Invertible ARMA Policies

• Invertible ARMA policies are defined by conditions that ensure the innovations of a model can be recovered as convergent causal linear combinations of past observations.
• They enable optimal forecasting, robust parameter estimation, and reliable identification of latent processes in both finite-dimensional and functional time series contexts.
• The methodology employs power series expansions, matrix criteria, and operator-theoretic conditions to verify invertibility across constant, time-varying, fractional, and Hilbert-space models.

Invertible ARMA policies refer to the rigorous mathematical and algorithmic conditions under which autoregressive moving average (ARMA) processes—possibly extended to time-varying, operator-valued, or fractional differenced forms—admit representations where innovations can be expressed as convergent causal linear functions of observed time series values. Invertibility is necessary for optimal forecasting, parameter estimation, and correct identification of latent processes in both finite-dimensional and functional time series contexts. The property is usually stated via power series expansions, matrix criteria, or operator-theoretic conditions, depending on the model class.

1. Definition, Fundamental Notion, and Representation

Invertibility in ARMA models requires that the innovation process $\varepsilon_t$ can be recovered as a convergent causal linear combination of observed series values. For classical ARMA($p$, $q$), the process

$$X_t = \sum_{j=1}^p \phi_j X_{t-j} + \varepsilon_t + \sum_{l=1}^q \theta_l \varepsilon_{t-l}$$

is invertible if the moving average polynomial $\Theta(z) = 1 + \theta_1 z + \cdots + \theta_q z^q$ has all its roots outside the closed unit disk $|z| \le 1$—equivalently, $\varepsilon_t = \sum_{h=0}^\infty \pi_h X_{t-h}$ with $\pi_h$ absolutely summable. Consequently, the MA($\infty$) expansion converges for $|z|<1$ and innovations are expressible in terms of observed values (Bhootna et al., 2023).

For time-varying ARMA (TV-ARMA) models, invertibility is characterized via explicit expansions:

$$y_t = \varphi(t) + \sum_{m=1}^p \varphi_m(t) y_{t-m} + u_t$$

$$u_t = \varepsilon_t + \sum_{\ell=1}^q \theta_\ell(t) \varepsilon_{t-\ell}$$

Invertibility is defined by the existence of an operator $G_t(B)$ such that

$$\varepsilon_t = G_t(B) [y_t - \varphi(t)], \quad G_t(B) \circ \Theta_t(B) = 1$$

where $\Theta_t(B)$ encodes the time-varying MA polynomial (Karanasos et al., 2021).

2. General Solution via Green-Function and Principal Determinant

Invertibility in TV-ARMA is determined by the absolute summability of coefficients arising from a banded Hessenbergian matrix representation. For the moving average part, define:

• The $k \times k$ principal MA matrix $\Theta(t,s)$ with time-varying coefficients.
• The principal determinant $\vartheta(t,s) = \det[\Theta(t,s)]$.
• Green-function coefficients $g_{t,j} = \vartheta(t, t-j)$, $j = 0, 1, \dots$.

The invertibility theorem states: If $\varphi_m(t)$ are bounded and

$\sum_{r=-\infty}^t |\vartheta(t,r)| < \infty$

the unique solution admits a Wold–Cramér decomposition with innovations rendered as:

$$\varepsilon_t = \sum_{r=-\infty}^t \delta_p(t,r) y_r - \sum_{r=-\infty}^t \vartheta(t,r) \varphi(r)$$

where $$\delta_p(t,r) = \vartheta(t,r) - \sum_{m=1}^p \vartheta(t,r+m) \varphi_m(r+m)$$, and invertibility is equivalent to

$$\sum_{j=0}^\infty |g_{t,j}| < \infty \text{ for each } t$$

This guarantees the causal recovery of innovations from observed values (Karanasos et al., 2021).

