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One extension to this weighting scheme is to assume a long-run variance level in addition to weighted squared return observations. The most frequently used model is an autoregressive conditional heteroskedasticity model, ARCH.

what is the long-run variance level in weighting schemes(ARCH)?

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Consider a GARCH($p,q$) model for the conditional variance of $(X_t)$ with $$\sigma_t^2=\omega+\sum_{j=1}^p \alpha_j X_{t-j}^2+\sum_{j=1}^q\beta_j\sigma_{t-j}^2,$$ where $\omega>0$ and $\alpha_j,\beta_j\geq0$ for all $j$. Then, the long-term average volatility level is given by $$\mathbb{V}\mathrm{ar}[X_t]=\frac{\omega}{1-\sum\limits_{j=1}^p \alpha_j -\sum\limits_{j=1}^q\beta_j}.$$ For positivity, we assume $\sum\limits_{j=1}^p \alpha_j +\sum\limits_{j=1}^q\beta_j<1$. Furthermore, $\lim\limits_{h\to\infty}\mathbb{E}[X^2_{t+h} \mid \mathcal{F}_t] = \mathbb{V}\mathrm{ar}[X_t]$. Thus, the effect of the given data vanishes when predicting the process and the process converges eventually to its stationary distribution. To sum up, ``GARCH models are mean reverting and conditionally heteroskedastic, but have a constant unconditional variance'' Engle (2001).

In the case of an ARCH($p$)-model, you can simply set $q=0$ and ignore the sums involving $\beta_j$.

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