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Consider the following GARCH(1,1) model for the volatility of asset returns ๐‘Ÿ ๐‘ก : ๐‘Ÿ ๐‘ก = ๐›ผ + ๐›ฝ ๐‘Ÿ ๐‘ก โˆ’ 1 + ๐œ€ ๐‘ก ๐œ€ ๐‘ก = โ„Ž ๐‘ก ๐‘ข ๐‘ก โ„Ž ๐‘ก = ๐œ‡ + ๐›ฟ โ„Ž ๐‘ก โˆ’ 1 + ๐œ™ ๐œ€ ๐‘ก โˆ’ 1 2 ๐”ผ ๐‘ก โˆ’ 1 ( ๐‘ข ๐‘ก ) = 0 ๐”ผ ๐‘ก โˆ’ 1 ( ๐‘ข ๐‘ก 2 ) = 1 You estimated the following values for the parameters Estimates Parameters ๐›ผ ๐›ฝ ๐œ‡ ๐›ฟ ๐œ™ Estimates 0.111 0.8122 0.0011 0.9321 0.0511 Assume that the last 2 observations of the return process are ๐‘Ÿ ๐‘‡ = 0.27 and ๐‘Ÿ ๐‘‡ โˆ’ 1 = 0.02 , and the value of the conditional variance in the last period of your sample is โ„Ž ๐‘‡ = 0.75 . Then what is the predicted value of the conditional variance โ„Ž ๐‘‡ + 1 in period ๐‘‡ + 1 ?

Options
A.โ„Ž ฬ‚ ๐‘‡ + 1 = 0.0729
B.โ„Ž ฬ‚ ๐‘‡ + 1 = 0.701216
C.There is not enough data to compute โ„Ž ฬ‚ ๐‘‡ + 1 .
D.โ„Ž ฬ‚ ๐‘‡ + 1 = 0.519615
E.โ„Ž ฬ‚ ๐‘‡ + 1 = 0.866025
F.โ„Ž ฬ‚ ๐‘‡ + 1 = 0.75
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Step-by-Step Analysis
To tackle this prediction problem, Iโ€™ll walk through how the next periodโ€™s conditional variance h_{T+1} is formed in a GARCH(1,1) setting as described by the given parameter estimates and the observed data. First, identify the components you need for the forecast. In a typical GARCH(1,1) framework adapted to the notation in the prompt, the next-period conditional variance is computed as: h_{T+1} = ฮผ + ฮด h_T + ฯ† ฮต_T^2, where: - ฮผ is the intercept term, - ฮด is the coefficient on the lagged variance h_T, - ฯ† is the coefficient on the squared shock ฮต_T^2, and - ฮต_T is th......Login to view full explanation

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