the power of unit root tests against nonlinear local …...pecially when the dgp is close to the...

The Power of Unit Root Tests AgainstNonlinear Local Alternatives∗

Matei Demetrescua and Robinson Kruseb

a University of Bonn† b CREATES, Aarhus University‡

May 23, 2011

Abstract

This article extends the analysis of local power of unit root tests in a nonlineardirection by considering local nonlinear alternatives and nonlinear tests. The fociare (i) on nonlinear smooth transition models and the related nonlinear unit roottest proposed by Kapetanios et al. (2003, Journal of Econometrics 112, 359-379),and (ii) on the comparison of different adjustment schemes for deterministic termsfor the considered nonlinear test. We provide asymptotic results which imply thatthe error variance has a severe impact on the behavior of the tests in the nonlinearcase; the reason for such behavior is the interplay of nonstationarity and nonlinear-ity. Moreover, we show that the nonlinearity of the data generating process can beasymptotically negligible when the error variance is moderate or large (comparedto the “amount of nonlinearity” of the data generating process), rendering the lin-ear Dickey-Fuller test more powerful than the nonlinear test. Should however theerror variance be small, the nonlinear test has better power against local alterna-tives. The theoretical findings of this paper explain previous results in the literatureobtained by simulation. Furthermore, our own simulation results suggest that theuser-specified adjustment scheme for deterministic components (e.g. OLS, GLS, orrecursive adjustment) is far more important for the power of unit root tests thanaccounting for nonlinearity, at least when the alternative is close to the null.

Key wordsStochastic trend, Local nonlinearity, Local power curve

JEL classificationC12 (Hypothesis Testing), C22 (Time-Series Models)

∗The authors would like to thank Niels Haldrup for helpful comments. Robinson Kruse gratefullyacknowledges financial support from CREATES funded by the Danish National Research Foundation.

†University of Bonn, Institute for Econometrics and Operations Research, Adenauerallee 24-42, D-53113 Bonn, Germany. E-mail address: [email protected] .

‡CREATES, Aarhus University, School of Economics and Management, Bartholins Alle 10, DK-8000Aarhus C, Denmark. E-mail address: [email protected] .

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1 Introduction

Nonlinear stationary time series models such as the smooth transition autoregressive[STAR] model (Teräsvirta, 1994) have become quite popular in applied time series econo-metrics. The exponential STAR [ESTAR] model is particularly successful in this respect:it has been applied to a variety of economic time series ranging from real (effective) ex-change rates (e.g. Taylor et al., 2001; Sarantis, 1999) real money balances (Sarno et al.,2003), business cycle data (e.g. Teräsvirta and Anderson, 1992) and inflation (e.g. Nobayet al., 2010) to price-dividend ratios, see McMillan (2007). Such series often exhibit non-linear regime-dependent dynamic behavior, depending on whether they are close to, oraway from, some equilibrium value; in particular, their tendency to revert to equilibriumis weak when the series is in the middle regime (i.e. close to equilibrium), but may gainstrength in the outer regimes; (E)STAR models are well-suited to capture this type ofnonlinear behavior.

The empirical question of relevance for such series is whether the outer regimes doindeed exhibit reversion to equilibrium. If not, the series under consideration, say yt, isactually integrated; the inner and outer regimes exhibit the same unit root behavior. Onthe contrary, if the series is mean-reverting in the outer regimes, the (E)STAR modelitself is globally stationary in spite of the nonstationary of the middle regime. The degreeof mean-reversion depends (i) on the distance between yt−1 and its equilibrium value, and(ii) on the shape of the exponential transition function which is controlled (in the ESTARmodel) by a single positive parameter, say γ. The parameter γ can be interpreted as thespeed of transition between inner and outer regimes: the larger γ, the tighter is the middleregime. The linear Dickey and Fuller (1979) [DF] test is consistent against generic ergodicalternatives, so one may use the usual DF test in order to discriminate between unit rootprocesses and STAR processes with mean-reverting outer regimes. However, Kapetanioset al. (2003) [KSS] and Kapetanios and Shin (2006) develop tests against specific nonlinearalternatives, and show that the power of unit root tests can significantly be improved innonlinear cases.

One focus of this paper is on the local power of unit root tests since the power undersequences of local alternatives gives insight into the behavior of test procedures when thealternative, although true, is close to the null. The DF test has nontrivial power against(linear) alternatives in 1/T neighborhoods of the unit root (Phillips, 1987). So the questionarises, how do nonlinear tests behave in the presence of such nearly integrated behavior?And how do linear and nonlinear tests behave when the data generating process [DGP]does indeed exhibit nonlinearities in addition to near integration?

The second focus is on the method used to remove deterministic components. Namely,the power under the alternative depends largely on the chosen adjustment procedure, es-

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pecially when the DGP is close to the null of a unit root. Ordinary least squares [OLS]adjustment for instance is known to be inefficient in the linear case (Elliott et al., 1996);it also induces a large bias in the estimator of the autoregressive parameter. The ap-plication of efficient adjustment procedures is thus worth considering. Along these lines,Kapetanios and Shin (2008) adopt local-to-unity generalized LS detrending procedures forunit root tests in the nonlinear STAR and self-exciting threshold autoregressive frame-works. In the case of a linear DGP, Leybourne et al. (2005) show recursive adjustment ofthe series (Shin and So, 2001; Taylor, 2002) and forward-reverse application of DF tests(Leybourne, 1995) to be of effectiveness comparable with local-to-unity GLS adjustment.Therefore, we additionally compare the efficiency of such alternative adjustment methodswhen considering local nonlinear alternatives.

In more detail, our contributions are as follows. We concentrate on the linear DF andnonlinear KSS unit root tests; after describing in Section 2 the setup of the paper, weanalyze in Section 3 the local power of the DF and KSS tests against the benchmark oflinear nearly integrated alternatives. We find the KSS test to have nontrivial power in thesame 1/T -neighborhoods as the DF test; the DF test is, however, locally more powerful.While this is not surprising given the linear nature of the considered alternatives, we findthe same to hold true for smooth transitions with nearly integrated outer regimes as well.In other words, the nonlinearity of the DGP has negligible effect on both the DF andthe KSS test when the outer regimes are close to the null. The reason for this counter-intuitive behavior of the tests is that near integration in the outer regimes leads to highunconditional variances of the process and hence to vanishing probability of hitting themiddle regime.

Thus, it might seem that unit root tests built against nonlinear alternatives have noadvantage over linear unit root tests when the alternative is near integration. This is afallacy, however, since the argument relies on fixed values of the parameters characterizingthe nonlinear behavior. We argue in Section 4 that such fixed-nonlinearity asymptoticsdoes not always describe the data well, as the error variance of nonlinear processes is oftenfound to be small compared to the sample size, see e.g. Taylor et al. (2001). The directeffect of small innovations variance is that the (effective) width of the middle regime isquite large, in fact large enough for the the probability of the process to hit the middleregime to be nonzero even in the limit. Small variances also lead to problems withidentifying the middle regime even when the model is stationary nonlinear; see Donaueret al., 2010. Hence, we discuss in Section 4 persistently nonlinear alternatives for whichthe variance of the process yt is comparable with the width of the middle regime and themiddle regime is hit with nonvanishing probability in the limit. We accomplish this byletting the width of the middle regime increase with the sample size at an appropriaterate so the error variance is comparatively small; this type of asymptotics ensures that,

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asymptotically, nonlinearity is not lost sight of. Under near integration with persistentnonlinearity, we confirm the superiority of the nonlinear KSS test compared to the DFtest when the same detrending procedure is applied. Most of the power gains, however,come from efficiently removing the deterministic component: e.g. the nonlinear KSS testwith OLS detrending is clearly dominated by the DF test with GLS detrending. Section5 concludes by addressing some implications of our results for applied work.

Before proceeding to the analysis, let us set some notation. Denote by W (s) thestandard Wiener process and by ”⇒” weak convergence in a space of càdlàg functionsendowed with a suitable norm. Let C be a generic constant whose value may changefrom equation to equation. Finally, the symbols “ d→” and “ p→” stand for convergence indistribution and convergence in probability.

2 Setup

2.1 Model and test statistics

The observed series yt is assumed to be generated according to the usual additive com-ponent model. We consider here the empirically more relevant cases of a constant and ofa linear trend. Since the procedures used here to adjust for deterministic components allresult in similar (w.r.t. the trend parameters) test statistics, we can go on as if the trendcoefficients are 0. So

Assumption 1 Let

∆yt = ϕyt−1G (yt−1; γ) + εt, t = 1, . . . , T,

with γ ∈ R+; the starting value satisfies y0 = 0 for ϕ = 0 and y0 = κ/√

1−(1+ϕ)2 for ϕ < 0,κ ∈ R.

