Autokorrelations

Autocorrelation, also known as serial correlation, is a statistical measure of the relationship between observations in a time series and the same observations at a different time lag. In time series analysis, autocorrelation at lag k quantifies the linear relationship between X_t and X_{t-k}. The autocorrelation coefficient at lag k is rho_k = Cov(X_t, X_{t-k}) / Var(X_t), assuming finite variance. For stationary processes, rho_0 = 1, and |rho_k| ≤ 1.

For a stochastic process with mean mu and variance sigma^2, the autocovariance function gamma_k = Cov(X_t, X_{t+k}). The autocorrelation function (ACF) is gamma_k / gamma_0. If the process is white noise, all autocorrelations for k ≠ 0 are zero. For ARIMA models, the ACF and the partial autocorrelation function (PACF) have characteristic shapes that help identify model orders and dependence structure.

Estimation of autocorrelation in finite samples yields the sample autocorrelation function (SACF). At lag k, the SACF is computed as the sum over t of (X_t - X_bar)(X_{t-k} - X_bar) divided by the sum of (X_t - X_bar)^2. Confidence intervals for the SACF are often obtained via Bartlett’s method for large samples. Hypothesis tests commonly assess H0: rho_k = 0.

Applications of autocorrelation include model selection, forecasting, and diagnosing time series data. It is used to detect non-random structure, seasonality, and persistence, and to assess stationarity and model adequacy. Interpretation should consider potential non-stationarity, trends, or changing variance, which can affect autocorrelation estimates. See also partial autocorrelation, Ljung-Box test, and stationarity.