ARMA processes, causality and invertibility

Let B shift a series backward by one time step. We write phi(B)(X_t−mu)=theta(B)epsilon_t, where phi(z)=1 minus the sum of autoregressive coefficients times powers of z and theta(z)=1 plus the sum of moving-average coefficients times powers of z. Noise has mean zero, positive finite variance and zero autocovariance at nonzero lags. Independence or Gaussianity is an additional assumption, not part of weak white noise by definition.

The constant mu is the stationary mean. An uncentered AR equation may instead include an intercept c=phi(1)mu when that quantity is nonzero. Thus in an AR(1), c=(1−phi)mu. A coefficient near one can make a modest intercept correspond to a large mean. Estimating the mean, conditioning on an initial value and assuming a stationary distribution are distinct choices that affect the likelihood and forecasts built from the model.

For coprime polynomials, a causal stationary ARMA solution with absolutely summable impulse coefficients exists exactly when every zero of phi lies outside the closed unit disk. Expanding theta divided by phi then gives a present-and-past noise representation. Invertibility of that representation requires every zero of theta outside the closed unit disk, allowing the inverse expansion to recover current noise from present and past centered observations. Roots on the unit circle require separate analysis and are excluded from these stable conditions.

A stationary solution need not be causal. An AR(1) coefficient with absolute value greater than one can have a stationary solution expressed using future noise when no unit-circle root occurs. That is not the forward innovation model usually intended for forecasting. Likewise, a finite moving average is stationary for any finite coefficients, even when it is not invertible. Calling every noninvertible MA process nonstationary would confuse recoverability of shocks with existence of second moments and time-invariant covariance.

For an AR(1), the polynomial root is one over its coefficient, so root modulus greater than one is equivalent to coefficient modulus less than one. For an MA(1) under our plus-sign convention, its root is minus one over theta, making invertibility equivalent to absolute theta less than one. The signs differ, but both conditions concern polynomial roots rather than a generic rule that coefficients must be positive.

Equivalent second-order descriptions can arise from noninvertible moving averages with adjusted noise variance. Choosing an invertible representation identifies a canonical innovation direction, but numerical root constraints do not prove that the fitted order or Gaussian distribution is correct. Residual dependence, changing parameters and omitted deterministic terms still need attention. A fitted equation should report its mean convention and initialization along with the coefficients so another analysis can reproduce its predictions.