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A unit-root AR(1) has no weakly stationary solution with nonzero white noise

Example

Let ϵ be centered white noise with variance σ2>0. The equation Xt−Xt−1=ϵt has AR polynomial ϕ(z)=1−z, whose root is z=1. It has no weakly stationary solution. Under Countable Choice, this is a boundary illustration for the unit-circle-root premise of the paired spectral-density theorem; it does not show that premise is essential, since there is no stationary solution to which the density conclusion could apply.

Facts & Assumptions

Given: A centered white-noise sequence ϵ with Var⁡(ϵt)=σ2>0, and the proposed AR(1) equation Xt−Xt−1=ϵt.

[A1]

Countable Choice is assumed only for the contextual comparison with the spectral-density theorem; the covariance contradiction does not use it.

[F1]

Linear processes, white noise, and ARMA models: white noise has zero covariance at nonzero lags, and the AR(1) equation has ϕ(z)=1−z.

[F2]

Strict and weak stationarity for integer-indexed time series and Autocovariance and sample autocovariance: under weak stationarity, γ(h)=Cov⁡(Xt+h,Xt) is independent of t.

[F3]

Moments, variance, and covariance on a probability space defines variance and covariance; Cauchy-Schwarz for random variables gives ∣γ(h)∣≤γ(0).

[F4]

The causal criterion in ARMA causality and invertibility root criteria is stated for a weakly stationary ARMA solution and excludes closed-unit-disk roots; under Countable Choice, Spectral density of a stationary ARMA solution excludes unit-circle roots for its density conclusion. This example compares root locations only and applies neither theorem to a nonexistent stationary solution.

Verification

1.1F1F2F3givenalgebra

Suppose a weakly stationary solution exists and write its autocovariance as γ. For every integer h≥1, white-noise covariance gives 0=Cov⁡(ϵt+h,ϵt). Substituting ϵs=Xs−Xs−1 and expanding the four covariances using [F2] yields 0=2γ(h)−γ(h−1)−γ(h+1).

2.1step 1.1algebra

The recurrence in step 1.1 says γ(h+1)−γ(h)=γ(h)−γ(h−1) for every h≥1. Induction makes these first differences constant, so γ(h)=γ(0)+h d for h≥0, where d=γ(1)−γ(0).

3.1A1F3F4step 2.1givenalgebra∎

By [F3], ∣γ(h)∣≤γ(0) for every h. The affine sequence in step 2.1 is bounded only if d=0, hence γ(1)=γ(0). Then σ2=Var⁡(Xt−Xt−1)=2γ(0)−2γ(1)=0, contradicting the given σ2>0. No weakly stationary solution exists. The polynomial root is z=1 by direct substitution, on the unit circle excluded by [F4]'s spectral-density premise under [A1]. Since the contradiction rules out a stationary solution, this only illustrates the boundary of that premise and does not establish its essentiality.

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