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(), ...)

Arguments

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().

…

additional arguments passed to the underlying optim().

Value

A list with components

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.

References

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

Examples

Run this code

# NOT RUN {
### 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
# }