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Presample Data

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Calculating Presample Data Using the garchfit Function to Generate User-Specified Presample Observations Automatically Generating Presample Observations

Calculating Presample Data

This section shows how garchfit automatically generates presample data for the conditional mean model and each of the supported conditional variance models. It also shows how to specify your own presample data. Maximum Likelihood Estimation discusses presample data required to initiate inverse filtering and evaluate the conditional loglikelihood objective function.

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Using the garchfit Function to Generate User-Specified Presample Observations

Use the time series column vector inputs PreInnovations, PreSigmas, and PreSeries to explicitly specify all required presample data. The following table summarizes the minimum number of rows required to successfully initiate the optimization process.

Garchfit Input Argument	Minimum Number of Rows GARCH(P,Q), GJR(P,Q)	EGARCH(P,Q)
PreInnovations	max(M,Q)	max(M,Q)
PreSigmas	P	max(P,Q)
PreSeries	R	R

If you specify at least one, but fewer than three, sets of presample data, garchfit does not attempt to derive presample observations for those you omit. When specifying your own presample data, include all data required for the given conditional mean and variance models. See the example Presample Data.

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Automatically Generating Presample Observations

If you do not specify presample data, garchfit automatically generates the required presample data.

Automatically Generating Presample Observations for Conditional Mean Models

For conditional mean models with an autoregressive component, garchfit assigns the R required presample observations (PreSeries) of y_t, the sample mean of Series. For models with a moving-average component, it sets the M required presample observations (PreInnovations) of ε_t to their expected value of zero. With this presample data, garchfit can infer the entire sequence of innovations for any general ARMAX conditional mean model, regardless of the conditional variance model you select.

garchfit attempts to eliminate the effect of transients in the presample data it generates. This effect parallels that in the simulation process described in Automatically Generating Presample Data. For an example of transient effects in the estimation process, see Inferring Residuals.

Automatically Generating Presample Observations for GARCH(P,Q) Models

Once garchfit computes the innovations, it assigns the sample mean of the squared innovations

to the required P and Q presample observations of and , respectively. See Hamilton [31] and Bollerslev [6].

Automatically Generating Presample Observations for GJR(P,Q) Models

garchfit also assigns the average squared innovation to all required presample observations of and . In addition, garchfit weights the Q presample observations of associated with the leverage terms by 0.5 (that is, the probability of a negative past residual).

Automatically Generating Presample Observations for EGARCH(P,Q) Models

garchfit also assigns the average squared innovation to all P presample observations of . In addition, it sets all Q presample observations of the standardized innovations z_t = ε_t/σ_t to zero and |z_t| = |ε_t|/σ_t to the mean absolute deviation. This has the effect of setting all Q presample ARCH and leverage terms to zero.

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Estimating Initial Parameters

Optimization Termination

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