 151-184. DOI: 10.1016/0304-4076(95)01736-4.

• Conrad, C., & Haag, B. R. (2006). Inequality constraints in the fractionally integrated GARCH model. Journal of Financial Econometrics, 4(3): 413-449. DOI: 10.1093/jjfinec/nbj015.

• Ding, Z., Granger, C. W. J., & Engle, R. F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1(1): 83-106. DOI: 10.1016/0927-5398(93)90006-D.

• Feng, Y., Gries, T., Letmathe, S., & Schulz, D. (2022). The smoots Package in R for Semiparametric Modeling of Trend Stationary Time Series. The R Journal, 14(1), 182-195. URL: https://journal.r-project.org/articles/RJ-2022-017/.

• Letmathe, S., Beran, J., & Feng, Y. (2023). An extended exponential SEMIFAR model with application in R. Communications in Statistics - Theory and Methods, 53(22), 7914–7926. DOI: 10.1080/03610926.2023.2276049.

• Nielsen, M. O., & Noel, A. L. (2021). To infinity and beyond: Efficient computation of ARCH($\infty$) models. Journal of Time Series Analysis, 42(3), 338–354. DOI: 10.1111/jtsa.12570.

• Tse, Y. K. (1998). The conditional heteroskedasticity of the yen-dollar exchange rate. Journal of Applied Econometrics, 13(1): 49-55. DOI: 10.1002/(SICI)1099-1255(199801/02)13:1<49::AID-JAE459>3.0.CO;2-O.

Examples

window.zoo <- get("window.zoo", envir = asNamespace("zoo"))
rt <- window.zoo(SP500, end = "2002-12-31")
model <- fiaparch(rt)
model

---