Volatility clustering and ARCH

When the variance is not constant

Returns are close to uncorrelated, yet they are not independent: the size of today's move tells you about the size of tomorrow's. Quiet weeks follow quiet weeks and turbulent ones follow turbulent ones. The ARCH model of Engle (1982) captures this by letting the variance of the shock depend on recent squared shocks: $r_t=\mu+\varepsilon_t$, $\varepsilon_t=\sigma_tz_t$ with $z_t$ standard normal, and $\sigma_t^{2}=\omega+\alpha_1\varepsilon_{t-1}^{2}+\dots+\alpha_q\varepsilon_{t-q}^{2}$. A large shock yesterday raises the variance today, and the shock itself has no memory in its sign. The unconditional variance of an ARCH(1) is $\omega/(1-\alpha)$ and it needs $\alpha<1$.

Even with normal shocks $z_t$, an ARCH model produces returns with fatter tails than a normal distribution, because a calm stretch and a turbulent stretch are mixed in one sample. This is why return histograms show excess kurtosis. To test for ARCH effects, take the residuals $\hat\varepsilon_t$ of a mean model, regress $\hat\varepsilon_t^{2}$ on its own $q$ lags and compute Engle's statistic $T R^{2}$, which is $\chi^{2}_q$ under the null of no ARCH. A simple alternative that needs no fitting is the exponentially weighted moving average, $\sigma_t^{2}=\lambda\sigma_{t-1}^{2}+(1-\lambda)r_{t-1}^{2}$, with $\lambda=0.94$ as the RiskMetrics default for daily data. Real returns also show a leverage effect: a fall raises the next variance more than a rise of the same size, which the plain ARCH cannot represent. Matching the long persistence of volatility takes many ARCH lags, and GARCH, covered later in this skill, does it with just a few parameters.

• ARCH($q$): the variance of today's shock is a constant plus a weighted sum of the last $q$ squared shocks.
• Engle's test: regress squared residuals on their lags and use $T R^{2}\sim\chi^{2}_q$.
• EWMA with $\lambda=0.94$ is a no-fit alternative; leverage: falls raise volatility more than rises.

Lab · what's graded

• The long-run variance15 pts
• Variance after a shock15 pts
• An EWMA update15 pts
• Engle's statistic15 pts
• The tails of an ARCH model10 pts
• Match the vocabulary15 pts
• Risk right after a shock15 pts