# Weighting schemes - Volatility

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)?

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 $$\beta_j$$.