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 $${\sigma }_1^2,{\sigma }_2^2,\dots ,{\sigma }_N^2$$ and a forecast horizon h, generate predictions for $${\sigma }_{N+1}^2,{\sigma }_{N+2}^2,\dots ,{\sigma }_{N+h}^2.$$

Let $${\widehat{\sigma }}_{t+1}^2$$ 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 $${\widehat{\sigma }}_{t+1}^2$$ that minimizes the conditional expected square loss,

$$E({\sigma }_{t+1}^2-{\widehat{\sigma }}_{t+1}^2|H_t).$$

Minimizing this loss function yields the MMSE forecast,

$${\widehat{\sigma }}_{t+1}^2=E({\sigma }_{t+1}^2|H_t)=E({\epsilon }_{t+1}^2|H_t).$$

EGARCH MMSE Forecasts

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

$$\log {\widehat{\sigma }}_{t+1}^2=E(\log {\sigma }_{t+1}^2|H_t).$$

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

$${\widehat{\sigma }}_{t+1}^2=\exp \left\{\log {\widehat{\sigma }}_{t+1}^2\right\}.$$

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

$$E({\sigma }_{t+1}^2)\ge \exp \{E(\log {\sigma }_{t+1}^2)\}.$$

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, either in a numeric array, table or timetable, and you can optionally specify presample conditional variances, either as a numeric array using the V0 name-value argument or a table or timetable using the Presample name-value 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, for example Y, use the last few observations of Y as presample responses, for example 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, for example 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, $${\epsilon }_t={\sigma }_t{z}_t,$$ where

$${\sigma }_t^2=\kappa +{\gamma }_1{\sigma }_{t-1}^2+{\alpha }_1{\epsilon }_{t-1}^2.$$

Given presample innovation $${\epsilon }_T$$ and presample conditional variance $${\sigma }_T^2,$$ forecasts are recursively generated as follows:

• $${\widehat{\sigma }}_{T+1}^2=\kappa +{\gamma }_1{\sigma }_T^2+{\alpha }_1{\epsilon }_T^2$$

• $${\widehat{\sigma }}_{T+2}^2=\kappa +{\gamma }_1{\widehat{\sigma }}_{T+1}^2+{\alpha }_1{\widehat{\sigma }}_{T+1}^2$$

• $${\widehat{\sigma }}_{T+3}^2=\kappa +{\gamma }_1{\widehat{\sigma }}_{T+2}^2+{\alpha }_1{\widehat{\sigma }}_{T+2}^2$$

$$⋮$$

Note that innovations are forecasted using the identity

$$E({\epsilon }_{t+1}^2|H_t)=E({\sigma }_{t+1}^2|H_t)={\widehat{\sigma }}_{t+1}^2.$$

This recursion converges to the unconditional variance of the process,

$${\sigma }_{\epsilon }^2=\frac{\kappa }{\left(1-{\gamma }_1-{\alpha }_1\right)}.$$

GJR Model

The forecast function generates MMSE forecasts for GJR models recursively.

Consider generating forecasts for a GJR(1,1) model, $${\epsilon }_t={\sigma }_t{z}_t,$$ where $${\sigma }_t^2=\kappa +{\gamma }_1{\sigma }_{t-1}^2+{\alpha }_1{\epsilon }_{t-1}^2+{\xi }_{1}I[{\epsilon }_{t-1}<0]{\epsilon }_{t-1}^2.$$ Given presample innovation $${\epsilon }_T$$ and presample conditional variance $${\sigma }_T^2,$$ forecasts are recursively generated as follows:

• $${\widehat{\sigma }}_{T+1}^2=\kappa +{\gamma }_1{\widehat{\sigma }}_T^2+{\alpha }_1{\epsilon }_T^2+{\xi }_{1}I[{\epsilon }_T<0]{\epsilon }_T^2$$

• $${\widehat{\sigma }}_{T+2}^2=\kappa +{\gamma }_1{\widehat{\sigma }}_{T+1}^2+{\alpha }_1{\widehat{\sigma }}_{T+1}^2+\frac{1}{2}{\xi }_1{\widehat{\sigma }}_{T+1}^2$$

• $${\widehat{\sigma }}_{T+3}^2=\kappa +{\gamma }_1{\widehat{\sigma }}_{T+2}^2+{\alpha }_1{\widehat{\sigma }}_{T+2}^2+\frac{1}{2}{\xi }_1{\widehat{\sigma }}_{T+2}^2$$

$$⋮$$

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

$$E({\epsilon }_{t+1}^2|H_t)=E({\sigma }_{t+1}^2|H_t)={\widehat{\sigma }}_{t+1}^2.$$

This recursion converges to the unconditional variance of the process,

$${\sigma }_{\epsilon }^2=\frac{\kappa }{\left(1-{\gamma }_1-{\alpha }_1-\frac{1}{2}{\xi }_1\right)}.$$

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, $${\epsilon }_t={\sigma }_t{z}_t,$$ where

$$\log {\sigma }_t^2=\kappa +{\gamma }_1\log {\sigma }_{t-1}^2+{\alpha }_1\left[\left|\frac{{\epsilon }_{t-1}}{{\sigma }_{t-1}}\right|-E\left\{\left|\frac{{\epsilon }_{t-1}}{{\sigma }_{t-1}}\right|\right\}\right]+{\xi }_1\frac{{\epsilon }_{t-1}}{{\sigma }_{t-1}}.$$

The form of the expected value term depends on the choice of innovation distribution, Gaussian or Student’s t. Given presample innovation $${\epsilon }_T$$ and presample conditional variance $${\sigma }_T^2,$$ forecasts are recursively generated as follows:

• $$\log {\widehat{\sigma }}_{T+1}^2=\kappa +{\gamma }_1\log {\sigma }_T^2+{\alpha }_1\left[\left|\frac{{\epsilon }_T}{{\sigma }_T}\right|-E\left\{\left|\frac{{\epsilon }_T}{{\sigma }_T}\right|\right\}\right]+{\xi }_1\frac{{\epsilon }_T}{{\sigma }_T}$$

• $$\log {\widehat{\sigma }}_{T+2}^2=\kappa +{\gamma }_1\log {\widehat{\sigma }}_{T+1}^2$$

• $$\log {\widehat{\sigma }}_{T+3}^2=\kappa +{\gamma }_1\log {\widehat{\sigma }}_{T+2}^2$$

$$⋮$$

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,

$$\log {\sigma }_{\epsilon }^2=\frac{\kappa }{\left(1-{\gamma }_1\right)}.$$

forecast returns the exponentiated forecasts, $$\exp \left\{\log {\widehat{\sigma }}_{T+1}^2\right\},\exp \left\{\log {\widehat{\sigma }}_{T+2}^2\right\},\dots ,$$ which have limit

$$\exp \left\{\frac{\kappa }{\left(1-{\gamma }_1\right)}\right\}.$$