2.2 Measures of Dependence

Measures of Dependence in Time Series

Measures of dependence describe how observations relate across time within a series and, in multivariate settings, across different series. They formalize persistence, predictability, and the internal structure of a stochastic process.

Given a time series $\{X_t\}$, its behavior is characterized by two fundamental quantities: the mean function ${\mu }_t=\mathbb{E}[X_t]$, which specifies the expected level at each time, and the autocovariance function ${\gamma }_X(s,t)=\text{Cov}(X_s,X_t)$, which captures linear dependence across time.

Normalizing autocovariance yields the autocorrelation function ${\rho }_X(s,t)$, which measures dependence on a unit-free scale. The partial autocorrelation function ${\varphi }_{hh}$ strips out the indirect dependence routed through intermediate observations and reports only the direct contribution at each lag, the diagnostic that pairs with the ACF for ARMA model identification. In the multivariate setting, the cross-covariance and cross-correlation functions extend these ideas to pairs of series $\{X_t\}$ and $\{Y_t\}$.