Periodic autoregressive model of order one ($X_t={\varphi }_{\nu }{X}_{t-1}+{\sigma }_{\nu }{\epsilon }_t$)

A centered model $X_t={\varphi }_{\nu }{X}_{t-1}+{\sigma }_{\nu }{\epsilon }_t$, with $\nu =t\mathrm{mod}S$, has season-dependent coefficients and white noise innovations. In the nondegenerate case, the causal stability condition is $|{\prod }_{\nu =1}^S{\varphi }_{\nu }|<1$. Individual coefficients can exceed one in modulus while the product remains stable. Its second-order law is a periodically correlated process.

For unit-variance innovations, orthogonality to the past givesConsequently ${\varphi }_{\nu }={\gamma }_{\nu }(1)/V_{\nu -1}$ and ${\sigma }_{\nu }^2=V_{\nu }-{\gamma }_{\nu }(1{)}^2/V_{\nu -1}$. Replacing autocovariance by matched sample estimates yields seasonal regression estimators. The unit innovation variance is necessary to identify the noise scale separately.