Package com.imsl.stat

Class GARCH

java.lang.Object

com.imsl.stat.GARCH

All Implemented Interfaces:

Serializable, Cloneable

public class GARCH
extends Object
implements Serializable, Cloneable

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

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 \(z_t\)'s are independent and identically distributed standard normal random variables,

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

and

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

The above model is denoted as GARCH(p, q). The \( \alpha_i, i=1,\ldots,q \) and the \( \beta_j, j=1,\ldots,p \) parameters will be called the ARCH and GARCH coefficients, respectively. When \(\beta_i = 0, i = 1,2,\ldots, p\), the above model reduces to ARCH(q) which was proposed by Engle (1982). That is, a GARCH(0,q) is equivalent to the ARCH(q) model. The non-negativity conditions on the parameters imply 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 maximal 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 (getlogLikelihood method), the Akaike Information Criterion (getAkaike method), and the variance-covariance matrix (getVarCovarMatrix method).

See Also:

• Example
• Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static class	GARCH.ConstrInconsistentException	The equality constraints are inconsistent.
static class	GARCH.EqConstrInconsistentException	The equality constraints and the bounds on the variables are found to be inconsistent.
static class	GARCH.NoVectorXException	No vector X satisfies all of the constraints.
static class	GARCH.TooManyIterationsException	Number of function evaluations exceeded 1000.
static class	GARCH.VarsDeterminedException	The variables are determined by the equality constraints.

Constructor Summary

Constructors

Constructor	Description
GARCH(int p, int q, double[] y, double[] xguess)	Constructor for GARCH.

Method Summary

Modifier and Type	Method	Description
final void	compute()	Computes estimates of the parameters of a GARCH(p,q) model.
double	getAkaike()	Returns the value of Akaike Information Criterion evaluated at the estimated parameter array.
double[]	getAR()	Deprecated. Use getGARCH() instead.
double[]	getARCH()	Returns the estimated values of the ARCH coefficients.
double[]	getGARCH()	Returns the estimated values of the GARCH coefficients.
double	getLogLikelihood()	Returns the value of Log-likelihood function evaluated at the estimated parameter array.
double[]	getMA()	Deprecated. Use getARCH() instead.
double	getSigma()	Returns the estimated value of sigma squared.
double[][]	getVarCovarMatrix()	Returns the variance-covariance matrix.
double[]	getX()	Returns the estimated parameter array, x.
void	setMaxSigma(double maxSigma)	Sets the value of the upperbound on the first element (sigma) of the array of returned estimated coefficients.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

GARCH

public GARCH(int p,
 int q,
 double[] y,
 double[] xguess)

Constructor for GARCH.

Parameters:

p - An int scalar containing the number of GARCH parameters.

q - An int scalar containing the number of ARCH parameters.

y - A double array containing the observed time series data.

xguess - A double array of length p + q + 1 containing the initial values for the parameter array.

Throws:

IllegalArgumentException - is thrown if the dimensions of y, and xguess are not consistent.

Method Details

compute

public final void compute()
                   throws GARCH.ConstrInconsistentException,
GARCH.EqConstrInconsistentException,
GARCH.NoVectorXException,
GARCH.TooManyIterationsException,
GARCH.VarsDeterminedException

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

Throws:

GARCH.ConstrInconsistentException - is thrown if the equality constraints are inconsistent.

GARCH.EqConstrInconsistentException - is thrown if the equality constraints and the bounds on the variables are found to be inconsistent.

GARCH.NoVectorXException - is thrown if no vector X satisfies all of the constraints.

GARCH.TooManyIterationsException - is thrown if the number of function evaluations exceeded 1000.

GARCH.VarsDeterminedException - is thrown if the variables are determined by the equality constraints.

setMaxSigma

public void setMaxSigma(double maxSigma)

Sets the value of the upperbound on the first element (sigma) of the array of returned estimated coefficients.

Parameters:

maxSigma - A double scalar containing the value of the upperbound on the first element (sigma) of the array of returned estimated coefficients. Default = 10.

getSigma

public double getSigma()

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.

Returns:

a double scalar containing the estimated value of sigma squared.

getAR

@Deprecated
public double[] getAR()

Deprecated.

Use getGARCH() instead.

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 NullPointerException exception.

Returns:

a double array of size p containing the estimated values of autoregressive (AR) parameters.

getGARCH

public double[] getGARCH()

Returns the estimated values of the GARCH coefficients. Note that the compute method must be invoked first before invoking this method. Otherwise, the method throws a NullPointerException exception.

Returns:

a double array of size p containing the estimated values of autoregressive (GARCH) parameters.

getMA

@Deprecated
public double[] getMA()

Deprecated.

Use getARCH() instead.

Returns the estimated values of the ARCH coefficients. Note that the compute method must be invoked first before invoking this method. Otherwise, the method throws a NullPointerException exception.

Returns:

a double array of size q containing the estimated values of the ARCH coefficients.

getARCH

public double[] getARCH()

Returns the estimated values of the ARCH coefficients. Note that the compute method must be invoked first before invoking this method. Otherwise, the method throws a NullPointerException exception.

Returns:

a double array of size q containing the estimated values of the moving average (ARCH) coefficients.

getVarCovarMatrix

public double[][] getVarCovarMatrix()

Returns the variance-covariance matrix. Note that the compute method must be invoked first before invoking this method. Otherwise, the method throws a NullPointerException exception.

Returns:

a double matrix of size p + q + 1 by p + q + 1 containing the variance-covariance matrix.

getAkaike

public double getAkaike()

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.

Returns:

a double scalar containing the value of Akaike Information Criterion evaluated at the estimated parameter array.

getLogLikelihood

public double getLogLikelihood()

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.

Returns:

a double scalar containing the value of Log-likelihood function evaluated at the estimated parameter array.

getX

public double[] 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 NullPointerException exception.

Returns:

a double array of size 1 + q + p containing the estimated values of sigma squared, the ARCH parameters, and the GARCH parameters.