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MMSE Forecasting of Conditional Variance Models

What Are MMSE Forecasts?

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 σ12,σ22,,σN2 and a forecast horizon h, generate predictions for σN+12,σN+22,,σN+h2.

Let σ^t+12 denote a forecast for the variance at time t + 1, conditional on the history of the process up to time t, Ht. The minimum mean square error (MMSE) forecast is the forecast σ^t+12 that minimizes the conditional expected square loss,

E(σt+12σ^t+12|Ht).

Minimizing this loss function yields the MMSE forecast,

σ^t+12=E(σt+12|Ht)=E(εt+12|Ht).

EGARCH MMSE Forecasts

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

logσ^t+12=E(logσt+12|Ht).

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

σ^t+12=exp{logσ^t+12}.

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

E(σt+12)exp{E(logσt+12)}.

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).

How forecast Generates MMSE Forecasts

The forecast function generates MMSE forecasts recursively. When you call forecast, you must specify presample responses Y0, and you can optionally specify presample conditional variances V0 using the 'V0' name-value pair argument. 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.

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.

GARCH Model

The forecast function generates MMSE forecasts for GARCH models recursively.

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

σt2=κ+γ1σt12+α1εt12.

Given presample innovation εT and presample conditional variance σT2, forecasts are recursively generated as follows:

  • σ^T+12=κ+γ1σT2+α1εT2

  • σ^T+22=κ+γ1σ^T+12+α1σ^T+12

  • σ^T+32=κ+γ1σ^T+22+α1σ^T+22

Note that innovations are forecasted using the identity

E(εt+12|Ht)=E(σt+12|Ht)=σ^t+12.

This recursion converges to the unconditional variance of the process,

σε2=κ(1γ1α1).

GJR Model

The forecast function generates MMSE forecasts for GJR models recursively.

Consider generating forecasts for a GJR(1,1) model, εt=σtzt, where σt2=κ+γ1σt12+α1εt12+ξ1I[εt1<0]εt12. Given presample innovation εT and presample conditional variance σT2, forecasts are recursively generated as follows:

  • σ^T+12=κ+γ1σ^T2+α1εT2+ξ1I[εT<0]εT2

  • σ^T+22=κ+γ1σ^T+12+α1σ^T+12+12ξ1σ^T+12

  • σ^T+32=κ+γ1σ^T+22+α1σ^T+22+12ξ1σ^T+22

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

E(εt+12|Ht)=E(σt+12|Ht)=σ^t+12.

This recursion converges to the unconditional variance of the process,

σε2=κ(1γ1α112ξ1).

EGARCH Model

The forecast function 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, εt=σtzt, where

logσt2=κ+γ1logσt12+α1[|εt1σt1|E{|εt1σt1|}]+ξ1εt1σt1.

The form of the expected value term depends on the choice of innovation distribution, Gaussian or Student’s t. Given presample innovation εT and presample conditional variance σT2, forecasts are recursively generated as follows:

  • logσ^T+12=κ+γ1logσT2+α1[|εTσT|E{|εTσT|}]+ξ1εTσT

  • logσ^T+22=κ+γ1logσ^T+12

  • logσ^T+32=κ+γ1logσ^T+22

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,

logσε2=κ(1γ1).

forecast returns the exponentiated forecasts, exp{logσ^T+12},exp{logσ^T+22},, which have limit

exp{κ(1γ1)}.

See Also

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