0.1 White Noise
A time series \(\{Z_t\}\) is white noise, denoted \(Z_t \sim WN(0,\sigma^2)\), if 1. \(\mathbb{E}Z_t = 0\) for all \(t\), 2. \(\mathbb{E}Z_t^2 = \sigma^2 < \infty\) for all \(t\), 3. \(\gamma_Z(h) = Cov(Z_t, Z_{t+h}) = 0\) for all \(h \neq 0\).
If in addition the \(\{Z_t\}\) are independent and identically distributed we write \(Z_t \sim IID(0,\sigma^2)\); if jointly Gaussian, \(Z_t \sim GWN(0,\sigma^2)\). White noise is the fundamental “innovation” or “shock” process from which MA, AR and ARMA models are built.
Every white noise process is weakly stationary with mean \(0\) and constant (“white”) spectral density \(f(u) = \sigma^2/2\pi\) across all frequencies — see Spectral Distributions.