The function G is often specified as smooth with values between 0 and 1; in particularG(y, ·) → 1 as y → ±∞. The errors εt satisfy standard assumptions in the literature.

Assumption 2 Let εt be an iid sequence such that E(εt) = 0, Var (εt) = σ2, and ∃δ > 0

with E |εt|4+δ < C < ∞.

The DGP given by Assumptions 1 and 2 exhibits in the nonlinear case smooth tran-sitions between three regimes: the middle regime (G = 0) is characterized by a unit rootand the process is driven a stationary AR(1) process by in the outer regimes (G = 1) aslong as ϕ < 0. In this sense, the process is partially nonstationary as one of the regimescontains a unit root while the other regimes are stationary.

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For fixed parameters ϕ and γ Kapetanios et al. (2003) establish stationarity of yt byusing the drift condition of Tweedie (1975). De Jong (2009) analyzes the stationarity prop-erties of such models under weak assumptions about the innovations. Roughly speaking,if the outer regimes are stationary, so is the process yt itself. In fact, the derivations ofthis paper hold under more general data generating processes (e.g. martingale differenceassumptions and short-run dynamics); but the aim of the paper is to give insight into thebehavior of unit root tests when nonlinearity interacts with near integration and not tohave state-of-the-art assumptions.

We are interested in modeling situations where the alternative, if true, is close to thenull. This takes us away from the fixed-ϕ and fixed-γ setup. Since the alternative ischaracterized by both ϕ and γ, there are two dimensions of the local interpretation ofthe (E)STAR model. First, one can define sequences of local alternatives for which theautoregressive parameters in the outer regimes tend to one, thereby implying a unit rootin all regimes which are identical in the limiting case. Second, if the shape parameterof the transition function γ tends to zero, the model becomes linear in the limit and thenonstationary middle regime dominates. Such reasoning applies to a variety of nonlin-ear models where one parameter is unidentified under the null. So it is required for ϕ

throughout the paper that

Assumption 3 Let ϕ = − cT

with c ≥ 0 fixed.

We shall write in fact ϕT to emphasize the local-to-unity character of the sequence ofalternatives examined. It is only natural to look at such neighborhoods of the null: theDF test has nontrivial power against this type of local alternatives, so the nonlinear KSStest is to be evaluated against this benchmark.

Turning our attention to the transition function G, we shall analyze three mutuallyexclusive cases resulting from the type of alternative we consider for γ.

In the case of a purely linear DGP (i.e. identical inner and outer regimes), we have

Assumption 4 Let G (y; γ) = 1 ∀y ∈ R and ∀γ ∈ R+.

Assumption 4 can be seen as the limiting case γ → ∞.

In the nonlinear case, the most common choice of a transition function is an exponen-tial, which leads to the popular Exponential STAR model

G1 (yt−1; γ) = 1− e−γ(yt−1−m)2

for some fixed shape parameter γ > 0 and location parameter m ∈ R. The parameter γ

can be interpreted as speed of transition between the inner and outer regimes; an equallyrelevant interpretation is provided by its proportionality with the width of the middleregime.

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A more flexible choice is given by the 2nd order logistic function,

G2 (yt−1; γ) =2

1 + e−γ(yt−1−m1)(yt−1−m2)− 1

which has two location parameters m1 ∈ R and m2 ∈ R satisfying m1 ≤ m2. Both tran-sition functions fulfill the following two low-level assumptions. Since both functions haveexponential tail behavior, we’ll (improperly) call ESTAR both resulting STAR models.

Assumption 5 Let γ > 0 fixed. Let G (y; γ) 7→ [0; 1] be a Lipschitz function such thaty2 (1−G (y; γ)) → 0 as y → ±∞ ∀γ > 0.

The condition y2 (1−G (y; γ)) → 0 implies that G (y; γ) → 1 as y → ±∞ at arate higher than quadratic; it is purely technical and fulfilled by virtually all populartransition functions, not only by the exponential or the 2nd order logistic. The functionG is actually asymptotically homogenous of order 0 in the sense of Park and Phillips(1999); see also Park and Phillips (2001) and Chang et al. (2001). It is this type ofbehavior that ultimately leads to asymptotic negligibility of the nonlinear part of theDGP; cf. the proof of Proposition 2.

Section 3 motivates the use of alternatives for which the width of the middle regimegrows with the sample size. Since the effective width is inversely proportional to thetransition speed γ, this is equivalent to requiring γ to converge to 0 at an appropriaterate as T → ∞ as described by the following

Assumption 6 Let γ = γT = g2/T . Let G (·) be a Lipschitz function with G (0) = 0 andG (y) → 1 as y → ±∞ such that ∀γ > 0

∣∣G (y; γ)− G(γy2)∣∣ ≤ Cγ (1 + |y|) .

The asymptotic homogeneity condition for G from Assumption 6 enables us to charac-terize the weak limit of the suitably normalized process yt. Note that the ESTAR transitionfunction satisfies Assumption 6 with G = 1 − exp {−z} , while for the 2nd order logisticfunction we have G = 2

1+exp{−z} − 1. Since a nearly integrated process has variance ofmagnitude O(t), there is a strong similarity to the practice of standardizing the parameterγ by the standard deviation of the transition variable; see e.g. Teräsvirta (2004, p. 229).Since the parameter γ enters the model exponentially, the rate γT = O(1/T) may appeararbitrary; see Park and Phillips (2001). But it follows from the derivations in the appendixthat a lower rate recovers nearly-integrated linearity in the limit, whereas a higher raterecovers the null hypothesis of integration. So 1/T neighborhoods are the relevant ones fornonlinear behavior, at least in conjunction with the asymptotic homogeneity conditionfrom Assumption 6.

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2.2 Test procedures

In the ESTAR framework, the null hypothesis of a unit root is parameterized by ϕ = 0,

and the alternative by ϕ < 0 together with γ > 0. Without deterministic components,the DF test tDF is based on the t statistic of ϕ from the OLS regression

∆yt = φDF yt−1 + ϵt, (1)

while the nonlinear KSS test tKSS relies on the t statistic of ϕ from the OLS regression

∆yt = φKSS y3t−1 + ϵt. (2)

In (2), the specific choice of y3t−1 as regressor is due to employing a Taylor series expansionto deal with the lack of identification of γ under the null. The asymptotic null distributionof the nonlinear KSS test statistic was derived by Kapetanios et al. (2003),

tKSSd→´ 1

0W 3 (s) dW (s)√´ 1

0W 6 (s) ds

. (3)

(Correct) Lag augmentation in case of serial dependence does not affect the asymptoticsunder the null; Rothe and Sibbertsen (2006) propose a nonparametric correction followingPhillips and Perron (1988).

Comparing regression (2) with regression (1), large values of yt−1 are more pronounced,while small ones are reduced; this is useful inasmuch as the middle regime (small |yt−1|)exhibits no mean reversion under a nonlinear alternative and there is little evidence infavor of stationarity to be obtained there. The nonlinear KSS test is expected to performbetter than the DF test under (ESTAR) nonlinear alternatives; see Kapetanios et al.(2003).

When deterministic terms are considered, the standard procedure is to apply OLS de-meaning and detrending procedures onto the series yt before computing the test statistics.Denote the resulting test statistics by tµDF , t

µKSS, t

τDF and tτKSS.

But efficient adjustment of deterministic terms when testing for unit roots is quiterelevant. Several adjustment schemes have proven to be useful in the purely linear case;see Leybourne et al. (2005). So we employ the following adjustment schemes for both thelinear DF and the nonlinear KSS tests and aim to find which (if any) is the most usefulin terms of power for the nonlinear test.

1. Local-to-unity GLS demeaning and detrending as proposed by Elliott et al.(1996) for the DF test and Kapetanios and Shin (2008) for the KSS test. For boththe linear and the nonlinear tests, the data are demeaned prior to application of the

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test in the following way: yµ;c = [y1, y2−(1−c/T)y1, ..., yT −(1−c/T)yt−1]′ is regressed

on zµ;c = [1, 1 − (1 − c/T), . . . , 1 − (1 − c/T)]′ in order to obtain GLS residuals yµ;ct ;in the linear case c = 7, while c = 9 in the nonlinear case. The GLS unit root testsare then based on yµ;ct . GLS detrending is carried out in analogous way, with zτ ;c

being zµ;ct with the additional column [1, 1 − c/T , 1 − 2c/T , . . . , 1 − c(T−1)/T ]′ (i.e. thequasi-differenced trend); in the linear case c = 13.5, while c = 17.5 in the nonlinearcase. Denote the resulting local-to-unity GLS unit root test statistics by tµ;cDF , t

µ;cKSS,

tτ ;cDF and tτ ;cKSS.

