In an ARIMA model, what do the stationarity and invertibility conditions require of its roots?

## Roots of the lag polynomials An ARMA model on a series `z` (already differenced, if it needed to be) can be written with the backshift operator `B`, where `B z_t = z_{t-1}`: `phi(B) z_t = theta(B) e_t` with `phi(B) = 1 - phi_1 B - ... - phi_p B^p` and `theta(B)` the analogous polynomial built from the moving-average coefficients. Both are ordinary polynomials in a complex variable, so both have roots, and the location of those roots is what the stationarity and invertibility conditions constrain. In this convention — polynomials in `B`, not in an eigenvalue — the requirement is that **all roots lie strictly outside the unit circle**, meaning modulus greater than one. Some texts factor the model differently and state the mirror-image condition on the reciprocal roots, which must then lie inside. Both say the same thing; state which convention you are using and the interviewer will follow, but mixing them mid-sentence is a genuine error. ### Stationarity, from the AR side For AR(1), `phi(B) = 1 - phi B` has the single root `B = 1/phi`, which sits outside the unit circle exactly when `|phi| < 1`. That is the familiar condition, and it maps onto three regimes: - `|phi| < 1`: shocks decay geometrically, the series reverts toward its mean, the variance is finite and constant, and long-horizon forecasts converge to that mean. - `|phi| = 1`: a unit root. The series is a random walk; shocks are permanent, the variance grows with time, and there is no mean to revert to. - `|phi| > 1`: explosive. Each shock is amplified, and forecasts diverge — essentially never what a real series does. When an estimation returns an autoregressive root sitting essentially on the circle, it is telling you that the series at the differencing level you chose is still not stationary. The model is straining to represent a random-walk-like level with a bounded parameter. ### Invertibility, from the MA side Invertibility is less intuitive but has a clean motivation. An invertible MA process can be rewritten as an infinite autoregression: the current value expressed in terms of past **observations** with geometrically declining weights. That matters because forecasting works from observed history — the shocks `e_t` are not data, they are reconstructed recursively, and that recursion only converges if the moving-average roots lie outside the unit circle. The second motivation is identification. For MA(1), the coefficients `theta` and `1/theta` generate exactly the same autocorrelation function, so the two are indistinguishable from the second-moment structure of the data. Requiring `|theta| < 1` picks one of the pair, and it is the one with the convergent autoregressive representation. Without the restriction the parameter simply is not identified. A moving-average estimate that lands exactly at the boundary — modulus one — deserves attention rather than a shrug. It means the fitted process cannot be inverted, the likelihood surface is flat against a wall at that point, and the reported standard error is not trustworthy. The usual reading is that the model has been asked to undo a transformation that was applied one time too many, and the specification should be reconsidered rather than accepted. ### Near-common roots: the over-parameterisation signal A distinct pathology appears when the autoregressive and moving-average polynomials share, or nearly share, a root. Because the model is `phi(B) z_t = theta(B) e_t`, a common factor cancels from both sides, and what looked like an ARMA(1,1) is really white noise, or an ARMA(2,2) is really an ARMA(1,1). Estimation on real data will not cancel it exactly; instead you see the symptoms: - coefficients with standard errors as large as, or larger than, the estimates themselves; - estimates that swing wildly when a few observations are added or removed; - an optimiser that converges slowly, to different points from different starting values, because the likelihood is nearly flat along the direction where the factors trade off; - roots of the two polynomials that, when you compute them, sit almost on top of each other. The fix is to reduce the order. The data is telling you the extra pair of terms is buying nothing, and a leaner model will produce steadier coefficients and better-behaved forecasts. ### Why this matters in practice The conditions are not academic hygiene. Stationarity of the autoregressive part is what makes long-horizon forecasts and their intervals meaningful; invertibility is what makes the forecast recursion converge from observed data; and the absence of near-common roots is what makes the estimated parameters mean anything at all. Checking the fitted roots takes a moment and catches a whole family of silent failures that a fit statistic will happily hide.