Admissibility of lagged state space models through the ARIMA form of the whole model · C26

D5 PBH reference: The largest modulus among the eigenvalues $\lambda$ of the expanded discount matrix for which the smallest singular values of $[\lambda I - \tilde D;\ \tilde w']$ and of $[\lambda I - \tilde D,\ \tilde g]$ both exceed $10^{-7}(\|\tilde D\|_2 + \|\tilde g\| + \|\tilde w\|)$ (Popov-Belevitch-Hautus rank tests), computed with numpy as pbh_max().

D6 cyclotomic cancellation: The procedure that computes the coefficients of $$\begin{aligned} \theta(B) &= \det(I - D L(B)) \\ \phi(B) &= \det(I - F L(B)) \end{aligned}$$ by an inverse discrete Fourier transform of their values at the $N$-th roots of unity ($N$ the smallest power of 2 above $\sum_j l_j$), and divides both by every cyclotomic polynomial $\Phi_n(B)$, $n$ dividing one of the lags, for as long as it divides both with a remainder at most $10^{-9}$ times the sum of absolute coefficients; the quotient of $\theta$ is the cancelled MA polynomial $\theta^*(B)$.

D7 step-down test: The Levinson-Durbin step-down (Schur-Cohn) recursion on $1 + c_1 B + \dots + c_n B^n$: for $j = n, \dots, 1$, $$\kappa_j = c_j, \qquad c_i \leftarrow \frac{c_i - \kappa_j c_{j-i}}{1 - \kappa_j^2};$$ the test passes when every $|\kappa_j| < 1$, which holds if and only if all roots lie outside the unit circle (the $\kappa_j$ are the partial autocorrelations of the autoregression with this polynomial). Source: Barndorff-Nielsen and Schou (1973).

D10 check of the whole model: Admissibility of a lagged SSOE model from the MA polynomial $$\theta(B) = \det(I - D L(B))$$ of its ARIMA form: $\theta$ is divided by its common factors with $\phi(B) = \det(I - F L(B))$ by cyclotomic cancellation, and the model is accepted when the cancelled polynomial $\theta^*$ has no root in $|B| \le 1$ and, for a model with ARIMA components, the AR polynomial of its ARIMA part is stationary; its largest mode modulus is $1/\min|B|$ over the roots of $\theta^*$. In the numpy prototype (admissibility.py, confirm.py, ets_arima.py) the roots come from numpy; in smooth the step-down test decides and bisection gives the modulus.

D39 exact gcd oracle: Admissibility by rational determinants and the exact greatest common divisor of $\theta$ and $\phi$, which cancels any shared factor (code/checker/exact_oracle.py of the checker P4).

D22 random ETS+ARIMA models: Random ADAM models combining additive ETS with ARIMA (ets_arima.py), 500 with ARIMA at lag 1 and 500 with seasonal ARIMA per seed (seed 2 for plans PL7 and PL8, seed 3 for plan PL9): ETS with a level; no, additive or damped trend; no or one seasonal lag $m$ in {4, 12}; smoothing parameters drawn from wide ranges, some fixed at 0; ARIMA with a stationary AR polynomial, differences and MA up to order 2 at lag 1 and 1 at lag $m$, 15% over-differenced (an MA root at one on the unit circle of a differenced lag); built in ADAM's state space form (one state per lag of the ARI and MA polynomials).

D23 ETS+ARIMA models of the checker's generator: Random ADAM ETS+ARIMA models from the generator of the checker P4 (code/checker/exact_oracle.py with lib.py), wider than the random ETS+ARIMA models: 250 models for each of seeds 21-23 and 31-34.

D29 admissibility decision: Whether a method accepts a model as admissible (decaying weights); compared between a method and a reference as the number of models on which the two differ.

D30 largest mode modulus: The largest modulus of the modes that decide admissibility: for the PBH reference the largest $|\lambda|$ of the reachable and observable eigenvalues of the expanded discount matrix, for the check of the whole model $1/\min|B|$ over the roots $B$ of the cancelled MA polynomial, for the per-lag check the largest $|\lambda|$ of its blocks; a model is admissible when it is below 1.

D26 comparison with a reference: Each model's admissibility decision, and its largest mode modulus where both find it inadmissible, computed by the method studied and by a reference on the same model (the PBH reference, the roots by numpy, the numpy prototype of the check of the whole model or an exact oracle); the result is the number of models whose decisions differ and the largest absolute difference of the modulus. In the planned tests, models whose reference modulus lies within 1e-8 of 1 are left out and counted.

D2 expanded discount matrix: The transition matrix $\tilde D$ of size $\sum_j l_j$ of the lag-1 system equivalent to $v_t = D v_{t-l}$, in which component $j$ carries itself and its $l_j - 1$ lagged copies; $\tilde F$, $\tilde g$ and $\tilde w$ are expanded in the same way, so that $\tilde D = \tilde F - \tilde g \tilde w'$.

D3 decaying weights: A lagged SSOE model has decaying weights when the weights $\tilde w' \tilde D^j \tilde g$ of the past observations in its fitted values tend to zero, that is, when every eigenvalue $\lambda$ of its expanded discount matrix that is reachable from $\tilde g$ and observable through $\tilde w$ has $|\lambda| < 1$; the largest such $|\lambda|$ is its largest mode modulus. A mode that $\tilde g$ does not reach but $\tilde w$ sees carries an initial state whose effect is not covered: with $|\lambda| > 1$ it grows.