2. Recursive demeaning and detrending. For recursive demeaning (So and Shin,1999), the levels are recursively demeaned, yµt−1 = yt−1 − 1

t

∑tj=1 yj, leading to

tµKSS =

∑Tt=2

(yµt−1

)3∆yt

σ√∑T

t=2

(yµt−1

)6For the test with recursive detrending, one uses recursive detrending, and the dif-ferences are demeaned, cf. Taylor (2002) and Rodrigues (2006):

tτKSS =

∑Tt=2

(yτt−1

)3 (∆yt −∆y

)σ√∑T

t=2

(yτt−1

)6with yτt−1 = yt−1 + 2

t−1

∑t−1j=1 yj −

6t(t−1)

∑t−1j=1 j yj. The DF tests are constructed

analogously and denoted by tµDF and tτDF ; see Shin and So (2001); Taylor (2002).The recursive adjustment schemes reduce the bias of the OLS estimator (when closeto the unit root) enhancing this way power properties, see the discussions in Shinand So (2001) and Taylor (2002).

3. The maximum of the forward-computed and backward-computed (i.e. ap-plied to y∗t = yT+1−t.) DF tests with usual demeaning and detrending, as well as thecorresponding nonlinear KSS tests. Denote the test statistics by tµ;max

DF , tµ;maxKSS , tτ ;max

DF

and tτ ;maxKSS . Strictly speaking, it is not a different adjustment scheme; but the max

test procedure has (apart from new critical values) power properties comparable tothe previous two modifications, at least in the linear case (Leybourne et al., 2005).

3 Nearly integrated STAR processes

Before discussing the nonlinear case, we review the benchmark of a linear local-to-unityalternative.

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3.1 Linear local-to-unity alternatives

We compare the local power of linear DF and nonlinear KSS tests under the sequenceof local linear alternatives from Assumption 3. Since the work of Phillips (1987) it isknown what the weak limit of the suitably normalized yt under such near integrationis; the following lemma summarizes it. To formulate the lemma, denote by Jc (s) thestandard Ornstein-Uhlenbeck [OU] process with “mean-reversion” parameter c, dJc (s) =−cJc (s) ds+ dW (s) with Jc (0) = 0 a.s., and let

Jc,κ (s) =

W (s) c = 0

κσ√2ce−cs + Jc (s) c > 0

.

This is nothing else than the standard OU process with starting value κσ√2c.

Lemma 1 Under Assumptions 1, 2, 3 and 4 (linearity), it holds that 1

σ√T

[sT ]∑j=1

εj;1

σ√Ty[sT ]

⇒ (W (s) ; Jc,κ (s))

as T → ∞.

Proof: omitted.

The limiting distribution of the DF test statistic under a sequence of local linear al-ternatives has already been derived in the relevant literature for all examined methods ofremoving deterministic components. Although this is not entirely the case for the KSStest (where only OLS and GLS adjustment have been considered sofar), the followingproposition gives the asymptotic behavior of the linear DF and nonlinear KSS test statis-tics only for the case without adjustment. The extensions for deterministic adjustment areactually particular cases of the results we give in Section 4 so there is no loss of generality.

Proposition 1 Under the assumptions of Lemma 1 (linearity), it holds as T → ∞ that

tDFd→´ 1

0Jc;κ (s) dW (s)√´ 1

0J2c;κ (s) ds

− c

√ˆ 1

0

J2c;κ (s) ds;

the analogous result holds for the nonlinear KSS test,

tKSSd→´ 1

0J3c;κ (s) dW (s)√´ 1

0J6c;κ (s) ds

− c

´ 1

0J4c;κ (s) ds√´ 1

0J6c;κ (s) ds

.

Proof: omitted.

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0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

OLS

c

Loca

l pow

er

DFKSS

Figure 1: Local Power of DF and KSS test against a linear alternative.

An analysis of the local power curves given in Figure 1 reveals that the DF testdominates the nonlinear KSS test for all c > 0.

This result was to be expected, since the nonlinear KSS test is after all misspecifiedwhen the DGP is linear, having replaced yt with y3t as a regressor. The nonlinear KSStest being explicitly built to take nonlinearities into account, the same reasoning pointstoward the nonlinear KSS dominating in terms of power when the DGP is nonlinear, sayas specified by Assumption 5. The argument fails however in the near-integration setup,see the following subsection.

3.2 Nonlinear local-to-unity alternatives

Consider now a fixed nonlinear local alternative, i.e. some fixed transition function which,according to Assumption 5, is not equal to 1 almost everywhere. It shape is howeverfixed and does not depend on the sample size T. In spite of the nonlinearity of this DGP,the following lemma shows that the limiting behavior of the suitably normalized yt is thesame as in the linear case.

Lemma 2 Under Assumptions 1, 2, 3 and 5 (fixed nonlinearity), it holds that 1

σ√T

[sT ]∑j=1

εj;1

σ√Ty[sT ]

⇒ (W (s) ; Jc,κ (s))

10

as T → ∞.

Proof: see the Appendix.

The explanation for this counter-intuitive result is as follows. With yt nearly integratedin the outer regimes, the levels yt have a relatively large variance even in small samplesand they practically never reach the middle regime. The nonlinearity is consequentlyrendered asymptotically irrelevant for the levels when keeping the shape of the transitionfunction fixed.

But the worst is yet to come: although the nonlinearity still enters the DF and KSStest statistics via ∆yt, the asymptotic distribution of both tests is not affected by thenonlinearity of the DGP either. In other words, the limiting behavior of the test statisticsunder local-to-unity nonlinear alternatives is identical to that of the test statistics underthe usual nearly-integrated linear alternative. Like for the case of a linear DGP in Sub-section 3.1, we prove the result for the case without deterministic terms only: consideringremoval of deterministic components may change the shape of the limiting distributions,but not the fact that the distributions are not affected by the (fixed) nonlinearity.

Proposition 2 Under the assumptions of Lemma 2 (fixed nonlinearity), it holds as T →∞ that

tDFd→´ 1

0Jc;κ (s) dW (s)√´ 1

0J2c;κ (s) ds

− c

√ˆ 1

0

J2c;κ (s) ds;

the analogous result holds for the nonlinear KSS test,

tKSSd→´ 1

0J3c;κ (s) dW (s)√´ 1

0J6c;κ (s) ds

− c

´ 1

0J4c;κ (s) ds√´ 1

0J6c;κ (s) ds

.


The results in Proposition 2 nicely confirm the findings of Choi and Moh (2007),who found by extensive Monte Carlo simulations that the power of unit root tests builtagainst nonlinear alternatives is mostly influenced by the distance to integration, i.e. bythe parameter ϕ, and the type of fixed nonlinearity does not matter too much. Our workcomplements their findings by theoretical results for the ESTAR data generating process.

The key insight is that nonlinearity as specified by Assumption 5 is really weak com-pared to the variance of the (even nonlinear) nearly integrated process. This can also beseen as an identification problem as the middle regime is simply ignored asymptotically.See also Donauer et al. (2010). Note that it is a problem of the DGP and not of theKSS test, since lack of identification of γ under the null is circumvented by using a Taylorseries approximation.

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0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

OLS

c

Loca

l pow

er

DFKSS

Figure 2: Local Power of DF and KSS test against a nonlinear ESTAR alternative.

To illustrate the implications of the proposition we plotted the local power curvesunder a fixed-nonlinearity DGP in Figure 2. The DF test dominates again the nonlinearKSS test for all c > 0. Moreover, the curves are almost identical to those in the linearcase, cf. Figure 1.

Should however the variance of the errors εt be small, the overall variability of yt isitself reduced; hitting the middle regime becomes more likely in this case and the nonlineardynamics will play a role under the local alternative as well. Moreover, error variancesestimated in empirical work are actually often rather small; see e.g. Taylor et al. (2001).So the type of quasi-linear asymptotics with fixed nonlinearity specified by Assumption 5does not always capture the salient features of the data.

For this reason we shall study in Section 4 asymptotics for which the middle regimeis not overlooked. We accomplish this by modeling the middle regime as having widthproportional to the variance of yt in order to mimic the effect of a small error variance.We call processes implied by Assumptions 3 and 6 persistently nonlinear. The followingsection deals with the local power of unit root tests under such a DGP.

