gsFit(formula = ~garch(1, 1), data, cond.dist = c("stableS0", "stableS1",
"stableS2", "gev", "gat", "norm", "std", "sstd", "skstd", "ged"),
include.mean = TRUE, algorithm = c("sqp", "sqp.restriction",
"nlminb", "nlminb+nm"), control = NULL,
tolerance = NULL, title = NULL, description = NULL)"stableS0" (stable in S0-parameterization), "stableS1" (stable in S1-parameterization), ""sqp", "sqp.restriction" (enforce stationarity) "sqp" and
"sqp.restriction" algorithms the control parameters are passed to the solnp function from list( TOLG = 1e-8, TOLSTABLE = 1e-2, TOLSTATIONARITY = 1e-3 ), where TOLG is the tolerance used to segsFit
returns a S4 object of class "GEVSTABLEGARCH" with the following slots:gsFit function.par - the estimated parameters;
llh - the estimated negative log-likelihood function;
hessian - the hessian matrix returned by the optimization algorithm;
ics - the value of the goodness-of-fit measures (AIC, AICc and BIC) (See Brockwell and Davis, 2002 for more details);
order - a list with the ARMA and GARCH/APARCH orders;
cond.dis - the conditional distribution;
se.coef - standard errors of the estimated parameters;
tValue - tValue of the estimated parameters;
matcoef - an organized matrix with the estimated parameters.The
The parameters will be interpreted according to the following equations (see Wurtz et al. ,2009) $$X_t = \mu + \sum_{i=1}^m a_i X_{t-i} + \sum_{j=1}^n b_j \varepsilon_{t-j} + \varepsilon_t$$ $$\varepsilon_t = \sigma_t z_t,\;\;\;z_t \stackrel{iid}{\sim} D(0,1)\;,\$$ $$\sigma_t^\delta = \omega + \sum_{i=1}^p \alpha_i(\varepsilon_{t-i}-\gamma_i|\varepsilon_{t-i}|)^\delta + \sum_{j=1}^q \beta_j \sigma_{t-j}^\delta$$ where $\mathcal{D}_{\vartheta}(0,1)$ is the density of the innovations with zero location and unit scale and $\vartheta$ are additional distributional parameters that describe the skew and the shape of the distribution.
Most software packages implement the estimation of GARCH models without imposing stationarity, but restricting the parameter set by appropriate bounds. This last approach was implemented in the "sqp", "nlminb" and "nlminb + nm". The first two algorithms search for the optimum value by restricting the parameter set to appropriate lower and upper bounds. The last implements a two step optimization procedure, which consists in starting the search by using the constrained routine nlminb and then performing another search using an unconstrained method (in our case the Nelder-Mead method implemented in the R base function optim). This approach was suggested by Wuertz et al. (2009) since in many cases it leads to an improved solution (in terms of the likelihood function of the data).
Finally, the "sqp.restriction" algorithm performs a constrained search to maximize the log-likelihood function in order to obtain an stationary model. It can be used to estimate models with the following conditional distributions: "stableS1", "gev", "gat", "skstd", "norm", "std" and "ged".
}
Since we want to estimate the parameters that better adjust real data to ARMA-APARCH models, the notion of stationarity is crucial. Usually, even when the data set is non-stationary in appearance, we still are able to apply transformation techniques so that the resulting time series can be reasonably modeled as a stationary process (see Brockwell and Davis (1996)).
The APARCH(p,q) model with finite variance innovations has a unique $\delta$-order stationary solution if and only if (see Ling and McAller (2002)) $$\sum_{i=1}^p E(|Z| - \gamma_i Z)^\delta \alpha_i +\sum_{j=1}^q\beta_j < 1,$$
The case of infinite variance has a somewhat different historical background. The first assumption made on stable distributions is that the index of stability $\alpha$ must be greater than one, because in this case the innovations have finite first moment. The second assumption is that they must have a $\delta$-moment finite, which means that we must restrict our model to $1 < \delta < \alpha$.
Diongue et al. (2008) showed that the APARCH(p,q) (all coefficients $\gamma_i = 0$) model has a strictly stationary solution if and only if $$\sum_{i=1}^p E(|Z| - \gamma_i Z)^\delta\alpha_i +\sum_{j=1}^q\beta_j < 1,$$
where $Z$ has distribution $S(\alpha,\beta;1)$ (stable in 1-parameterization). The estimation of stationary models is mainly dependent on the time taken to evaluate the expression for the asymmetric stable distribution $E(|Z| - \gamma_i Z)^\delta$.
The moment expression used to verify the stationarity condition is given by
$$E(|Z| - \gamma Z)^\delta = \frac{ (1 - \gamma )^\delta\tilde{\sigma}^{\delta+1}\Gamma(\delta+1)\Gamma(-\frac{\delta}{\alpha}) } { \alpha\tilde{\sigma}\Gamma\bigg[ \bigg(\frac{1}{2}+\tilde{\beta}\frac{k(\alpha)}{2\alpha}\bigg)(-\delta)\bigg] \Gamma\bigg[ \frac{1}{2}- \tilde{\beta}\frac{k(\alpha)}{2\alpha} + \bigg(\frac{1}{2} + \tilde{\beta}\frac{k(\alpha)}{2\alpha}\bigg)(\delta+1)\bigg] }$$
$$+ \frac{ (1 + \gamma )^\delta\tilde{\sigma}^{\delta+1}\Gamma(\delta+1)\Gamma(-\frac{\delta}{\alpha}) } { \alpha\tilde{\sigma}\Gamma\bigg[ \bigg(\frac{1}{2}-\tilde{\beta}\frac{k(\alpha)}{2\alpha}\bigg)(-\delta)\bigg] \Gamma\bigg[ \frac{1}{2} + \tilde{\beta}\frac{k(\alpha)}{2\alpha} + \bigg(\frac{1}{2} - \tilde{\beta}\frac{k(\alpha)}{2\alpha}\bigg)(\delta+1)\bigg] },$$
where
$$\tilde{\sigma} = \bigg[1 + \beta^2 \tan^2(\frac{\alpha\pi}{2})\bigg]^{\frac{1}{2\alpha}},$$
$$k(\alpha) = \left{ \begin{array}{rl} \alpha & , \alpha < 1, \ \alpha-2 & , \alpha > 1, \end{array} \right.$$
and
$$\tilde{\beta} = \left{ \begin{array}{rl} \frac{2}{\pi\alpha}\arctan(\beta\tan(\frac{\alpha\pi}{2})) & , 0 < \alpha < 1, \ \frac{2}{\pi(\alpha-2)}\arctan(\beta\tan(\frac{\pi(\alpha-2)}{2})) & , 1 < \alpha < 2. \end{array} \right.$$
Notice that this expression is only valid for the stable distribution in 1-parameterization, and thus cannot be used to the estimate models with stable distribution in 0 or 2 parameterizations.
The stationarity restriction implemented inside TOL.STATIONARITY,
where the variable TOL.STATIONARITY is set to 0.001 by default. For numerical reasons, the value of this variable cannot be greater than 0.05.
}
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gsSelect# This examples uses the dem2gbp dataset to estimate
# an ARMA(1,1)-GARCH(1,1) with GEV conditional distribution.
data(dem2gbp)
x = dem2gbp[, 1]
gev.model = gsFit(data = x , formula = ~garch(1,1), cond.dist = "gev")Run the code above in your browser using DataLab