# 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 $\pm$?

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.