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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 log-likelihood objective function.
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 Specifying Presample Data.
If you do not specify presample data, garchfit automatically generates the required presample data.
For conditional mean models with an autoregressive component, garchfit assigns the R required presample observations (PreSeries) of yt, the sample mean of Series. For models with a moving-average component, it sets the M required presample observations (PreInnovations) of εt 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 Presample Data and Transient Effects.
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 [22] and Bollerslev [6].
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).
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
to zero and
to the mean absolute deviation.
This has the effect of setting all Q presample
ARCH and leverage terms to zero.
| Initial Parameter Estimates | Termination Criteria and Optimization Results | |
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