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4 Persistently nonlinear processes

4.1 Asymptotic results

We now examine the asymptotic behavior of the partial sums of ∆yt under the persis-tent nonlinearity assumption. To characterize it, let X (s) be the diffusion given by thestochastic differential equation

dX (s) = −cX (s)G(g2σ2X2 (s)

)ds+ dW (s) (4)

with starting value X (0) = κσ√2c. We then have the following lemma.

Lemma 3 Under Assumptions 1, 2, 3 and 6 (persistent nonlinearity), it holds that 1

σ√T

[sT ]∑j=1

εj;1

σ√Ty[sT ]

⇒ (W (s) ;X (s))

as T → ∞.


Note that the limiting process X (s) depends on the variance σ2 even when the startingvalue is 0; this is an important difference to the linear case.

The distributions of the linear DF and nonlinear KSS tests under the sequence of localalternatives specified by Assumption 6 is then as follows.

Proposition 3 Under the assumptions of Lemma 3 (persistent nonlinearity), it holdsthat

tDFd→´ 1

0X (s) dW (s)√´ 1

0X2 (s) ds

− c

´ 1

0X2 (s)G (g2σ2X2 (s)) ds√´ 1

0X2 (s) ds

and

tKSSd→´ 10X3 (s) dW (s)√´ 1

0X6 (s) ds

− c

´ 1

0X4 (s)G (g2σ2X2 (s)) ds√´ 1

0X6 (s) ds

as T → ∞.


The extension to usual demeaning or detrending is straightforward: the involved lim-iting are simply replaced with their demeaned or detrendend versions. But Proposition 3allows one to compare the behavior of the studied tests under the different assumptionsabout the nonlinearity. Examining namely the results for linearity and fixed nonlinearity,

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we observe the DF and KSS tests to have nontrivial local power in the same type ofneighborhoods of the null.

Since the power functions are not available in closed form, Subsection 4.2 comparesthe local power properties of the two tests via Monte Carlo simulation. The comparisonwill take different ways of adjusting for deterministic components into account. To thisend, the following proposition derives the asymptotic behavior of the KSS when the datais adjusted for deterministic components as discussed in Subsection 2.2. The results forthe DF test can be obtained analogously. To state the proposition, define the recursivelydemeaned version of the diffusion X (s) ,

Xµ (s) =

0 a.s. s = 0

X (s)− 1s

´ s

0X (r) dr s > 0

;

the recursively detrended version is

Xτ (s) =

0 a.s. s = 0

X (s) + 2s

´ s

0X (r) dr − 6

s2

´ s

0rX (r) dr s > 0

and let

M c =(1 + c)X (1) + c2

´ 1

0sX (s) ds

1 + c+ c2

3

.

We then have the following

Proposition 4 Under the assumptions of Lemma 3 (persistent nonlinearity), it holds asT → ∞ :

a) for local-to-unity GLS adjustment

tµ,cKSS

d→´ 1

0X3 (s) dW (s)√´ 1

0X6 (s) ds

− c

´ 1

0X4 (s)G (g2σ2X2 (s)) ds√´ 1

0X6 (s) ds

and

tτ,cKSS

d→´ 1

0

(X (s)− sM c

)3dW (s)−M c

´ 1

0

(X (s)− sM c

)3ds√´ 1

0

(X (s)− sM c

)6ds

−c

´ 1

0

(X (s)− sM c

)3X (s)G (g2σ2X2 (s))√´ 1

0

(X (s)− sM c

)6ds

;

14

b) for recursive adjustment

tµKSS

d→

´ 1

0

(Xµ (s)

)3dW (s)√´ 1

0

(Xµ (s)

)6ds

− c

´ 1

0

(Xµ (s)

)3X (s)G (g2σ2X2 (s)) ds√´ 1

0

(Xµ (s)

)6ds

and

tτKSSd→

´ 1

0

(Xτ (s)

)3dW (s)−W (1)

´ 1

0

(Xτ (s)

)3ds√´ 1

0

(Xτ (s)

)6ds

−c

´ 1

0

(Xτ (s)

)3X (s)G (g2σ2X2 (s)) ds−

´ 1

0

(Xτ (s)

)3ds´ 1

0X (s)G (g2σ2X2 (s)) ds√´ 1

0

(Xτ (s)

)6ds

.

c) for the max test without adjustment

tmaxKSS

d→ max

tKSS;−tKSS − 3

´ 1

0X2 (s) ds√´ 1

0X6 (s) ds

with tKSS from Proposition 3 being expressed in terms of the same diffusion X (s) ; for themax test with OLS adjustment, replace the process X(s) with its demeaned and detrendedversions.


As expected, the behavior of the KSS test under GLS demeaning is the same as inthe case without deterministics, which is not true under detrending; see also Kapetaniosand Shin (2008). In contrast, the distributions of the nonlinear KSS test with recursiveadjustment and of the max KSS test change for both demeaning and detrending. Thefollowing subsection compares the local power of the DF and the KSS unit root tests fromProposition 4 and the respective effect of the employed adjustment procedures.

4.2 Local power curves

The following setup for the analysis of local power is considered: The sample size equalsT = 1, 000 and the data generating process is given by

∆yt = ϕyt−1(1− exp(−γy2t−1)) + εt, t = 2, ..., T, y0 = 0,

15

5 10 15 20

0.00

0.05

0.10

0.15

0.20

0.25

OLS

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.00

0.05

0.10

0.15

0.20

0.25

GLS

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.00

0.05

0.10

0.15

0.20

0.25

REC

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.00

0.05

0.10

0.15

0.20

0.25

MAX

g2

Loca

l pow

erDFKSS

Figure 3: Local Power against ESTAR, σ2 = 1, c = 5.

with ϕ = −c/T , γ = g2/T , εt ∼ N(0, σ2). Size of the unit root tests is consideredwhenever g2 = 0, power of tests is considered whenever g2 > 0. For the local alternatives,we specify c = {1, 2, ..., 50} and g2 = {0, 1, 2, ..., 20}. Table 1 in the Appendix reportsnumerical results for the selection c = {1, 5, 10, 20, 50} and g2 = {0, 1, 5, 10, 20} whileFigures 3-6 cover the whole range of g2 while c is fixed. Regarding the variance of theerrors, we consider the following scenarios: σ2 = {0.1, 0.25, 0.5, 1, 2, 5}. The number ofreplications is 5,000 for each single Monte Carlo experiment.

Figures 3-6 highlight certain findings by selecting some specific parameter settings.A more general picture can be obtained by inspecting Table 1. Each figure shows fourdifferent mean adjustment schemes for two different tests, namely the DF (straight line)and the KSS test (dashed line). Following our theoretical analysis, we consider OLS, GLSand recursive adjustment as well as the MAX procedure by Leybourne (1995).

In Figures 3 and 4, the error variance is fixed at σ2 = 1, while c takes values 5 and20, respectively. A comparison between the linear DF and the nonlinear KSS test foreach single adjustment scheme reveals that there is no clear ranking. When adjustmentschemes are compared, it becomes evident that the GLS approach is most promising.Recursive adjustment and the MAX procedure are performing very similar and both arebetter in terms of local power than their OLS counterparts. The OLS adjustment leadsto the lowest local power overall. These findings imply, amongst other things, that a KSStest with OLS adjustment is dominated by a DF test with any other adjustment scheme

16

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

OLS

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

GLS

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

REC

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

MAX

g2

Loca

l pow

er

DFKSS


5 10 15 20

0.00

0.10

0.20

0.30

OLS

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.00

0.10

0.20

0.30

GLS

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.00

0.10

0.20

0.30

REC

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.00

0.10

0.20

0.30

MAX

g2

Loca

l pow

er

DFKSS


17

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

OLS

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

GLS

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

REC

g2

Loca

l pow

er

DFKSS

5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

MAX

g2

Loca

l pow

erDFKSS


than OLS. Hence, the results suggest that it may be more important to carefully treat thedeterministic part of the model than to account for nonlinearity when it comes to unitroot testing. This conclusion is further supported by the circumstance that, for a givenadjustment scheme, the local power curve of the two tests are close to each other. Thismeans that even if the KSS test is performing better than the DF test in some situations(e.g. Figure 4 upper left cell, OLS adjustment), the gains in terms of local power aresmall.

Figures 5 and 6 are similar to the previous two, the difference is an error variance of 5instead of 1. The numerical results support our theoretical findings and further indicatethat for a fixed value of c, the degree of nonlinearity (measured by g2) does not have alarge impact on the local power. In fact, after a certain threshold the curves become flat.In addition, the results suggest that the DF test dominates the KSS test in nearly allcases. One exception is found in Figure 5 (upper left cell, OLS adjustment) where bothtests perform similar. Similar to the previous setup with σ2 = 1, the GLS approach turnsout to be most efficient. Recursive adjustment and the MAX procedure are ranked second,while the OLS adjustment performs rather poorly in these settings as well. Moreover, itis evident from the results that the local power is hardly influenced by a change in theshape parameter of the transition function. A change in the autoregressive parameterseems to be more important. These findings nicely corroborate the results found in Choiand Moh (2007).

