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Use the usual normalization that ${\epsilon }_t$ is mean-zero unit-variance white noise orthogonal to the past, and extend $\varphi (\nu ),\sigma (\nu )$ periodically in $\nu$. A nondegenerate causal periodic autoregressive model of order one requires $|{\prod }_{\nu =1}^T\varphi (\nu )|<1$. Write $V_{\nu }={\gamma }_{\nu }(0)$, with season indices understood modulo $T$.

For $h\ge 1$, multiply the recursion by $X_{nT+\nu -h}$ and use orthogonality of the current innovation to the past. The periodic Yule-Walker equations areAt lag zero, expansion of the squared recursion givesIn particular, ${\gamma }_{\nu }(1)=\varphi (\nu )V_{\nu -1}$, so an equivalent variance equation is $V_{\nu }=\varphi (\nu ){\gamma }_{\nu }(1)+\sigma (\nu {)}^2$. Negative lags are obtained from the given symmetry ${\gamma }_{\nu }(-h)={\gamma }_{\nu +h}(h)$.

Replacing covariances by their sample versions givesTake the nonnegative scale and require a positive estimated predecessor variance. Using matched seasonal sample covariance matrices makes the quantity under the square root nonnegative by Cauchy-Schwarz inequality; with unmatched covariance estimates one can instead fit seasonal ordinary least squares and use the nonnegative residual variance. Statistical consistency of the covariance estimates gives statistical consistency of these plug-in estimators. If the driving noise variance were not normalized to $1$, the equations would identify $\sigma (\nu {)}^2\mathrm{Var}({\epsilon }_t)$, not the two scales separately.