pedantic
Autoregressive conditional heteroskedasticity (ARCH) is a statistical model used to analyze volatility in time series in order to forecast future volatility. ARCH modeling shows that periods of high volatility are followed by more high volatility and periods of low volatility are followed by more low volatility.
ARCH models recognized that past financial data influences future data — that is the definition of autoregressive. The conditional heteroskedasticity portion of ARCH simply refers to the observable fact that volatility in financial markets is non-constant.
The ARCH/GARCH model then can find serial correlation in the volatility and capture it by the parametrization of the variance equation.
The best identification tool may be a time series plot of the series. It’s usually easy to spot periods of increased variation sprinkled through the series.
Homoscedastic return (== returns with constant variance) decomposition:
$$ \begin{align*}r_{t} & \equiv \mu_{t-1} + \epsilon_{t}, \quad \epsilon_t :\sim i.i.d.(0,\sigma^2_{\epsilon})\end{align*} $$
Heteroscedastic return (== return with variable variance) decomposition:
$$ \begin{align*}r_t &= \mu_{t-1} + \epsilon_t; \quad \epsilon_t \equiv \sigma_{t-1} \, \times \, \eta_t, \; \eta_t :\sim i.i.d.(0,1) \\\\\mu_{t-1} &:= E [r_t | \mathcal{F}{t-1}] \\\\\sigma^2{t-1} &:= Var(r_t|\mathcal{F}_{t-1}) \end{align*} $$
We think of unexpected return innovations $\epsilon_t$ to be the product of two components, volatility $\sigma_{t-1}$ (expected squared deviations from the mean), and fundamental return shock $\eta_t$ (innovation, White Noise)
Different ARCH and GARCH models do assume a slightly different parametric structure for the calculation of $\sigma_{t-1}$.
$$ \begin{align*} \mu_{t-1} &:=\phi_{0}+\underbrace{\sum_{i=1}^{p}\phi_{i}r_{t-i}+\sum_{i=1}^{q}\theta_{i} \epsilon_{t-i}}{\text{ARMA(p,q)}} \underbrace{+\sum{i=1}^{k} \beta_i x_{it-1}}_{\text{k factor risk premiums}} \end{align*} $$
(k-factor risk premiums are optional)