Statistical Analysis & Time Series

Weak (covariance) stationarity

A process {Xₜ} is weakly stationary if (1) E[Xₜ] is constant, (2) Var(Xₜ) is finite and constant, and (3) Cov(Xₜ, Xₜ₊ₖ)

Strict stationarity

The joint distribution of (X_(t₁),…,X_(tₖ)) equals that of (X_(t₁+h),…,X_(tₖ+h)) for all h, k, and time indices. Strict

Autocorrelation function (ACF)

ρ(k) = Cov(Xₜ, Xₜ₊ₖ) / Var(Xₜ). Measures linear dependence between observations separated by lag k. Always satisfies |ρ(

Partial autocorrelation function (PACF)

PACF at lag k is the correlation between Xₜ and Xₜ₊ₖ after removing the linear effect of the intermediate lags Xₜ₊₁,…,Xₜ

ACF/PACF identification of AR and MA

AR(p): ACF tails off (decays); PACF cuts off after lag p. MA(q): ACF cuts off after lag q; PACF tails off. ARMA(p,q): bo