Daily equity returns show almost no autocorrelation at any lag, while squared returns are strongly autocorrelated out to twenty lags. Explain what single process generates both facts, and say what it implies for forecasting returns against forecasting variance.

Daily equity returns show almost no autocorrelation at any lag, while squared returns are strongly autocorrelated out to twenty lags. Explain what single process generates both facts, and say what it implies for forecasting returns against forecasting variance.

Approach: Write the return as a scale-mixture with a predictable scale and an unpredictable sign, then compute the autocovariance of the level and of the square separately.

Returns are a martingale difference sequence with a persistent conditional variance, so they are serially uncorrelated while their squares inherit the persistence of the variance process. Write r_t = sigma_t*z_t with z_t iid mean zero and unit variance and sigma_t measurable at t - 1. Then E[r_t*r_{t-k}] = E[sigma_t*r_{t-k}*E[z_t | past]] = 0, giving a flat return autocorrelation at every lag. The square is r_t^2 = sigma_t^2*z_t^2, and if sigma_t^2 follows a persistent recursion such as a GARCH with alpha + beta near 0.98, the squares carry that autocorrelation out to many lags. This is volatility clustering, and uncorrelated is a strictly weaker statement than independent. In practice today's large move raises tomorrow's expected variance while saying nothing about tomorrow's direction, so variance is forecastable from the same data that leaves the sign unforecastable.

Follow-up: Given returns with zero autocorrelation and strong squared-return autocorrelation, what does the sample autocorrelation of absolute returns add over that of squared returns?

Key concepts: volatility clustering, conditional variance, martingale difference, GARCH.