The power of unit root tests against nonlinear local alternatives

Abstract

This article extends the analysis of local power of unit root tests in a nonlinear direction by considering local nonlinear alternatives and tests built specifically against stationary nonlinear models. In particular, we focus on the popular test proposed by Kapetanios et al. (2003, Journal of Econometrics 112, 359-379) in comparison to the linear Dickey-Fuller test. To this end, we consider different adjustment schemes for deterministic terms. We provide asymptotic results which imply that the error variance has a severe impact on the behaviour of the tests in the nonlinear case; the reason for such behaviour is the interplay of non-stationarity and nonlinearity. In particular, we show that nonlinearity of the data generating process can be asymptotically negligible when the error variance is moderate or large (compared to the 'amount of nonlinearity'), rendering the linear test more powerful than the nonlinear one. Should however the error variance be small, the nonlinear test has better power against local alternatives. We illustrate this in an asymptotic framework of what we call persistent nonlinearity. The theoretical findings of this article explain previous results in the literature obtained by simulation. Furthermore, our own simulation results suggest that the user-specified adjustment scheme for deterministic components (e.g. OLS, GLS, or recursive adjustment) has a much higher impact on the power of unit root tests than accounting for nonlinearity, at least under local (linear or nonlinear) alternatives.ASSUMPTION 2. Let e t be an iid sequence such that E(e t ) ¼ 0, Varðe t Þ ¼ r 2 , and 9d > 0 with E e t j j 4þd < C < 1:LOCAL POWER OF NONLINEAR UNIT ROOT TESTS

A Monte Carlo Investigation of Unit Root Tests and Long Memory in Detecting Mean Reversion in I(0) Regime Switching, Structural Break, and Nonlinear Data

“…Following the analysis ofTaylor et al [2001], inTable 5, I analyze stationary exponential smooth transition models with In all but one case, GLS de-trending in both regressions results in higher rejection rates than both the inf-t test and the test ofKruse [2011] with T = 200. These findings are very much in line withDemetrescu and Kruse [2013], who find power gains with GLSdetrending when the initial condition is set to zero. The inf-t test results in higher relative power with T = 500, producing either the highest or second highest power in every case, and also tends to perform well when long memory tests yield relatively low power.…”

Unit roots, non-linearities and structural breaks

“…Adjustment of deterministic terms can be handled in a similar way as in the case of linear alternatives. Kapetanios and Shin (2008) suggest GLS-adjustment, while Demetrescu and Kruse (2012) compare also recursive adjustment and the MAX-procedure by Leybourne (1995) in a local-to-unity framework. The …ndings suggest that GLS-adjustment performs best in the absence of a non-zero initial condition.…”

“…Similar to the case of linear models, OLS adjustment proves to work best, when the initial condition is more pronounced. Another …nding of Demetrescu and Kruse (2012) is that a combination of unit root tests against linear and nonlinear alternatives would be a successful (union-of-rejections) strategy.…”

On the Transmission of Memory in Garch‐in‐Mean Models

“… We are grateful to Robinson Kruse for sharing his code for simulating the GLS‐demeaned DF test (see Demetrescu and Kruse, ).…”