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A common objective of conditional variance modeling is generating
forecasts for the conditional variance process over a future time
horizon. That is, given the conditional variance process
and a forecast horizon *h*,
generate predictions for

Let
denote a forecast for the variance
at time *t* + 1, conditional on the history of the
process up to time *t*, *H _{t}*.
The minimum mean square error (MMSE) forecast is the forecast
that minimizes the conditional
expected square loss,

Minimizing this loss function yields the MMSE forecast,

For the EGARCH model, the MMSE forecast is found for the log conditional variance,

For conditional variance forecasts
of EGARCH processes, `forecast` returns the exponentiated
MMSE log conditional variance forecast,

This results in a slight forecast bias because of Jensen's inequality,

As an alternative to MMSE forecasting, you can conduct Monte Carlo simulations to forecast EGARCH processes. Monte Carlo simulations yield unbiased forecasts for EGARCH models. However, Monte Carlo forecasts are subject to Monte Carlo error (which you can reduce by increasing the simulation sample size).

The `forecast` method generates MMSE forecasts
recursively. When you call `forecast`, you can specify
presample responses (`Y0`) and presample conditional
variances (`V0`) using name-value arguments. If the
model being forecasted includes a mean offset—signaled by a
nonzero `Offset` property—`forecast` subtracts
the offset term from the presample responses to create presample innovations.

To begin forecasting from the end of an observed series, say `Y`,
use the last few observations of `Y` as presample
responses `Y0` to initialize the forecast. The minimum
number of presample responses needed to initialize forecasting is
stored in the property `Q` of a model object.

When specifying presample conditional variances `V0`,
the minimum number of presample conditional variances needed to initialize
forecasting is stored in the property `P` for GARCH(*P*,*Q*)
and GJR(*P*,*Q*) models. For EGARCH(*P*,*Q*)
models, the minimum number of presample conditional variances needed
to initialize forecasting is max(*P*,*Q*).

Note that for all variance models, if you supply at least max(*P*,*Q*)
+ *P* presample response observations (`Y0`), `forecast` infers
any needed presample conditional variances (`V0`)
for you. If you supply presample observations, but less than max(*P*,*Q*)
+ *P*, `forecast` sets any needed
presample conditional variances equal to the unconditional variance
of the model.

If you do not provide any presample innovations, then for GARCH
and GJR models, `forecast` sets any necessary presample
innovations equal to the unconditional standard deviation of the model.
For EGARCH models, `forecast` sets the presample
innovations equal to zero.

The `forecast` method generates MMSE forecasts
for GARCH models recursively.

Consider generating forecasts for a GARCH(1,1) model, where

Given presample innovation and presample conditional variance forecasts are recursively generated as follows:

Note that innovations are forecasted using the identity

This recursion converges to the unconditional variance of the process,

The `forecast` method generates MMSE forecasts
for GJR models recursively.

Consider generating forecasts for a GJR(1,1) model, where Given presample innovation and presample conditional variance forecasts are recursively generated as follows:

Note that the expected value of the indicator is 1/2 for an innovation process with mean zero, and that innovations are forecasted using the identity

This recursion converges to the unconditional variance of the process,

The `forecast` method generates MMSE forecasts
for EGARCH models recursively. The forecasts are initially generated
for the log conditional variances, and then exponentiated to forecast
the conditional variances. This results in a slight forecast bias.

Consider generating forecasts for an EGARCH(1,1) model, where

The form of the expected value term depends on
the choice of innovation distribution, Gaussian or Student's *t*.
Given presample innovation
and
presample conditional variance
forecasts
are recursively generated as follows:

Notice that future absolute standardized innovations and future innovations are each replaced by their expected value. This means that both the ARCH and leverage terms are zero for all forecasts that are conditional on future innovations. This recursion converges to the unconditional log variance of the process,

`forecast` returns the exponentiated forecasts,
which have limit

`egarch` | `forecast` | `forecast` | `forecast` | `garch` | `gjr`

- Assess the EGARCH Forecast Bias Using Simulations
- Forecast a Conditional Variance Model
- Forecast GJR Models

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