Suppose that the monthly log returns of GE stock, measured in percentages, follows a smooth threshold GARCH \((1,1)\) model. For the sampling period from January 1926 to December 1999, the fitted model is

\[
\begin{aligned}
r_{t} & =1.06+a_{t}, \quad a_{t}=\sigma_{t} \epsilon_{t} \\
\sigma_{t}^{2} & =0.103 a_{t-1}^{2}+0.952 \sigma_{t-1}^{2}+\frac{1}{1+\exp \left(-10 a_{t-1}\right)}\left(4.490-0.193 \sigma_{t-1}^{2}\right)
\end{aligned}
\]

where all of the estimates are highly significant, the coefficient 10 in the exponent is fixed a priori to simplify the estimation, and \(\left\{\epsilon_{t}\right\}\) are iid \(N(0,1)\). Assume that \(a_{888}=16.0\) and \(\sigma_{888}^{2}=50.2\), what is the 1 -step ahead volatility forecast \(\widehat{\sigma}_{888}^{2}(1)\) ? Suppose instead that \(a_{888}=-16.0\), what is the 1-step ahead volatility forecast \(\widehat{\sigma}_{888}^{2}(1) ?\)