Suppose that the monthly log returns of GE stock, measured in percentages, follow a smooth threshold IGARCH

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Suppose that the monthly log returns of GE stock, measured in percentages, follow a smooth threshold IGARCH \((1,1)\) model. For the sampling period from January 1926 to December 2008, the fitted model is

\[ \begin{aligned} r_{t} & =1.14+a_{t}, \quad a_{t}=\sigma_{t} \epsilon_{t} \\ \sigma_{t}^{2} & =0.119 a_{t-1}^{2}+0.881 \sigma_{t-1}^{2}+\frac{1}{1+\exp \left(-10 a_{t-1}\right)}\left(4.276-0.084 \sigma_{t-1}^{2}\right) \end{aligned} \]

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 \(a_{996}=-5.06\) and \(\sigma_{996}^{2}=50.5\). What is the 1-step-ahead volatility forecast \(\widehat{\sigma}_{996}(1)\) ? Suppose instead that \(a_{996}=5.06\). What is the 1 -step-ahead volatility forecast \(\widehat{\sigma}_{996}(1)\) ?

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