Syntax

C#
[SerializableAttribute] public class GARCH

Visual Basic (Declaration)
<SerializableAttribute> _ Public Class GARCH

Visual C++
[SerializableAttribute] public ref class GARCH

Remarks

The Generalized Autoregressive Conditional Heteroskedastic (GARCH) model is defined as

$y_t = z_t \sigma _t$

$\sigma _t^2 = \sigma ^2 + \sum\limits_{i = 1}^p {\beta _i \sigma _{t - i}^2 + } \sum\limits_{i = 1}^q {\alpha _i y_{t - i}^2 }$

where 's are independent and identically distributed standard normal random variables,

$\sigma > 0,\beta _i \ge 0,\alpha _i \ge 0{\rm{ }}$

and

$\sum\limits_{i = 1}^p {\beta _i } + \sum\limits_{i = 1}^q {\alpha _i} \lt1$

The above model is denoted as GARCH(p, q). The p is the autoregressive lag and the q is the moving average lag. When $\beta_i = 0, i = 1,2,\ldots, p$ , the above model reduces to ARCH(q) which was proposed by Engle (1982). The nonnegativity conditions on the parameters implied a nonnegative variance and the condition on the sum of the $\beta_i$ 's and $\alpha_i$ 's is required for wide sense stationarity.

In the empirical analysis of observed data, GARCH(1,1) or GARCH(1,2) models have often found to appropriately account for conditional heteroskedasticity (Palm 1996). This finding is similar to linear time series analysis based on ARMA models.

It is important to notice that for the above models positive and negative past values have a symmetric impact on the conditional variance. In practice, many series may have strong asymmetric influence on the conditional variance. To take into account this phenomena, Nelson (1991) put forward Exponential GARCH (EGARCH). Lai (1998) proposed and studied some properties of a general class of models that extended linear relationship of the conditional variance in ARCH and GARCH into nonlinear fashion.

The maximum likelihood method is used in estimating the parameters in GARCH(p,q). The log-likelihood of the model for the observed series $\left\{ {Y_t} \right\}$ with length m is

$\log (L) = \frac{m}{2}\log (2\pi ) - \frac{1}{2}\sum\limits_{t = 1}^m {y_t^2 /\sigma _t^2 - \frac{1}{2}\sum\limits_{t = 1}^m {\log \sigma _t^2 } } ,$

${\rm{where}} \,\,\, \sigma _t^2 = \sigma ^2 + \sum\limits_{i = 1}^p {\beta _i \sigma _{t - i}^2 } + \sum\limits_{i = 1}^q {\alpha _i y_{t - i}^2 } .$

In the model, if q = 0, the model GARCH is singular such that the estimated Hessian matrix H is singular.

The initial values of the parameter array $x[\,\,]$ entered in array xguess[ ] must satisfy certain constraints. The first element of xguess refers to sigma and must be greater than zero and less than MaxSigma. The remaining p+q initial values must each be greater than or equal to zero but less than one.

To guarantee stationarity in model fitting,

$\sum\limits_{i = 1}^{p + q} {x(i)} \lt 1,$

is checked internally. The initial values should be selected from the values between zero and one. The value of Akaike Information Criterion is computed by

${\rm{ 2 \times log (L) + 2 \times (p + q + 1),}}$

where log(L) is the value of the log-likelihood function at the estimated parameters.

In fitting the optimal model, the class MinConGenLin, is modified to find the maximal likelihood estimates of the parameters in the model. Statistical inferences can be performed outside of the class GARCH based on the output of the log-likelihood function (LogLikelihood property), the Akaike Information Criterion (Akaike property), and the variance-covariance matrix (GetVarCovarMatrix method).

Inheritance Hierarchy

System..::.Object
Imsl.Stat..::.GARCH

Syntax

Remarks

Inheritance Hierarchy

See Also