0.1 Causality and Invertibility
An \(ARMA(p,q)\) process \(\phi(B)X_t = \theta(B)Z_t\) is causal (a causal function of \(\{Z_t\}\)) if there exist coefficients \(\{\psi_j\}\) with \(\sum_{j=0}^\infty |\psi_j| < \infty\) such that \[ X_t = \sum_{j=0}^\infty \psi_j Z_{t-j} = \psi(B)Z_t. \] Causality means \(X_t\) depends only on present and past shocks, never future ones — required for the process to be interpretable as being generated over time and hence for real-time forecasting.
The process is invertible if there exist \(\{\pi_j\}\) with \(\sum_{j=0}^\infty|\pi_j|<\infty\) such that \[ Z_t = \sum_{j=0}^\infty \pi_j X_{t-j} = \pi(B)X_t. \] Invertibility means the driving noise can be recovered from (an infinite history of) the observed series — required for consistent one-step-ahead prediction and residual-based estimation.
For \(\phi(B)X_t=\theta(B)Z_t\) with \(\phi,\theta\) sharing no common zeroes: - The process is causal iff \(\phi(z) \neq 0\) for all \(|z|\leq 1\) (roots of \(\phi\) outside the unit circle). - The process is invertible iff \(\theta(z)\neq 0\) for all \(|z|\leq 1\) (roots of \(\theta\) outside the unit circle).
Consequences: - \(MA(q)\) processes are always causal but only invertible when \(\theta(z)\)’s roots lie outside the unit circle (see Moving Average Processes). - \(AR(p)\) processes are always invertible but only causal (stationary) when \(\phi(z)\)’s roots lie outside the unit circle (see Autoregressive Processes). - Any non-invertible \(MA\) model has an observationally equivalent invertible representation with the same ACF (obtained by flipping roots inside the unit circle to their reciprocals), which is why estimation routines always return the invertible solution.