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

The type double function is imsls_d_garch.

Pointer to the parameter array x[] of length p + q + 1 containing the estimated values of sigma squared, followed by the q ARCH parameters, and the p GARCH parameters.

The Generalized Autoregressive Conditional Heteroskedastic (GARCH) model for a time series {wt} is defined as

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

The above model is denoted as GARCH(p,q). The βi and αi coefficients will be referred to as GARCH and ARCH coefficients, respectively. When βi = 0, i = 1,2,…, p, the above model reduces to ARCH(q) which was proposed by Engle (1982). The nonnegativity conditions on the parameters imply a nonnegative variance and the condition on the sum of the βi’s and αi’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 {wt} with length m is

Thus log(L) is maximized subject to the constraints on the αi, βi, and σ.

In this model, if q = 0, the GARCH model is singular since the estimated Hessian matrix is singular.

The initial values of the parameter vector x entered in vector xguess must satisfy certain constraints. The first element of xguess refers to σ2 and must be greater than zero and less than max_sigma. The remaining p+q initial values must each be greater than or equal to zero and sum to a value less than one.

To guarantee stationarity in model fitting,

is checked internally. The initial values should selected from values between zero and one.

AIC is computed by

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

Statistical inferences can be performed outside the function GARCH based on the output of the log-likelihood function (a), the Akaike Information Criterion (aic), and the variance-covariance matrix (var).

The data for this example are generated to follow a GARCH(p,q) process by using a random number generation function sgarch. The data set is analyzed and estimates of sigma, the ARCH parameters, and the GARCH parameters are returned. The values of the Log-likelihood function and the Akaike Information Criterion are returned from the optional arguments IMSLS_A and IMSLS_AIC.