The Unit-Root Revolution Revisited: Where Do Non-Standard Sampling Distributions and Related Conundrums Stem From?

Spanos Aris ()
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Spanos Aris: Department of Economics, 1757 Virginia Tech , Blacksburg, USA

Journal of Time Series Econometrics, 2025, vol. 17, issue 2, 69-117

Abstract: The primary objective of the paper is twofold. First, to answer the question posed in the title by arguing that the conundrums: [C1] the non-standard sampling distributions, [C2] the low power of unit-root tests for α 1 ∈ [0.9, 1], and [C3] their size distortions, [C4] issues in handling Y 0, and [C5] the framing of H 0 and H 1 in testing α 1 = 1, as well as [C6] two competing parametrizations for the AR(1) models, (B) Y t = α 0 + α 1 Y t−1 + ɛ t , (C) Y t = α 0 + γt + α 1 Y t−1 + ɛ t , stem from viewing these models as aPriori Postulated (aPP) stochastic difference equations driven by the error process { ε t , t ∈ N ≔ ( 1,2 , … , n , … ) } $\left\{{\varepsilon }_{t}, t\in \mathbb{N}{:=}\left(1,2,\dots ,n,\dots \right)\right\}$ . Second, to use R.A. Fisher’s model-based statistical perspective to unveil the statistical models implicit in each of the AR(1): (B)-(C) models, specified entirely in terms of probabilistic assumptions assigned to the observable process { Y t , t ∈ N } $\left\{{Y}_{t}, t\in \mathbb{N}\right\}$ underlying the data y 0, which is all that matters for inference. The key culprit behind [C1]–[C6] is the presumption that the AR(1) nests the unit root [UR(1)] model when α 1 = 1, which is shown to belie Kolmogorov’s existence theorem as it relates to { Y t , t ∈ N } $\left\{{Y}_{t}, t\in \mathbb{N}\right\}$ . Fisher’s statistical perspective reveals that the statistical AR(1) and UR(1) models are grounded on (i) two distinct processes { Y t , t ∈ N } $\left\{{Y}_{t}, t\in \mathbb{N}\right\}$ , with (ii) different probabilistic assumptions and (iii) statistical parametrizations, (iv) rendering them non-nested, and (v) their respective likelihood-based inferential components are free from conundrums [C1]–[C6]. The claims (i)–(v) are affirmed by analytical derivations, simulations, as well as proposing a non-stationary AR(1) model that nests the related UR(1) model, where testing α 1 = 1 relies on likelihood-based tests free from conundrums [C1]–[C6].

Keywords: stochastic difference equations; Dickey-Fuller unit root testing; non-standard sampling distributions; Brownian motion process; statistical model and its adequacy; non-nested models (search for similar items in EconPapers)
JEL-codes: C12 C18 C22 C32 C51 C52 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1515/jtse-2024-0008

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