Calculating the Fourth Moment of an ARCH(2) Process

Summary

The document derives the conditional variance specification for an ARCH(2) process with normally distributed innovations and sets up the calculation of its unconditional fourth moment. Expanding the squared conditional variance produces terms involving the fourth moments of lagged observations, the unconditional second moment, and a cross-moment between adjacent squared observations. The author substitutes the stationary variance expression for the second-moment terms, then asks how to handle the remaining cross-moment.

The key analytical issue is that stationarity makes the marginal moments at different dates equal, but it does not make the product of squared observations independent or equal to a product of their expectations. The cross-moment therefore requires its own derivation, typically using the process’s conditional structure and a stationarity relation. The document states the setup and assumptions but does not supply that derivation or the resulting fourth-moment condition, so it serves as a focused moment-calculation question rather than a complete ARCH(2) solution.

Key ideas

• The ARCH(2) fourth moment involves both lagged fourth moments and a cross-moment of squared observations.
• Stationarity equates marginal moments across time but does not determine the cross-moment by itself.
• The cross-moment must be derived using the dependence structure of the ARCH process.
• The document assumes normal innovations and finite variance and fourth moments but leaves the calculation unresolved.

Tags

# Fourth moment of ARCH(2)


# Fourth moment of ARCH(2)












I am studying the ARCH(2) process given by

$$X_t = \sqrt{h_t} \varepsilon_t$$

where

$$h_t = \alpha_0 + \alpha_1 X_{t-1} ^2 + \alpha_2 X_{t-2} ^2$$

and $\varepsilon_t$ follows $N(0,1)$. Assuming the process is stationary both in variance and the fourth moment, such that

$$E(X_t ^2) = E(X_{t-1}^2) = E(X_{t-2} ^2),$$

how can I work on the term $E( X_{t-1} ^2 X_{t-2} ^2)$ in the $E(X_t ^4)$ expression?

What I have done so far is

$$ \begin{align*} E(X_t ^4 ) &= E (E(X_t ^4 | \mathcal{F}_{t-1}))= E(\varepsilon_t ^4 | \mathcal{F}_{t-1}) E [(\alpha_0 + \alpha_1 X_{t-1} ^2 + \alpha_2 X_{t-2} ^2 ) ^2]\\ &= E(\varepsilon_t ^4 | \mathcal{F}_{t-1}) E[(\alpha_0 ^2 + \alpha_1 ^2 X_{t-1} ^4 + \alpha_2 ^2 X_{t-2} ^4 + 2 \alpha_0 \alpha_1 X_{t-1} ^2 + 2 \alpha_0 \alpha_2 X_{t-2} ^2 \\ &+ 2 \alpha_1 \alpha_2 X_{t-1} ^2 X_{t-2} ^2 ] \\ &= E(\varepsilon_t ^4 | \mathcal{F}_{t-1}) (\alpha_0 ^2 + \alpha_1 ^2 E(X_{t-1} ^4 )+ \alpha_2 ^2 E(X_{t-2} ^4) + 2 \alpha_0 \alpha_1 E(X_{t-1} ^2) + 2 \alpha_0 \alpha_2 E(X_{t-2} ^2) \\ & + 2 \alpha_1 \alpha_2E( X_{t-1} ^2 X_{t-2} ^2)) \\ &= E(\varepsilon_t ^4 | \mathcal{F}_{t-1}) (\alpha_0 ^2 + \alpha_1 ^2 E(X_{t-1} ^4 )+ \alpha_2 ^2 E(X_{t-2} ^4)) \\ & \left(+ 2 \alpha_0 \alpha_1 \frac{\alpha_0}{1- \alpha_1 - \alpha_2 } + 2 \alpha_0 \alpha_2 \frac{\alpha_0}{1- \alpha_1 - \alpha_2 } + 2 \alpha_1 \alpha_2E( X_{t-1} ^2 X_{t-2} ^2)\right) \end{align*}$$

since $E(X_t ^2 ) = \frac{\alpha_0}{1- \alpha_1 - \alpha_2}$.

How do I treat the treat $E(X_{t-1} ^2 X_{t-2} ^2)$?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.