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The fourth moment is $\mathbb{E}[X^4]$. Its finiteness is equivalent to square integrability of $X^2$. For centered variables, dividing it by the squared variance gives kurtosis.

For finite fourth moment and positive variance, kurtosis is $\mathbb{E}[(X-\mathbb{E}X{)}^4]/\mathrm{Var}(X{)}^2$. The normal distribution has kurtosis $3$. A random scale can increase kurtosis even when the conditional distribution is normal.

For ${\alpha }_0>0$ and $0<{\alpha }_2<1$, the model $X_t=\sqrt{{\alpha }_0+{\alpha }_2{X}_{t-2}^2}{\epsilon }_t$ splits into independent even-time and odd-time chains in its stationary causal solution. With standard normal noise, $\mathbb{E}{X}_t^2={\alpha }_0/(1-{\alpha }_2)$, and its fourth moment is finite exactly when $3{\alpha }_2^2<1$.

Two useful features of autoregressive conditional heteroscedasticity are persistent changes in conditional scale and excess unconditional kurtosis. Its time-varying conditional variance can explain volatility clustering, where large absolute returns occur in groups even when signed returns have little autocorrelation. A homoscedastic autoregressive moving-average model has a fixed innovation variance.

Also, a conditional normal distribution with a random scale is a Gaussian scale mixture. Its unconditional kurtosis can exceed $3$, or its fourth moment can be infinite. A Gaussian autoregressive moving-average model remains jointly Gaussian and cannot reproduce this effect. In the particular lag-two ARCH process, persistence of the squared scale occurs within each parity subsequence; the two parity subsequences are independent.

First justify the dependence structure rather than assume that adjacent squares behave like an ordinary lag-one ARCH model. Write $Z_t=X_t^2$. Iterating its nonnegative recurrence givesThe product coefficient has expected value ${\alpha }_2^m$, so it tends to zero in probability. The stationary $Z_{t-2m}$ are a tight family; consequently the remainder tends to zero in probability, without needing independence between that remainder's factors. The increasing partial sums therefore give the representationThis proves the parity decomposition of a lag-two ARCH process: even and odd observations are functions of disjoint noise families. It also shows that the positive scale ${\sigma }_t$ is a function of past noise, independent of ${\epsilon }_t$. The conditional expectation of $X_t$ given past noise is zero, and the second-moment recursion is $m_2={\alpha }_0+{\alpha }_2{m}_2$. independence of the parity families gives the adjacent-square covariance. ThusTo establish finiteness of the fourth moment before using its recursion, note that the $L^2$ norm of the $k$th term in the positive series for $Z_t$ is ${\alpha }_0\sqrt{3}({\alpha }_2\sqrt{3}{)}^k$. The triangle inequality in L2 space makes the series square-integrable when $3{\alpha }_2^2<1$. independence of the current noise and past scale then yieldsand henceIf $3{\alpha }_2^2\ge 1$, a finite $m_4$ would make $(1-3{\alpha }_2^2)m_4=3{\alpha }_0^2+6{\alpha }_0{\alpha }_2{m}_2>0$, which is impossible. Its fourth moment is then infinite. The adjacent-square product remains integrable by parity independence, even in that case.

The useful autoregressive model is for the squares, not for signed observations. Put $Y_t=X_t^2$ andThenThe errors form a martingale difference sequence relative to the noise history, because the current standardized noise is independent of the past. When the fourth moment is finite they have finite variance and are uncorrelated across distinct times, although their conditional variance depends on the regressor. In centered form, $Y_t-m_2={\alpha }_2(Y_{t-2}-m_2)+u_t$.

For ordinary least squares, use the $n-2$ response-regressor pairs $y_t=X_t^2$, $r_t=X_{t-2}^2$, $t=3,\dots ,n$. With their separate means $\bar{y}$ and $\bar{r}$, minimize ${\sum }_{t=3}^n(y_t-a-b{r}_t{)}^2$. If the regressor sum of squares is positive, the estimators areThe two means use the matched pairs; replacing them indiscriminately by a single full-sample mean is not the exact least-squares formula. The noise-series representation supplies an ergodic stationary process. If $3{\alpha }_2^2<1$, finite regressor second moments and the error's zero conditional expectation justify the usual population regression and statistical consistency argument. The observations still define a finite-sample least-squares fit outside that moment range, but the ordinary finite-variance justification must not be claimed there. If parameter constraints are required, minimize the same criterion subject to $a>0$ and $0<b<1$, rather than assert that unconstrained estimates automatically satisfy them.