14 Nonlinear Models: ARCH And Volatility
Every model so far has assumed constant variance. Financial series violate this plainly: quiet periods and turbulent periods cluster, so the variance is itself serially dependent even when the series is uncorrelated. ARCH models that directly.
Definition 14.0.1. A process follows an ARCH\((p)\) model if \[X_t = \sigma _t\,\varepsilon _t,\qquad \sigma _t^{2} = \alpha _0 + \sum _{i=1}^{p}\alpha _i X_{t-i}^{2},\] where \(\varepsilon _t\) is independent white noise with unit variance, \(\alpha _0>0\) and \(\alpha _i\geq 0\). The GARCH\((p,q)\) extension adds lagged variances, \[\sigma _t^{2} = \alpha _0 + \sum _{i=1}^{p}\alpha _i X_{t-i}^{2} + \sum _{j=1}^{q}\beta _j \sigma _{t-j}^{2}.\]
Result 14.0.2. For the ARCH\((1)\) model with \(0\leq \alpha _1<1\): \[E(X_t) = 0,\qquad \Var (X_t) = \frac {\alpha _0}{1-\alpha _1}, \qquad \Corr \left (X_t,\,X_{t+\tau }\right ) = 0 \ \text { for } \tau \neq 0,\] but \(X_t^{2}\) is autocorrelated, following an AR\((1)\) in the squares.
Proof. Conditioning on the past, \(E\left (X_t \mid \mathcal {F}_{t-1}\right ) = \sigma _t E(\varepsilon _t) = 0\), so \(E(X_t)=0\) by the tower property and \(X_t\) is uncorrelated with anything earlier — which gives the zero autocorrelation.
For the variance, \(E\left (X_t^{2}\right ) = E\left (\sigma _t^{2}\right ) = \alpha _0 + \alpha _1E\left (X_{t-1}^{2}\right )\). Stationarity makes both expectations equal, so \(E\left (X_t^{2}\right )(1-\alpha _1) = \alpha _0\), giving the stated variance and requiring \(\alpha _1<1\) for it to be finite and positive.
Writing \(\eta _t = X_t^{2}-\sigma _t^{2}\), which has mean zero, the definition rearranges to \[X_t^{2} = \alpha _0 + \alpha _1 X_{t-1}^{2} + \eta _t ,\] an AR\((1)\) for the squared series. □
Remark. The last line is the whole point. The series itself is uncorrelated — it passes every test built on the autocorrelation function — while its squares are strongly correlated. A series can therefore be unpredictable in level and entirely predictable in volatility, which is what makes returns look random while their risk does not.
This is also the practical diagnostic: plot the ACF of the squared residuals after fitting an ARMA model. If it is flat the constant-variance assumption stands; if it decays slowly, an ARCH or GARCH term is needed. Nothing in the ordinary residual ACF would have revealed it.
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