18

5 Concluding remarks

The paper examines the local power of linear and nonlinear unit root tests under a se-quence of nearly integrated alternatives. They are either linear, nonlinear, or persistentlynonlinear within the STAR framework. The need to investigate persistently nonlinearalternatives arises from the interplay of near integration and nonlinearity: under fixednonlinearity, the near-integrated behavior of the outer regimes dominates asymptoticallyunless the variance of the process is comparable to the width of the middle (unit root)regime.

We derive asymptotic distributions of various unit root test statistics and study theircorresponding local power curves. Under linear and fixed nonlinear nearly-integratedalternatives, the linear DF test dominates the nonlinear KSS test in terms of local power.On the contrary, under persistent nonlinearity, capturing e.g. small variance of the errors,the nonlinear KSS test is more powerful. Furthermore, we find that the efficient removalof deterministic components (e.g. by GLS local-to-unity adjustment) is more important(in terms of local power) than using a specifically nonlinear test. Considering our findings,it might be a more successful strategy to combine linear and nonlinear tests (both withefficient detrending) following e.g. the suggestion of Harvey et al. (2009), rather than toset exclusively on the DF or the KSS test.

While the paper focuses on the univariate exponential STAR model containing a unitroot regime, it is likely that our findings hold for other nonlinear DGPs exhibiting near in-tegration. The findings of Choi and Moh (2007) support the conjecture for other nonlinearDGPs. Furthermore, Kapetanios et al. (2006) address the issue of nonlinear cointegration.The findings of this paper are expected to extend to nonlinear cointegration tests as well,as long as nonlinearity enters the system trough the error correction mechanism and notvia the cointegrating relations only.

Appendix

Auxiliary results

Lemma 4 Under Assumptions 1, 2, 3 and 6, it holds as T → ∞ that

1.1

T

T∑t=2

(∆yt)4 p→ µ4 = E

(ε4t).

2.1

T 4

T∑t=2

y6t−1d→ σ6

ˆ 1

0

X6 (s) ds

19

3.1

T 2

T∑t=2

y3t−1εtd→ σ4

ˆ 1

0

X3 (s) dW (s)

4.1

T 2

T∑t=2

y2t−1 (∆yt)2 d→ σ4

ˆ 1

0

X2 (s) ds

5.1

T 2

T∑t=2

yt−1 (∆yt)3 p→ 0.

Note that, under a fixed nonlinear alternative, the results hold with X (s) replaced byJc;κ (s) .

Proof of Lemma 4

1. The item follows with yt = Op

(√T)

(cf. Lemma 3), the sequence of equalities

1

T

T∑t=2

(∆yt)4 =

1

T

T∑t=2

(εt + ϕTyt−1G (yt−1; γT ))4 =

1

T

T∑t=2

ε4t + op (1) ,

and the Law of Large Numbers.

2. Follows from Lemma 3 with the continuous mapping theorem [CMT].

3. Follows with Theorem 2.2 from Kurtz and Protter (1991), whose conditions (inparticular the martingale difference property of yt−1εt) are easily checked to holdgiven Lemma 3.

4. Write

1

T 2

T∑t=2

y2t−1 (∆yt)2 =

1

T 2

T∑t=2

y2t−1

((∆yt)

2 − σ2)+ σ2 1

T 2

T∑t=2

y2t−1.

Should the first term on the r.h.s. vanish as T → ∞, the result follows with the

20

CMT and Lemma 3. Now,

1

T 2

T∑t=2

y2t−1

((∆yt)

2 − σ2)

=1

T 2

T∑t=2

y2t−1

((εt + ϕTyt−1G (yt−1; γT ))

2 − σ2)

=1

T 2

T∑t=2

y2t−1

(ε2t − σ2 + 2εtϕTyt−1G (yt−1; γT ) + (ϕTyt−1G (yt−1; γT ))

2)=

1

T 2

T∑t=2

y2t−1

(ε2t − σ2

)+

2ϕT

T 2

T∑t=2

y3t−1G (yt−1; γT ) εt +ϕ2T

T 2

T∑t=2

y4t−1G2 (yt−1; γT ) ;

note that y2t−1 (ε2t − σ2) is an md sequence with variance t (µ4 − σ4) under the iid

assumption, so the first summand on the r.h.s. vanishes as T → ∞, as do thesecond and the third thanks to the fact that G is bounded, and the result follows.(If εt is a md sequence with conditional heteroskedasticity, cumulant summability

conditions are required to establish that E

((1T 2

∑Tt=2 y

2t−1 (ε

2t − σ2)

)2)→ 0; see

e.g. Demetrescu, 2010.)

5. Note that

1

T 2

T∑t=2

yt−1 (∆yt)3 =

1

T 2

T∑t=2

yt−1

(ε3t − E

(ε3t))

+ E(ε3t) 1

T 2

T∑t=2

yt−1;

the second term on the r.h.s. vanishes in probability. The result follows with the iidproperty of the errors implying E |yt−1 (ε

3t − E (ε3t ))| = E |yt−1|E |ε3t − E (ε3t )| where

E |yt−1| < C√T ∀1 ≤ t ≤ T because E

(y2t−1

)≤ Ct ∀t, as can easily be checked by

induction.

Proofs

Proof of Lemma 2

The case c = 0 is trivial. For c > 0, define the auxiliary process yt by

∆yt = ϕT yt−1 + εt

with y0 = y0 and note that the weak limit of y[sT ]/√σ2T is precisely Jc,κ (s) since

yt =t∑

j=1

(1 + ϕT )t−j εj + y0 (1 + ϕT )

t

21

and thus

1

σ√Tyt =

1

σ√T

t∑j=1

(1 + ϕT )t−j εj +

κ

σ√2c

((1 + ϕT )

1ϕT

)−c tT+ o (1) .

It then suffices to show thatyt = yt +Op (1)

uniformly in t to establish the desired result. Begin by writing

|yt − yt| ≤ |yt−1 − yt−1|+ |∆yt −∆yt| ;

and note that

∆yt = ϕT yt−1 + ϕT (yt−1 − yt−1)− ϕTyt−1 (1−G (yt−1; γ)) + εt

or|∆yt −∆yt| ≤ |ϕT | |yt−1 − yt−1|+ |ϕTyt−1 (1−G (yt−1; γ))| ,

hence|yt − yt| ≤ (1 + |ϕT |) |yt−1 − yt−1|+ |ϕTyt−1 (1−G (yt−1; γ))| .

Considering the assumption on the tail behavior of 1−G, it follows that

sup1≤t≤T

|ϕTyt−1 (1−G (yt−1; γ))| ≤c supγ∈R+,y∈R |y (1−G (y; γ))|

T=

C

T

uniformly in t for a suitable constant C. With y0 = y0, we then have for all t that

|yt − yt| ≤t∑

j=1

(1 + |ϕT |)jC

T

or

|yt − yt| ≤t∑

j=1

((1 + |ϕT |)

1

|ϕT |)j|ϕT | C

T.

Now, (1 + |ϕT |)1

|ϕT | → e for |ϕT | → 0 so (1 + |ϕT |)1

|ϕT | is bounded; considering thatj |ϕT | ≤ c, we finally obtain

|yt − yt| ≤t∑

j=1

C

T,

as required for the result.

22

Proof of Lemma 3

Weak convergence of standardized partial sums of εt to Wiener process is a standardresult in the literature under this paper’s assumptions.

To prove the remaining results, note that the function G (·) is uniformly Lipschitz sothe solution X(s) of the stochastic differential equation (4) exists uniquely.

Then,

∆yt = ϕTyt−1G(g2σ2 y

2t−1

σ2T

)+ εt + ϕTyt−1 (G− G) ;

since |G− G| ≤ CT(1 + |yt−1|) , we have that

∆yt = ϕTyt−1G(g2σ2 y

2t−1

σ2T

)+ εt +Op

(T−1

)where the Op term is uniform in t. Hence yt−1 will have the same weak limit as yt givenby

∆yt

σ√T

= cyt−1

σ√TG(g2σ2 y

2t−1

σ2T

)1

T+

εt

σ√T

with y0 = y0.

Weak convergence of standardized y[sT ] to X(s) (jointly with T−0.5∑[sT ]

j=1 εj) followsfrom the work of Gikhman and Skorokhod (1969), leading to the desired result; see Kush-ner (1974) for the same convergence under similar conditions for G (·) .

