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Derivation of the Infinite AR Representation and $\pi$-weights for an Invertible MA(q) Process

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The Formal Theorem

For an invertible Moving Average process of order qq, denoted as Yt=∑i=0qθiZt−i Y_t = \sum_{i=0}^q \theta_i Z_{t-i} where θ0=1 \theta_0 = 1 and Zt∼WN(0,σ2) Z_t \sim WN(0, \sigma^2) , there exists an equivalent infinite Autoregressive representation Yt=∑j=1∞πjYt−j+Zt Y_t = \sum_{j=1}^{\infty} \pi_j Y_{t-j} + Z_t . The coefficients πj \pi_j are uniquely determined by the MA coefficients θi \theta_i through the relation: 1−∑j=1∞πjxj=1∑i=0qθixi 1 - \sum_{j=1}^{\infty} \pi_j x^j = \frac{1}{\sum_{i=0}^q \theta_i x^i} for ∣x∣<1 |x| < 1 .

Analytical Intuition.

Picture a complex, multi-layered tapestry representing our time series Yt Y_t . This tapestry is woven using a finite set of recent shocks Zt Z_t (an MA(q) process). Our quest is to find an alternative way to describe this same tapestry, not by referencing the recent shocks, but by referencing *all* past values of the tapestry itself – an infinite Autoregressive (AR) representation. This is like finding a hidden symmetry where the pattern at any point can be perfectly predicted by an infinite summation of its predecessors. The π \pi weights are the secret 'thread values' that translate past tapestry patterns into the current one, ensuring the description remains consistent and the MA process, crucially, is invertible.
CAUTION

Institutional Warning.

The convergence of the infinite sum of πjYt−j \pi_j Y_{t-j} and the precise condition for invertibility (roots of the MA polynomial outside the unit circle) are common stumbling blocks.

Academic Inquiries.

01

What does it mean for an MA(q) process to be invertible?

An MA(q) process is invertible if its MA polynomial has all its roots strictly outside the unit circle. This ensures a unique, stable representation as an infinite AR process, and that past innovations can be uniquely recovered from the observed series.

02

How are the π \pi coefficients derived formally?

The derivation involves equating the generating functions of the MA and AR processes. The MA process generating function is Θ(x)=∑i=0qθixi \Theta(x) = \sum_{i=0}^q \theta_i x^i , and the AR process is Φ(x)=1−∑j=1∞πjxj \Phi(x) = 1 - \sum_{j=1}^{\infty} \pi_j x^j . For equivalence, Φ(x) \Phi(x) must be the reciprocal of Θ(x) \Theta(x) multiplied by σ2 \sigma^2 (or normalized to have π0=1 \pi_0 = 1 for the AR representation of Yt/σ Y_t/\sigma ).

03

Can any MA(q) process be represented as an infinite AR process?

Yes, *if* the MA(q) process is invertible. Non-invertible MA processes can be represented as infinite AR processes, but the AR representation will not be unique, and the π \pi coefficients might not converge.

04

Why is the AR(infinity) representation useful?

It provides an alternative perspective on MA processes, allowing us to use AR-based estimation techniques. It also highlights the connection between MA and AR models and is fundamental in understanding the dynamics of more complex time series models like ARMA.

Standardized References.

  • Definitive Institutional SourceBrockwell, Peter J., and Richard A. Davis. Introduction to Time Series and Forecasting.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Derivation of the Infinite AR Representation and $\pi$-weights for an Invertible MA(q) Process: Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/time-series-analysis/derivation-of-the-infinite-ar-representation-and---pi--weights-for-an-invertible-ma-q--process

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