GARCH Models

GARCH models are used to model time series where the volatility or variance of the errors is not constant. This is often the case in financial time series.

The simplest models only have auto-regressive terms and are called ARCH(q) models, where q is the autoregressive order. A GARCH(p,q) model is a generalization of an ARCH(q) model, with p additional terms corresponding to past conditional variances.

GARCH models are implemented by the GarchModel class.

The GarchModel class has two constructors. The first takes two arguments and is used to construct an ARCH(q) model. The first argument is a Vector<T> that contains the time series data. The second argument is the autoregressive order.

var model1 = new GarchModel(y, 2);

Dim model1 = New GarchModel(y, 2)

No code example is currently available or this language may not be supported.

let model1 = new GarchModel(y, 2)
//

The second constructor takes three arguments and is used to construct a GARCH(p,q) model. The first argument is once again the Vector<T> that contains the time series data. The remaining parameters are the number of conditional variance terms p and the autoregressive order q.

There are several choices for the distribution of the error terms or innovations. The distribution can be set through the InnovationDistribution property. This property is of type GarchInnovationDistribution, which can have the following values:

When the degrees of freedom is fixed, it must be set using the StudentTDegreesOfFreedom property. The following example creates a GARCH(1,1) model. The innovation distribution is a student-t distribution with 10 degrees of freedom:

var model2 = new GarchModel(y, 1, 1);
model2.InnovationDistribution = GarchInnovationDistribution.StudentTWithFixedDegreesOfFreedom;
model2.StudentTDegreesOfFreedom = 10;

Dim model2 = New GarchModel(y, 1, 1)
model2.InnovationDistribution = GarchInnovationDistribution.StudentTWithFixedDegreesOfFreedom
model2.StudentTDegreesOfFreedom = 10

No code example is currently available or this language may not be supported.

let model2 = new GarchModel(y, 1, 1)
model2.InnovationDistribution <- GarchInnovationDistribution.StudentTWithFixedDegreesOfFreedom
model2.StudentTDegreesOfFreedom <- 10.0
//

The Compute method estimates the parameters of the model using conditional maximum likelihood estimation.

The parameters of the model can be retrieved through the Parameters collection. The first p parameters are the coefficients of the autoregressive components. The next q parameters are the coefficients of the moving average components. If the degrees of freedom of the student-t distribution of the innovations was estimated also, it is returned as the last parameter. The coefficients can also be retrieved separately through the GarchParameters and ArchParameters properties.

The members of these collections are of type Parameter<T>, and can be used to obtain a wide range of information about the computed values, including the standard error, significance tests and confidence intervals.

model1.Fit();
Console.WriteLine("Constant: {0}", model2.Parameters[0].Value);
Console.WriteLine("ARCH(1): {0}", model2.Parameters[1].Value);
Console.WriteLine("GARCH(1): {0}", model2.Parameters[2].Value);
Console.WriteLine(model2.ParameterValues);

model2.Fit()
Console.WriteLine("Constant: { 0 }", model2.Parameters(0).Value)
Console.WriteLine("ARCH(1): { 0 }", model2.Parameters(1).Value)
Console.WriteLine("GARCH(1): { 0 }", model2.Parameters(2).Value)
Console.WriteLine(model2.ParameterValues)

No code example is currently available or this language may not be supported.

model1.Fit()
printfn "Constant: %f" model2.Parameters.[0].Value
printfn "ARCH(1): %f" model2.Parameters.[1].Value
printfn "GARCH(1): %f" model2.Parameters.[2].Value
Console.WriteLine(model2.ParameterValues)
//

Because a GARCH model is computed by directly maximizing the likelihood function, it does not have the same range of diagnostic values available for linear regression models. Still, a number of standard measures are available.

The LogLikelihood method returns the logarithm of the likelihood of the computed model. The GetAkaikeInformationCriterion method returns the Akaike Information Criterion (AIC) value. This is commonly used to compare different models. The GetBayesianInformationCriterion method returns the Bayesian Information Criterion (BIC) value, which is sometimes used instead of the AIC.

var ll = model2.LogLikelihood;
var aic = model2.GetAkaikeInformationCriterion();
var bic = model2.GetBayesianInformationCriterion();

Dim ll = model2.LogLikelihood
Dim aic = model2.GetAkaikeInformationCriterion()
Dim bic = model2.GetBayesianInformationCriterion()

No code example is currently available or this language may not be supported.

let ll = model2.LogLikelihood
let aic = model2.GetAkaikeInformationCriterion()
let bic = model2.GetBayesianInformationCriterion()
//

Once the model has been computed, the Forecast method can then be used to forecast new values. This method has three overloads.

Without arguments, the method returns the one step ahead forecast based on the computed model. With a single argument, it computes a point forecast the specified number of steps ahead. It returns a Vector<T> that contains the point forecast for the specified number of periods.

var forecast = model2.Forecast(4);

Dim forecast = model2.Forecast(4)

No code example is currently available or this language may not be supported.

let forecast = model2.Forecast(4)
//

The third overload takes two additional arguments that specify pre-sample data. This is data from time periods before the current observation period. The first is a Vector<T> that contains the pre-sample responses. The second is a Vector<T> that contains the pre-sample innovations. Either or both of these can be null. In that case, the values computed during model estimation are used.

Reference