Proof of Proposition 2

Under fixed nonlinearity, we have for the DF test that

tDF =

∑Tt=2 yt−1

(εt − c

Tyt−1G (yt−1; γ)

)σ√∑T

t=2 y2t−1

;

since σ is consistent (the proof is straightforward, cf. the proof of Proposition 3), it followsthat

tDF =1

σ2T

∑Tt=2 yt−1εt√

1σ2T 2

∑Tt=2 y

2t−1

− c

√√√√ 1

σ2T 2

T∑t=2

y2t−1 − c1

σ2T 2

∑Tt=2 y

2t−1 (1−G (yt−1; γ))√1

σ2T 2

∑Tt=2 y

2t−1

+ op (1) .

Since y2 (1−G (y; ·)) is bounded at ±∞, the Lipschitz assumption implies that it isbounded over R. The third term on the r.h.s. hence vanishes as T → ∞. The proof isanalogous for the nonlinear KSS test.

23


Begin with the DF test and show φDF ,

φDF =

∑Tt=2 yt−1∆yt∑T

t=2 y2t−1

, (5)

to be superconsistent. For γT = g2/T , we have along the lines of the proof of Lemma 3that

T∑t=2

yt−1∆yt =T∑t=2

yt−1εt − c1

T

T∑t=2

y2t−1G(g2

Ty2t−1

)+Op

(1

T 2

∑y3t−1

);

furthermore, G is bounded, yt = Op

(√T)

and yt−1εt is a martingale difference, so thenumerator of the r.h.s. of (5) is Op(T ). The denominator in (5) is of exact order Op (T

2),establishing superconsistency of φDF and consistency of σ2 = T−1

∑Tt=2(∆yt− φDFyt−1)

2.

Then,

tDF =

∑Tt=2 yt−1∆yt

σ√∑T

t=2 y2t−1

=1

σ2T

∑Tt=2 yt−1εt√

1σ2T 2

∑Tt=2 y

2t−1

− c

1σ2T 2

∑Tt=2 y

2t−1G

(g2σ2 y

2t−1

σ2T

)√

1σ2T 2

∑Tt=2 y

2t−1

+ op (1) .

Convergence follows with the CMT and Lemma 4 item 4. For the nonlinear KSS test, theresult follows similarly.


a) local-to-unity GLS We have from Elliott et al. (1996) that

yµt = yt − µ

with µ the GLS estimator of µ under an assumed linear local alternative ρ = 1− cT. This

leads to

yµt = yt −y1 +

cT(yT − y1) +

c2

T 2

∑Tt=2 yt−1

1 + c2

T 2 (T − 1)= yt +Op

(T−0.5

).

Moreover, ∆yµt = ∆yt, so the asymptotics of the GLS-demeaned KSS test can only bethat of the KSS test without deterministics.

In the case of detrending, we obtain on the one hand

yτt = yt −t

T(1 + c+ c2

3

) (yT − y1 +c

T

T∑t=2

(t− 1)∆yt +c

T

T∑t=2

yt−1 +c2

T 2

T∑t=2

(t− 1) yt−1

)+Op

(T−0.5

)24

leading with∑T

t=2 (t− 1)∆yt = (T − 1) (yT − y1)−∑T

t=2 yt−1 and the CMT to

1

σ√Tyτt ⇒ X (s)− s

(1 + c)X (1) + c2´ 1

0sX (s) ds

1 + c+ c2

3

= X (s)− sM c.

On the other hand we have that

∆yτt = ∆yt −(1 + c) (yT − y1) +

c2

T 2

∑Tt=2 (t− 1) yt−1

T(1 + c+ c2

3

) + op(T 0.5

)

The result follows again with the CMT, since ∆yt = ϕyt−1G(γy2t−1

)+Op (T

−1) .

b) recursive detrending Like in the proof of Proposition 3, it follows from Assumption6 that

tµNL =

∑Tt=2

(yµt−1

)3εt

σ√∑T

t=2

(yµt−1

)6 − c

T

∑Tt=2

(yµt−1

)3yt−1G

(γTy

2t−1

)σ√∑T

t=2

(yµt−1

)6 + op (1) .

Using Proposition 2 in Born and Demetrescu (2011) together with Lemma 3, we havethat

1

σ√Tyµ[sT ] ⇒ Xµ (s)

jointly with weak convergence of the normalized partial sums of εt. Proposition 2.2 inKurtz and Protter (1991) and the CMT lead to the desired result.

For the test with recursive detrending, we have

tτNL =

∑Tt=2

(yτt−1

)3(εt − ε)

σ√∑T

t=2

(yτt−1

)6 − c

T

∑Tt=2

(yτt−1

)3 (yt−1G

(γTy

2t−1

)− y G (γTy2)

)σ√∑T

t=2

(yτt−1

)6 + op (1) ;

the arguments used for the demeaning case apply analogously to complete the proof.

c) max test The reverse nonlinear KSS test is given by

trKSS =

∑1t=T−1 y

3t+1 (yt − yt+1)

σ√∑1

t=T−1 y6t+1

with σ2 = T−1∑1

t=T−1

(−∆yt+1 − φr

DFy3t+1

)2 and φrDF =

∑1t=T−1 y

3t+1(yt−yt+1)∑1

t=T−1 y6t+1

.

Now,1∑

t=T−1

y3t+1 (yt − yt+1) = −T∑t=2

y3t∆yt = −T∑t=2

(yt−1 +∆yt)3 ∆yt

25

so

1∑t=T−1

y3t+1 (yt − yt+1) = −T∑t=2

(y3t−1∆yt + 3y2t−1 (∆yt)

2 + 3yt−1 (∆yt)3 + (∆yt)

4) .Then, by making use of Lemma 4 items 1 and 5, we obtain that

φrDF = −

∑Tt=2 y

3t εt∑T

t=2 y6t

+c

T

∑Tt=2 y

4tG(γTy

2t−1

)∑Tt=2 y

6t

−∑T

t=2 3y2t−1 (∆yt)

2∑Tt=2 y

6t

+ op (1) ;

note that the 2nd term on the r.h.s. vanishes since hence, by using Lemma 4 again, weobtain

T 2φrDF

d→−´ 1

0X3 (s) dW (s) + c

´ 1

0X4 (s)G (g2σ2X2 (s)) ds− 3

´ 1

0X2 (s) ds

σ2´ 1

0X6 (s) ds

.

As a direct consequence of this convergence rate, the residuals and the residual varianceestimator are consistent. Moving on to the t statistic, we obtain

trKSS = −1

σ4T 2

∑Tt=2 y

3t∆yt

σσ

√1

σ6T 4

∑Tt=2 y

6t

and correspondingly

trKSSd→

−´ 1

0X3 (s) dW (s) + c

´ 1

0X4 (s)G (g2σ2X2 (s)) ds− 3

´ 1

0X2 (s) ds√´ 1

0X6 (s) ds

.

For the max test with OLS adjustment, simply replace the processes with their demeanedand detrended versions to complete the result.

References

Born, B. and M. Demetrescu (2011). Recursive Adjustment for General DeterministicComponents and Improved Tests for the Cointegration Rank. Working paper, Depart-ment of Economics, University of Bonn.

Chang, Y., J. Y. Park, and P. C. B. Phillips (2001). Nonlinear Econometric Modelswith Cointegrated and Deterministically Trending Regressors. The Econometrics Jour-nal 4 (1), 1–36.

Choi, C. Y. and Y. K. Moh (2007). How Useful are Tests for Unit-Root in Distinguishing

26

Unit-Root Processes from Stationary but Non-Linear Processes? The EconometricsJournal 10 (1), 82–112.

Demetrescu, M. (2010). On the Dickey-Fuller Test with White Standard Errors. StatisticalPapers 51 (1), 11–25.

Dickey, D. A. and W. A. Fuller (1979). Distribution of the Estimators for AutoregressiveTime Series with a Unit Root. Journal of the American Statistical Association 74 (366),427–431.

Donauer, S., F. Heinen, and P. Sibbertsen (2010). Identification Problems in ESTAR Mod-els and a New Model. Diskussionspapiere der Wirtschaftswissenschaftlichen Fakultät,Universität Hannover.

Elliott, G., T. J. Rothenberg, and J. H. Stock (1996). Efficient Tests for an AutoregressiveUnit Root. Econometrica 64 (4), 813–836.

Gikhman, I. I. and A. V. Skorokhod (1969). Introduction to the Theory of StochasticProcesses. Saunders, Philadelphia.

