fit_GARCH_11: Fast(er) and Numerically More Robust Fitting of GARCH(1,1) Processes

Description

Fast(er) and numerically more robust fitting of GARCH(1,1) processes according to Zumbach (2000).

Usage

fit_GARCH_11(x, init = NULL, sig2 = mean(x^2), delta = 1,
             distr = c("norm", "st"), control = list(), ...)
tail_index_GARCH_11(innovations, alpha1, beta1,
                    interval = c(1e-6, 1e2), ...)

Value

fit_GARCH_11():

coef:: estimated coefficients $\alpha_0$, $\alpha_1$, $\beta_1$ and, if distr = "st" the estimated degrees of freedom.
logLik:: maximized log-likelihood.
counts:: number of calls to the objective function; see ?optim.
convergence:: convergence code ('0' indicates successful completion); see ?optim.
message:: see ?optim.
sig.t:: vector of length $n$ giving the conditional volatility.
Z.t:: vector of length $n$ giving the standardized residuals.

tail_index_GARCH_11():

The tail index $alpha$ estimated by Monte Carlo via McNeil et al. (2015, p. 576), so the $alpha$ which solves $$E({(\alpha_1Z^2 + \beta_1)}^{\alpha/2}) = 1$$, where $Z$ are the innovations. If no solution is found (e.g. if the objective function does not have different sign at the endpoints of interval), NA is returned.

Arguments

x

vector of length $n$ containing the data (typically log-returns) to be fitted a GARCH(1,1) to.

init

vector of length 2 giving the initial values for the likelihood fitting. Note that these are initial values for $z_{corr}$ and $z_{ema}$ as in Zumbach (2000).

sig2

annualized variance (third parameter of the reparameterization according to Zumbach (2000)).

delta

unit of time (defaults to 1 meaning daily data; for yearly data, use 250).

distr

character string specifying the innovation distribution ("norm" for N(0,1) or "st" for a standardized $t$ distribution).

control

see ?optim().

innovations

random variates from the innovation distribution; for example, obtained via rnorm() or rt(, df = nu) * sqrt((nu-2)/nu) where nu are the d.o.f. of the $t$ distribution.

alpha1

nonnegative GARCH(1,1) coefficient $alpha[1]$ satisfying $alpha[1] + beta[1] < 1$.

beta1

nonnegative GARCH(1,1) coefficient $beta[1]$ satisfying $alpha[1] + beta[1] < 1$.

interval

initial interval for computing the tail index; passed to the underlying uniroot().

...

fit_GARCH_11():: additional arguments passed to the underlying optim().

tail_index_GARCH_11():

additional arguments passed to the underlying uniroot().

Author

Marius Hofert

References

Zumbach, G. (2000). The pitfalls in fitting GARCH (1,1) processes. Advances in Quantitative Asset Management 1, 179--200.

McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.

Examples

Run this code

### Example 1: N(0,1) innovations ##############################################

## Generate data from a GARCH(1,1) with N(0,1) innovations
library(rugarch)
uspec <- ugarchspec(variance.model = list(model = "sGARCH",
                                          garchOrder = c(1, 1)),
                    distribution.model = "norm",
                    mean.model = list(armaOrder = c(0, 0)),
                    fixed.pars = list(mu = 0,
                                      omega = 0.1, # alpha_0
                                      alpha1 = 0.2, # alpha_1
                                      beta1 = 0.3)) # beta_1
X <- ugarchpath(uspec, n.sim = 1e4, rseed = 271) # sample (set.seed() fails!)
X.t <- as.numeric(X@path$seriesSim) # actual path (X_t)

## Fitting via ugarchfit()
uspec. <- ugarchspec(variance.model = list(model = "sGARCH",
                                           garchOrder = c(1, 1)),
                     distribution.model = "norm",
                     mean.model = list(armaOrder = c(0, 0)))
fit <- ugarchfit(uspec., data = X.t)
coef(fit) # fitted mu, alpha_0, alpha_1, beta_1
Z <- fit@fit$z # standardized residuals
stopifnot(all.equal(mean(Z), 0, tol = 1e-2),
          all.equal(var(Z),  1, tol = 1e-3))

## Fitting via fit_GARCH_11()
fit. <- fit_GARCH_11(X.t)
fit.$coef # fitted alpha_0, alpha_1, beta_1
Z. <- fit.$Z.t # standardized residuals
stopifnot(all.equal(mean(Z.), 0, tol = 5e-3),
          all.equal(var(Z.),  1, tol = 1e-3))

## Compare
stopifnot(all.equal(fit.$coef, coef(fit)[c("omega", "alpha1", "beta1")],
                    tol = 5e-3, check.attributes = FALSE)) # fitted coefficients
summary(Z. - Z) # standardized residuals


### Example 2: t_nu(0, (nu-2)/nu) innovations ##################################

## Generate data from a GARCH(1,1) with t_nu(0, (nu-2)/nu) innovations
uspec <- ugarchspec(variance.model = list(model = "sGARCH",
                                          garchOrder = c(1, 1)),
                    distribution.model = "std",
                    mean.model = list(armaOrder = c(0, 0)),
                    fixed.pars = list(mu = 0,
                                      omega = 0.1, # alpha_0
                                      alpha1 = 0.2, # alpha_1
                                      beta1 = 0.3, # beta_1
                                      shape = 4)) # nu
X <- ugarchpath(uspec, n.sim = 1e4, rseed = 271) # sample (set.seed() fails!)
X.t <- as.numeric(X@path$seriesSim) # actual path (X_t)

## Fitting via ugarchfit()
uspec. <- ugarchspec(variance.model = list(model = "sGARCH",
                                           garchOrder = c(1, 1)),
                     distribution.model = "std",
                     mean.model = list(armaOrder = c(0, 0)))
fit <- ugarchfit(uspec., data = X.t)
coef(fit) # fitted mu, alpha_0, alpha_1, beta_1, nu
Z <- fit@fit$z # standardized residuals
stopifnot(all.equal(mean(Z), 0, tol = 1e-2),
          all.equal(var(Z),  1, tol = 5e-2))

## Fitting via fit_GARCH_11()
fit. <- fit_GARCH_11(X.t, distr = "st")
c(fit.$coef, fit.$df) # fitted alpha_0, alpha_1, beta_1, nu
Z. <- fit.$Z.t # standardized residuals
stopifnot(all.equal(mean(Z.), 0, tol = 2e-2),
          all.equal(var(Z.),  1, tol = 2e-2))

## Compare
fit.coef <- coef(fit)[c("omega", "alpha1", "beta1", "shape")]
fit..coef <- c(fit.$coef, fit.$df)
stopifnot(all.equal(fit.coef, fit..coef, tol = 7e-2, check.attributes = FALSE))
summary(Z. - Z) # standardized residuals

Run the code above in your browser using DataLab