GARCH Class

GARCH Class

Computes estimates of the parameters of a GARCH(p,q) model.

Inheritance Hierarchy

SystemObject
Imsl.StatGARCH

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class GARCH

<SerializableAttribute>
Public Class GARCH

[SerializableAttribute]
public ref class GARCH

[<SerializableAttribute>]
type GARCH =  class end

The GARCH type exposes the following members.

Constructors

	Name	Description
	GARCH	Constructor for GARCH.

Top

Methods

	Name	Description
	Compute	Computes estimates of the parameters of a GARCH(p,q) model.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetAR	Returns the estimated values of autoregressive (AR) parameters. Note that the Compute method must be invoked first before invoking this method. Otherwise, the method throws a NullReferenceException exception.
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetMA	Returns the estimated values of moving average (MA) parameters. Note that the Compute method must be invoked first before invoking this method. Otherwise, the method throws a NullReferenceException exception.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	GetVarCovarMatrix	Returns the variance-covariance matrix. Note that the Compute method must be invoked first before invoking this method. Otherwise, the method throws a NullReferenceException exception.
	GetX	Returns the estimated parameter array, x. Note that the Compute method must be invoked first before invoking this method. Otherwise, the method throws a NullReferenceException exception.
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

Top

Properties

	Name	Description
	Akaike	Returns the value of Akaike Information Criterion evaluated at the estimated parameter array. Note that the Compute method must be invoked first before invoking this method. Otherwise, the return value is 0.
	LogLikelihood	Returns the value of Log-likelihood function evaluated at the estimated parameter array. Note that the Compute method must be invoked first before invoking this method. Otherwise, the return value is 0.
	MaxSigma	The value of the upperbound on the first element (sigma) of the array of returned estimated coefficients.
	Sigma	Returns the estimated value of sigma squared. Note that the Compute method must be invoked first before invoking this method. Otherwise, the return value is NaN.

Top

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).

Reference

Imsl.Stat Namespace

Other Resources

Example