Harvey, D. I., S. J. Leybourne, and A. M. R. Taylor (2009). Unit Root Testing in Prac-tice: Dealing with uncertainty over the trend and initial condition. Econometric The-ory 25 (3), 587–636.

de Jong, R. M. (2009). Nonlinear Time Series Models and Weakly Dependent Innovation.Working paper, Department of Economics, The Ohio State University.

Kapetanios, G. and Y. Shin (2006). Unit Root Tests in Three-Regime SETAR Models.Econometrics Journal 9 (2), 252–278.

Kapetanios, G. and Y. Shin (2008). GLS Detrending-Based Unit Root Tests in NonlinearSTAR and SETAR Models. Economics Letters 100 (3), 377–380.

Kapetanios, G., Y. Shin, and A. Snell (2003). Testing for a Unit Root in the NonlinearSTAR Framework. Journal of Econometrics 112 (2), 359–379.

Kapetanios, G., Y. Shin, and A. Snell (2006). Testing for Cointegration in NonlinearSmooth Transition Error Correction Models. Econometric Theory 22 (02), 279–303.

Kurtz, T. G. and P. Protter (1991). Weak Limit Theorems for Stochastic Integrals andStochastic Differential Equations. Annals of Probability 19 (3), 1035–1070.

Kushner, H. J. (1974). On the Weak Convergence of Interpolated Markov Chains to aDiffusion. The Annals of Probability 2 (1), 40–50.

27

Leybourne, S., T. H. Kim, and P. Newbold (2005). Examination of Some More PowerfulModifications of the Dickey-Fuller Test. Journal of Time Series Analysis 26 (3), 355–369.

Leybourne, S. J. (1995). Testing for Unit Roots Using Forward and Reverse Dickey-FullerRegressions. Oxford Bulletin of Economics and Statistics 57 (4), 559–571.

McMillan, D. G. (2007). Bubbles in the Dividend-Price Ratio? Evidence from an asym-metric exponential smooth-transition model. Journal of Banking & Finance 31 (3),787–804.

Nobay, B., I. Paya, and D. A. Peel (2010). Inflation Dynamics in the U.S.: Global butnot local mean reversion. Journal of Money, Credit and Banking 42 (1), 135–150.

Park, J. Y. and P. C. B. Phillips (1999). Asymptotics for Nonlinear Transformations ofIntegrated Time Series. Econometric Theory 15 (3), 269–298.

Park, J. Y. and P. C. B. Phillips (2001). Nonlinear Regressions with Integrated TimeSeries. Econometrica 69 (1), 117–161.

Phillips, P. C. B. (1987). Towards a Unified Asymptotic Theory for Autoregression.Biometrika 74 (3), 535–547.

Phillips, P. C. B. and P. Perron (1988). Testing for a Unit Root in Time Series Regression.Biometrika 75 (2), 335–346.

Rodrigues, P. M. M. (2006). Properties of Recursive Trend-Adjusted Unit Root Tests.Economics Letters 91 (3), 413–419.

Rothe, C. and P. Sibbertsen (2006). Phillips-Perron-Type Unit Root Tests in the Nonlin-ear ESTAR Framework. Allgemeines Statistisches Archiv 90 (3), 439–456.

Sarantis, N. (1999). Modeling Non-Linearities in Real Effective Exchange Rates. Journalof International Money and Finance 18 (1), 27–45.

Sarno, L., M. P. Taylor, and D. A. Peel (2003). Nonlinear Equilibrium Correction inU.S. Real Money Balances, 1869-1997. Journal of Money, Credit and Banking 35 (5),787–799.

Shin, D. W. and B. S. So (2001). Recursive Mean Adjustment for Unit Root Tests. Journalof Time Series Analysis 22 (5), 595–612.

So, B. S. and D. W. Shin (1999). Recursive Mean Adjustment in Time-Series Inferences.Statistics & Probability Letters 43 (1), 65–73.

28

Taylor, A. M. R. (2002). Regression-Based Unit Root Tests With Recursive Mean Ad-justment for Seasonal and Nonseasonal Time Series. Journal of Business & EconomicStatistics 20 (2), 269–281.

Taylor, M. P., D. A. Peel, and L. Sarno (2001). Nonlinear Mean-Reversion in Real Ex-change Rates: Toward a solution to the Purchasing Power Parity puzzles. InternationalEconomic Review 42 (4), 1015–1042.

Teräsvirta, T. (1994). Specification, Estimation, and Evaluation of Smooth TransitionAutoregressive Models. Journal of the American Statistical Association 89 (425), 208–218.

Teräsvirta, T. (2004). Smooth Transition Regression Modeling. In H. Lütkepohl andM. Krätzig (Eds.), Applied Time Series Econometrics. Cambridge University Press.

Teräsvirta, T. and H. M. Anderson (1992). Characterizing Nonlinearities in BusinessCycles Using Smooth Transition Autoregressive Models. Journal of Applied Economet-rics 7 (S1), S119–S136.

Tweedie, R. L. (1975). Sufficient Conditions for Ergodicity and Recurrence of MarkovChains on a General State Space. Stochastic Processes and their Applications 3 (4),385–403.

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Table 1: Size and local power

OLS GLS REC MAXσ2 c g2 DF KSS DF KSS DF KSS DF KSS0.1 1 0 4.9 5.0 5.0 5.0 4.7 4.4 4.8 4.5

1 5.0 4.7 5.2 5.5 4.6 4.8 4.8 4.85 4.9 4.7 5.3 5.3 5.0 4.8 5.5 4.6

10 5.1 5.1 5.0 5.7 4.8 5.0 4.9 5.120 5.2 5.1 5.5 5.7 5.3 5.0 5.2 4.9

5 0 5.0 4.9 5.3 5.3 4.9 4.8 5.1 4.61 5.0 5.0 5.3 5.5 4.9 4.7 5.1 4.85 6.0 5.4 5.7 5.8 5.3 5.3 5.4 5.2

10 5.3 5.1 5.8 6.1 5.2 5.6 5.5 5.520 5.9 5.7 5.3 6.0 5.2 5.2 5.3 5.2

10 0 4.8 4.7 5.3 5.2 4.7 4.9 5.0 5.11 5.3 5.2 5.2 5.3 5.1 5.3 5.4 5.45 5.7 5.3 5.5 5.5 5.2 5.3 5.4 5.2

10 6.1 6.0 5.4 5.5 5.5 5.5 5.5 5.120 6.8 6.6 6.3 6.9 6.5 6.4 6.4 6.3

20 0 4.9 4.9 5.6 5.4 5.0 4.7 5.4 4.81 5.6 5.1 5.1 5.2 5.3 5.1 5.6 4.75 6.1 5.5 5.5 5.8 5.3 5.5 5.4 5.4

10 6.5 6.3 5.7 6.4 6.2 6.5 6.2 6.320 7.5 7.4 6.7 7.8 7.9 8.1 7.5 7.4

50 0 4.8 5.2 5.2 5.6 4.9 4.8 5.0 4.81 5.8 5.5 5.6 5.7 5.4 5.2 5.6 5.05 7.0 6.9 6.0 6.9 6.3 7.1 6.4 6.9

10 8.1 7.3 7.6 8.4 8.8 9.4 8.8 8.220 9.2 9.3 9.9 13.5 12.2 12.0 12.1 12.2

0.25 -1 0 5.0 5.0 5.3 5.2 4.8 4.7 4.9 4.81 5.2 5.5 5.7 5.8 5.3 5.3 5.5 5.35 5.4 5.4 5.3 5.6 5.4 4.9 5.6 4.9

10 5.5 5.1 5.7 5.4 5.4 5.1 5.5 4.920 6.1 5.6 5.5 5.7 5.8 5.7 5.8 5.9

-5 0 5.0 4.7 5.3 5.3 5.1 5.1 5.1 4.81 5.6 5.5 5.2 5.6 5.5 5.3 5.6 5.25 6.3 6.3 6.1 5.8 5.7 6.0 5.8 6.1

10 7.1 6.5 6.2 6.8 6.4 6.9 6.6 6.620 7.3 7.8 7.8 9.2 8.5 9.0 8.7 9.0

-10 0 5.1 4.8 5.3 5.3 4.7 4.7 5.0 4.71 5.9 6.2 5.8 6.3 5.6 5.5 5.7 5.95 6.9 7.0 5.9 7.0 6.6 7.3 6.4 6.9

10 7.7 7.2 7.4 8.8 8.6 9.3 9.0 8.320 9.2 9.5 10.5 13.9 12.7 12.3 12.7 13.1

-20 0 5.0 4.9 5.1 5.2 4.8 4.8 5.0 4.71 6.1 6.2 5.5 6.7 5.7 5.9 6.1 6.15 8.0 7.9 8.0 9.5 9.5 9.5 9.0 8.9

