View source: R/fit_ARMA_GARCH.R
fit_GARCH_11 | R Documentation |
Fast(er) and numerically more robust fitting of GARCH(1,1) processes according to Zumbach (2000).
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), ...)
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
( |
control |
see |
innovations |
random variates from the innovation distribution;
for example, obtained via |
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 |
... |
|
fit_GARCH_11()
:estimated coefficients alpha[0],
alpha[1], beta[1] and, if
distr = "st"
the estimated degrees of freedom.
maximized log-likelihood.
number of calls to the objective function; see
?optim
.
convergence code ('0' indicates successful
completion); see ?optim
.
see ?optim
.
vector of length n giving the conditional volatility.
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[1] * Z^2 + β[1])^(α/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.
Marius Hofert
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.
fit_ARMA_GARCH()
based on rugarch.
### 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
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