Why GARCH Forecasts Multiply Volatility by a Signed Innovation

Summary

In an ARMA-GARCH model, the conditional forecast combines the ARMA component with a volatility scale derived from the GARCH variance and a random innovation. The question asks why the formula writes a plus sign instead of explicitly showing both plus and minus outcomes when taking the square root of variance.

The answer is that the innovation is itself a random variable, assumed here to follow a standard normal distribution, so it can take either positive or negative values. Multiplying this signed draw by the nonnegative conditional standard deviation already permits outcomes on both sides of the ARMA component; a separate ± symbol is unnecessary. Without the innovation term, the forecast would be deterministic and would omit modeled randomness. This explanation depends on the stated innovation assumption and is a clarification of notation, not a broader discussion of alternative innovation distributions or model specification.

Key ideas

• The square root of the GARCH variance scales the innovation to produce a conditional shock.
• A standard normal innovation can be positive or negative, so the formula needs no separate plus-or-minus symbol.
• The ARMA component provides the forecast center, while the innovation adds randomness around it.
• The explanation assumes a standard normally distributed innovation.

Tags

# Double sign for the error term in an ARMA-GARCH model


# Double sign for the error term in an ARMA-GARCH model












Why in an ARMA-GARCH model for a stationary series $r$ (without $c$ for simplicity) is $r_{forecast} = ARMA + \sqrt{GARCH} \cdot inn$ and not $r_{forecast} = ARMA \pm \sqrt{GARCH} \cdot inn$? The latter would seem to be even more intuitive. Does the GARCH process'root square not produce the double result $\pm$?

## Answer by Fr1 (score 1, accepted)

https://quant.stackexchange.com/a/41525

That is because the innovation $inn$ is a standard normally distributed random variable and therefore it can actually take positive or negative values. Without the innovation it would be a deterministic model (i.e. no risk). So, de facto it is +/- but mathematically you have to write $+\sqrt{GARCH} \cdot inn$ because $inn$ itself contributes the sign and can be either positive or negative.

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