0.1 ARIMA(p,d,q) Processes
Many series are non-stationary (trending, no fixed mean level) yet become stationary after differencing. Denote the differencing operator \(\nabla = 1-B\) and its \(d\)-fold application \[ \nabla^d X_t = (1-B)^d X_t. \]
\(\{X_t\}\) follows an integrated ARMA process, \(ARIMA(p,d,q)\), if \(W_t := \nabla^d X_t\) is a stationary, causal ARMA(p,q) process. Equivalently, \[ \phi(B)(1-B)^d X_t = \theta(B)Z_t. \] \(d\) is the order of integration, often written \(X_t \sim I(d)\).
Example. A quadratic trend plus stationary noise, \(X_t = bt^2 + S_t\), is non-stationary, but \(\nabla^2 X_t = 2b + (S_t - 2S_{t-1}+S_{t-2})\) is stationary — this is the general mechanism: polynomial trends of degree \(d\) are eliminated by \(d\)-th order differencing.
In practice, the order \(d\) is chosen empirically (by inspecting the series/ACF for trend, and confirmed by Unit Root Tests such as the Augmented Dickey-Fuller or KPSS test) before an \(ARMA(p,q)\) model is fit to \(\nabla^d X_t\).
Related generalizations: - SARIMA(p,d,q)(P,D,Q)_s — adds seasonal differencing/ARMA structure. - ARFIMA(p,d,q) — allows fractional \(d \in (-0.5, 0.5)\), capturing long-range dependence without full unit-root non-stationarity. - Cointegration — the multivariate analogue, where individual \(I(1)\) series combine into a stationary linear combination.
0.2 References
- Time Series Analysis
- [[pstat274-time-series-summary-notes]]