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The primary objective of the paper is twofold. , 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 ∈ [0.9, 1], and [C3] their size distortions, [C4] issues in handling , and [C5] the framing of and in testing = 1, as well as [C6] two competing parametrizations for the AR(1) models, (B) = + + , (C) = + + + , viewing these models as aPriori Postulated (aPP) stochastic difference equations driven by the error process . , 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 underlying the data , 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 = 1, which is shown to belie Kolmogorov’s existence theorem as it relates to . Fisher’s statistical perspective reveals that the statistical AR(1) and UR(1) models are grounded on (i) two distinct processes , with (ii) different probabilistic assumptions and (iii) statistical parametrizations, (iv) rendering them , 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 = 1 relies on likelihood-based tests free from conundrums [C1]–[C6].

In this article we introduce efficient Wald tests for testing the null hypothesis of the unit root against the alternative of the fractional unit root. In a local alternative framework, the proposed tests are locally asymptotically equivalent to the optimal Robinson Lagrange multiplier tests. Our results contrast with the tests for fractional unit roots, introduced by Dolado, Gonzalo, and Mayoral, which are inefficient. In the presence of short range serial correlation, we propose a simple and efficient two-step test that avoids the estimation of a nonlinear regression model. In addition, the first-order asymptotic properties of the proposed tests are not affected by the preestimation of short or long memory parameters.

This paper presents a new test for fractionally integrated (FI) processes. In particular, we propose a testing procedure in the time domain that extends the well-known Dickey-Fuller approach, originally designed for the I(1) versus I(0) case, to the more general setup of FI(d0) versus FI(d1), with $d_1 < d_0$. When d0 = 1, the proposed test statistics are based on the OLS estimator, or its t-ratio, of the coefficient on Δd1yt-1 in a regression of Δ yt on Δd1yt-1 and, possibly, some lags of Δ yt. When d1 is not taken to be known a priori, a pre-estimation of d1 is needed to implement the test. We show that the choice of any T1/2-consistent estimator of d1 ∈ [0,1) suffices to make the test feasible, while achieving asymptotic normality. Monte-Carlo simulations support the analytical results derived in the paper and show that proposed tests fare very well, both in terms of power and size, when compared with others available in the literature. The paper ends with two empirical applications.

Recent approaches in unit root testing have taken into account the influences of initial conditions and data trend breaks via pre-testing and union of rejection testing strategies. This paper reviews existing methods, extends the methods of (Harvey, D. I., S. J. Leybourne, and A. M. R. Taylor. 2012b. “Unit Root Testing under a Local Break in Trend.” 167:140–167), and integrates these techniques to create a comprehensive testing strategy. Even when presented with nuisance parameters such as initial conditions and data breaks, this new strategy holds promising asymptotic and finite sample properties.

In this article we examine the impact of data-based lag-length estimation on the behavior of the augmented Dickey-Fuller (ADF) test for a unit root. We derive conditions under which the ADF test converges to the distribution tabulated by Dickey and Fuller and verify that these conditions are satisfied by several commonly employed lag-selection strategies. Simulation evidence indicates that the performance of the ADF test is considerably improved when the lag length is selected from the data. An application to inventory series illustrates that inference about a unit root can be very sensitive to the method of lag-length selection.