> [!tldr] > **Autoregressive Conditional Heteroscedastic (ARCH) models** are a type of volatility models, i.e. models for the residual term $(Y_{t})$ under a grander model for a time series $(X_{t})$: $X_{t}=\mathbb{E}[X_{t} ~|~ \mathcal{F}_{t-1}]+Y_{t}.$ > It has the form $\mathrm{Var}(X_{t})=\mathrm{Var}(Y_{t}):=\sigma^{2}_{t}=\omega+\sum_{i=1}^{k}\alpha_{i}Y^{2}_{t-i},$i.e. the weighted average of past observations (squared), within a window of size $k$. For stability, we assume $\sum_{i}a_{i}<1$, and $\omega,a_{i}>0, \forall i$. With this model, we can model $(Y_{t})$ to take a variety of distributions (each scaled by $\sigma_{t}$), not just the standard Gaussian errors: for example, a standardized t-distribution will allow the model to take heavier tails than a Gaussian error. The unconditional variance of $Y_{t}$ is $\sigma^{2}_{t}=\frac{\omega}{1-\sum_{i}\alpha_{i}}.$