Long-Run Variance in ARCH and GARCH Models

Summary

This note explains how a GARCH model combines a baseline variance level with lagged squared observations and past conditional variances. When the coefficients satisfy the stated stationarity condition, the unconditional variance is the baseline term divided by one minus the sum of the ARCH and GARCH coefficients. For an ARCH model, the formula reduces by omitting the lagged conditional-variance terms.

The note also describes the long-horizon forecast: expected future squared returns approach the unconditional variance as the forecast horizon grows, so the influence of current data fades. This provides a concise account of mean reversion in conditional variance. The result depends on the coefficient restriction and the model assumptions; it does not establish that a fitted ARCH or GARCH model is suitable for any particular market or return series.

Key ideas

• The GARCH unconditional variance equals the baseline variance divided by one minus the sum of variance persistence coefficients.
• The stationarity condition requires the combined ARCH and GARCH coefficients to sum to less than one.
• An ARCH model uses the same expression without lagged conditional-variance terms.
• Long-horizon expected squared returns converge to the unconditional variance under the stated model conditions.

Tags

# Weighting schemes - Volatility


# 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)?

## Answer by Kevin (score 0, accepted)

https://quant.stackexchange.com/a/48636

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 $\beta_j$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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