Time Series: AR(1) and Stationarity

Most of this track has assumed independent observations. Prices, spreads, rates and volatility are ordered in time, and today's value usually carries information about tomorrow's. The first tool for measuring that is autocorrelation.

For a series $X_t$ with constant mean $\mu$, the autocovariance at lag $k$ is $\gamma_k = \text{Cov}(X_t, X_{t-k})$, and the autocorrelation function (ACF) is

so $\rho_0 = 1$ and $\rho_k$ is the correlation between the series and a copy of itself shifted $k$ steps. The sample version replaces $\gamma_k$ with $\frac{1}{n}\sum_t (x_t - \bar x)(x_{t-k} - \bar x)$.

How big does a sample autocorrelation have to be before you care? If the series is really white noise (independent, constant variance), each $\hat\rho_k$ is roughly $N(0, 1/n)$. The usual band is $\pm 2/\sqrt{n}$. With 400 daily returns the band is $\pm 0.1$, so a lag-1 autocorrelation of $0.08$ is consistent with no predictability at all.

Daily stock returns typically sit inside that band at every lag. Their squares do not: $r_t^2$ shows positive autocorrelation that decays slowly over weeks and months. Returns can be close to uncorrelated while volatility is highly predictable, which is why "returns are unpredictable" and "volatility clusters" are both true.