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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>ε</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mspace width="0.3333em"/> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> <m:mo>≔</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1,2</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mo>…</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \ε ₜ, t∈ N:=(1,2, ,n, )\ . 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>Y</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mspace width="0.3333em"/> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> ₜ, t∈ N\ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>Y</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mspace width="0.3333em"/> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> ₜ, t∈ N\ . Fisher’s statistical perspective reveals that the statistical AR(1) and UR(1) models are grounded on (i) two distinct processes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>Y</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mspace width="0.3333em"/> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> </m:mrow> <m:mo str