3. Specialized Cases: Constant-Coefficient ARMA and Fractional Extensions

For constant MA coefficients, the time-varying machinery reduces to familiar results:

$$\theta_\ell(t) \equiv \theta_\ell \implies \vartheta(t,t-j) \equiv g_j$$

and invertibility is classically characterized by root locations of the MA polynomial. This is extended in Humbert generalized fractional differenced ARMA (“HARMA”) processes, where fractional-difference operators defined via generating functions of Humbert polynomials impose additional conditions:

• All AR and MA polynomial roots outside $|z| \leq 1$
• The fractional kernel $(1 - m u z + z^m)^\nu$ must admit a binomial expansion converging for $|z| < 1$, enforced by $|m u - 1| < 1$ and $|\nu| < \tfrac{1}{2}$ (Bhootna et al., 2023).

Both the classical and fractional ARMA invertibility conditions ensure the associated infinite MA and AR expansions are absolutely convergent.

4. Finite-Sample Verification and Computational Criteria

Practical invertibility testing in TV-ARMA relies on linear algebraic recursions. Given a sample path of MA coefficients up to time $T$:

• Construct the principal matrix $\Theta(T,s)$ for lag span $k=T-s$
• Compute determinants $\vartheta(T,T-j)$ and Green coefficients $g_{T,j}$ via either direct computation or recursive formula
• Accumulate the partial sum $S_k = \sum_{j=0}^k |g_{T,j}|$:
  • If $S_k$ stabilizes as $k$ increases, the process is effectively invertible.
  • If $S_k$ grows without bound, or $g_{T,j}$ fail to decay, invertibility fails (Karanasos et al., 2021).

This criterion extends to generalized fractional ARMA via verification of polynomial root positions and bounding of fractional kernel parameters (Bhootna et al., 2023).

5. Functional and Hilbert-Space Extensions

For functional time series (e.g., Hilbert-space-valued ARMA), invertibility is governed by operator polynomial invertibility. If $A(z)$ and $B(z)$ are operator polynomials, for $X_t$ in a separable Hilbert space $H$, invertibility holds if $A(z)$ is invertible for $|z|\leq 1$:

$$A(z)^{-1} = \sum_{k=0}^\infty \psi_k z^k \text{ converges in operator norm}$$

and the infinite-moving-average representation

$X_t = \sum_{j=0}^\infty \Phi_j \varepsilon_{t-j}$

is absolutely convergent. Estimation of the causal operators is achieved via regularized Yule–Walker equations with explicit asymptotic error bounds given operator smoothness and eigenvalue spectrum conditions (Kühnert et al., 2024).

6. Impact, Applications, and Significance

Invertibility is essential for estimation, forecasting, and identification in time series analysis:

• Enables optimal linear forecasting from finite past data.
• Guarantees convergence and stability of Wold-type MA expansions.
• Underpins model selection and parameter fitting regimes (e.g., maximum likelihood or minimum contrast under invertibility constraints).
• The explicit toolkit—principal determinants, Green functions, operator polynomials, and associated recursions—offers a unified, generalizable approach for ARMA models with time-varying, fractional, or functional extensions.

These frameworks have been applied in econometric analyses, including the characterization of inflation persistence in U.S. macroeconomic data (Karanasos et al., 2021).

7. Summary Table: Invertibility Criteria Across Model Classes

Model Type	Invertibility Criterion	Key Parameter Constraints
Classical ARMA	$\Theta(z) \neq 0$ for $|z|\le 1$	All MA roots outside unit disk
TV-ARMA	$\sum_{j=0}^\infty |g_{t,j}| < \infty$	Absolute summability of Green coeffs
HARMA (Fractional)	AR & MA roots outside $|z|\le 1$; $|mu-1|<1$, $|\nu|<\tfrac{1}{2}$	Polynomial/root location; fractional kernel
Functional (Hilbert)	$A(z)$ invertible, $\sum \|\Phi_j\|<\infty$	Operator polynomial spectrum

Invertibility in ARMA policies is thus a foundational requirement, with rigorous conditions now established for a broad variety of model classes, including time-varying, fractional-differenced, and infinite-dimensional settings (Karanasos et al., 2021, Bhootna et al., 2023, Kühnert et al., 2024).