autocovariances

Autocovariances are a set of quantities that quantify how a time series covaries with itself at different time lags. For a stochastic process {X_t} with mean mu, the autocovariance at lag h is defined as gamma(h) = Cov(X_t, X_{t+h}) = E[(X_t - mu)(X_{t+h} - mu)].

If the process is weakly stationary, gamma(h) does not depend on the time t and satisfies gamma(-h) = gamma(h) for all h, with gamma(0) = Var(X_t). The function gamma(h) captures the dependence structure of the series over time. The autocorrelation function, which standardizes autocovariances, is rho(h) = gamma(h) / gamma(0), provided gamma(0) > 0.

Estimating autocovariances from data typically uses a sample version. Given a sample x1, x2, ..., xn with sample mean x_bar, the sample autocovariance at lag h is often written as gamma_hat(h) = (1/(n - h)) sum_{t=1}^{n-h} (x_t - x_bar)(x_{t+h} - x_bar) (conventions vary, with some using n in the denominator). The sample autocovariance gives an empirical view of how observations separated by h steps co-vary.

Special cases and properties illustrate their use. For white noise with variance sigma^2, gamma(h) = 0 for h ≠ 0. For an AR(1) process X_t = phi X_{t-1} + e_t with |phi|<1, gamma(h) = phi^{|h|} gamma(0). Autocovariances form the basis of many analyses, including model identification, forecasting, and spectral density estimation, where the spectral density is the Fourier transform of gamma(h).