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2. Let $X_{t}=\sigma_{t} Z_{t}$ be a $\operatorname{GARCH}(1,1)$, where $\left\{Z_{t}\right\}$ is a sequence of independent standard normal random variables, and
$\sigma_{t}^{2}=\omega+\alpha X_{t-1}^{2}+\beta \sigma_{t-1}^{2},$
where $\omega, \alpha$ and $\beta$ are positive known constants. Assume that $\beta<1$.
2.1. Show that the $h-$ step-ahead forecast $\sigma_{t+h \mid t}^{2}$ of $\sigma_{t}^{2}$ can be written as
$\sigma_{t+h \mid t}^{2}=\omega \sum_{j=0}^{h-2}(\alpha+\beta)^{j}+(\alpha+\beta)^{h-1} \sigma_{t+1}^{2}, \quad \text { for } \quad h>1$
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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

To show the $h$-step-ahead forecast $\sigma_{t+h|t}^{2}$, we start with the given GARCH(1,1) model: $\sigma_{t}^{2}=\omega+\alpha X_{t-1}^{2}+\beta \sigma_{t-1}^{2}$

step 2

We iterate the GARCH(1,1) model for $h$ steps ahead, starting from time $t+1$ to $t+h$

step 3

For $h=2$, we have $\sigma_{t+2|t}^{2}=\omega+\alpha X_{t+1}^{2}+\beta \sigma_{t+1}^{2}$. Since $X_{t+1}^{2}=\sigma_{t+1}^{2}Z_{t+1}^{2}$ and $Z_{t+1}$ is independent of past information, the expected value of $X_{t+1}^{2}$ given information up to time $t$ is $\sigma_{t+1}^{2}$

step 4

Thus, $\sigma_{t+2|t}^{2}=\omega+\alpha \sigma_{t+1}^{2}+\beta \sigma_{t+1}^{2} = \omega+(\alpha+\beta)\sigma_{t+1}^{2}$

step 5

For $h=3$, we have $\sigma_{t+3|t}^{2}=\omega+\alpha X_{t+2}^{2}+\beta \sigma_{t+2}^{2}$. Applying the expectation and using the result from step 4, we get $\sigma_{t+3|t}^{2}=\omega+(\alpha+\beta)\sigma_{t+2|t}^{2}$

step 6

Substituting the expression from step 4 into step 5, we get $\sigma_{t+3|t}^{2}=\omega+(\alpha+\beta)(\omega+(\alpha+\beta)\sigma_{t+1}^{2})$

step 7

Simplifying, we obtain $\sigma_{t+3|t}^{2}=\omega+(\alpha+\beta)\omega+(\alpha+\beta)^{2}\sigma_{t+1}^{2}$

step 8

Continuing this process, we find that for any $h>1$, $\sigma_{t+h|t}^{2}$ can be written as $\sigma_{t+h|t}^{2}=\omega \sum_{j=0}^{h-2}(\alpha+\beta)^{j}+(\alpha+\beta)^{h-1}\sigma_{t+1}^{2}$

Answer

The $h$-step-ahead forecast $\sigma_{t+h|t}^{2}$ of $\sigma_{t}^{2}$ can be written as $\sigma_{t+h|t}^{2}=\omega \sum_{j=0}^{h-2}(\alpha+\beta)^{j}+(\alpha+\beta)^{h-1}\sigma_{t+1}^{2}$, for $h>1$.

Key Concept

Iterating the GARCH(1,1) model

Explanation

The $h$-step-ahead forecast is derived by iteratively applying the GARCH(1,1) model equation, taking expectations, and simplifying the resulting expression.

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