10 9.3 9.0 11.1 15.9 13.3 13.3 13.5 13.020 11.7 11.8 18.1 28.6 19.9 18.3 20.9 19.3

-50 0 4.9 4.8 5.1 5.4 4.9 4.5 5.1 4.71 7.2 6.6 6.2 6.9 6.6 7.2 6.9 7.05 10.5 11.0 13.3 20.1 16.0 15.3 16.3 16.3

10 12.7 13.9 25.8 40.4 23.4 21.9 25.1 24.820 17.2 20.8 52.4 69.9 34.4 33.4 36.6 37.6

Data generating process is given by: ∆yt = ϕyt−1(1 − exp(−γy2t−1)) + εt, withϕ = −c/T , γ = g2/T , εt ∼ N(0, σ2). OLS, GLS, REC and MAX stand for OLSdemeaning, GLS demeaning, recursive mean adjustment and the maximum procedureby Leybourne (1995). DF and KSS denote the linear Dickey-Fuller test and the non-linear Kapetanios et al. (2003) test, respectively. Size of tests is considered wheneverg2 = 0, power of tests is considered whenever g2 > 0.30

OLS GLS REC MAXσ2 c g2 DF KSS DF KSS DF KSS DF KSS0.5 1 0 5.1 5.0 5.4 5.4 4.9 4.9 5.0 4.8

1 5.4 5.3 5.8 6.0 5.0 5.2 5.3 5.15 5.4 5.4 5.6 6.1 5.6 5.5 5.8 5.3

10 5.9 6.1 6.1 6.1 5.8 6.1 5.8 6.020 5.7 5.9 6.3 7.0 6.4 6.6 6.3 6.4

5 0 5.1 5.2 5.5 5.6 5.0 4.6 5.0 4.61 6.1 5.8 5.6 6.3 5.8 5.6 5.8 5.55 7.1 7.2 7.7 8.7 8.5 8.3 8.3 8.3

10 8.6 8.3 10.1 12.6 11.4 11.3 11.3 10.820 8.9 8.7 14.1 17.8 14.6 13.1 15.3 13.6

10 0 5.3 5.2 5.5 5.6 5.3 5.0 5.7 5.01 6.4 6.7 5.3 6.6 6.2 6.4 5.9 6.15 9.1 8.7 10.5 14.3 12.6 12.1 12.7 12.3

10 10.7 10.8 16.2 24.0 17.1 16.5 18.2 16.820 12.5 13.3 28.5 38.2 24.0 20.8 25.6 23.0

20 0 5.2 5.1 5.6 5.7 5.3 5.1 5.5 5.11 7.6 7.3 7.3 8.4 8.0 8.9 8.1 8.15 11.4 11.7 19.1 29.0 19.3 18.0 20.6 19.5

10 14.6 15.6 36.4 51.3 27.2 25.1 29.4 28.620 20.5 24.6 65.3 72.4 41.9 38.2 44.4 42.2

50 0 5.2 5.0 5.4 5.2 5.0 4.7 5.1 4.91 10.1 10.3 11.3 16.0 13.9 14.1 14.3 14.15 17.7 21.6 52.7 70.3 35.4 33.6 38.1 38.4

10 29.0 40.0 86.4 91.1 58.4 54.9 60.8 61.420 57.1 67.4 99.1 98.0 88.0 79.0 89.9 84.0

1 1 0 5.3 5.9 5.2 5.3 5.4 5.1 5.5 5.11 5.4 5.6 5.4 5.7 5.2 5.2 5.4 5.25 5.9 5.6 6.3 6.6 6.3 6.6 6.5 6.0

10 5.7 5.9 7.6 8.2 6.8 6.7 7.0 6.520 6.0 5.6 8.3 8.0 7.3 7.2 7.7 6.9

5 0 4.9 5.2 5.2 5.4 4.9 5.0 5.0 4.91 6.6 6.4 7.4 8.1 7.5 7.8 7.7 7.55 8.9 8.6 14.0 18.6 14.3 12.7 14.9 13.9

10 9.3 10.1 19.8 23.3 17.3 14.9 18.6 16.920 10.3 11.1 25.8 27.8 20.2 17.6 21.4 19.2

10 0 5.3 5.3 5.4 5.8 5.1 4.8 5.2 4.91 8.4 8.4 9.6 12.3 11.2 11.3 11.2 11.55 12.9 13.2 28.7 37.8 24.2 21.2 25.7 23.1

10 15.5 17.1 45.8 50.1 31.7 27.9 33.7 30.320 20.3 21.3 61.4 56.2 40.8 33.6 43.1 36.0

20 0 4.9 5.1 5.4 5.7 5.1 4.8 5.2 4.81 11.0 11.7 15.0 23.4 17.5 16.0 18.2 17.65 20.3 25.0 65.2 73.0 41.6 37.7 44.7 42.5

10 32.5 38.5 88.9 84.9 63.5 53.4 66.0 58.920 52.4 52.1 97.1 89.3 83.1 68.5 85.3 70.7

50 0 5.1 5.2 5.2 5.1 4.8 5.0 4.9 5.01 15.9 18.1 41.9 60.5 30.3 28.5 32.6 32.55 56.2 67.2 99.2 98.2 88.5 78.8 90.3 83.9

10 91.0 88.5 100.0 99.6 99.5 93.9 99.7 95.420 99.6 96.2 100.0 99.9 100.0 98.5 100.0 98.6

Data generating process is given by: ∆yt = ϕyt−1(1 − exp(−γy2t−1)) + εt, withϕ = −c/T , γ = g2/T , εt ∼ N(0, σ2). OLS, GLS, REC and MAX stand for OLSdemeaning, GLS demeaning, recursive mean adjustment and the maximum procedureby Leybourne (1995). DF and KSS denote the linear Dickey-Fuller test and the non-linear Kapetanios et al. (2003) test, respectively. Size of tests is considered wheneverg2 = 0, power of tests is considered whenever g2 > 0.

31

OLS GLS REC MAXσ2 c g2 DF KSS DF KSS DF KSS DF KSS2 1 0 5.2 5.0 5.3 5.6 5.2 4.9 5.4 4.9

1 5.6 6.0 6.4 6.7 6.0 6.1 6.1 6.25 5.9 6.0 7.4 8.0 7.0 7.3 7.1 7.3

10 5.9 5.8 7.9 8.1 7.0 6.6 7.3 6.420 6.0 6.3 7.8 8.2 7.8 7.3 8.1 7.3

5 0 5.0 5.0 5.2 5.4 5.0 4.8 5.1 4.91 8.7 8.6 12.2 16.2 13.5 11.8 13.4 12.55 10.7 10.9 25.0 26.1 20.3 17.7 21.8 18.3

10 11.6 11.4 30.1 28.4 22.3 18.5 23.7 20.120 11.4 11.7 31.4 28.0 23.2 18.5 24.8 20.1

10 0 4.9 5.2 5.2 5.8 4.6 4.7 5.0 4.91 12.3 12.6 23.9 34.1 21.9 20.3 23.5 21.05 19.7 21.0 61.4 56.8 40.6 33.5 43.0 36.0

10 25.0 24.1 70.4 57.0 48.5 37.6 50.6 39.520 27.9 25.3 73.5 55.9 52.5 39.0 54.6 40.1

20 0 5.3 4.8 5.1 5.5 5.0 4.7 5.2 4.61 18.3 21.4 55.1 67.4 36.5 33.8 39.0 37.55 52.2 51.3 97.5 89.0 83.2 67.5 85.3 69.6

10 70.0 59.3 99.2 89.8 92.6 75.6 94.0 75.620 79.3 60.7 99.6 88.3 95.5 77.3 96.3 75.7

50 0 5.3 5.5 5.0 5.3 5.0 5.1 5.1 5.01 45.2 58.3 97.4 96.9 79.5 71.2 82.2 77.55 99.8 96.0 100.0 99.9 100.0 98.5 100.0 98.4

10 100.0 98.1 100.0 99.9 100.0 99.3 100.0 99.420 100.0 98.3 100.0 99.9 100.0 99.6 100.0 99.4

5 1 0 4.8 4.7 4.9 4.8 4.8 4.7 4.9 4.51 6.4 6.5 8.0 8.5 7.9 7.4 8.6 7.45 6.2 6.2 8.3 8.7 7.6 6.9 8.0 7.7

10 6.1 6.0 8.9 8.9 7.9 7.1 8.2 7.420 6.1 5.9 8.7 8.4 7.7 7.0 8.0 7.2

5 0 5.4 4.9 5.3 5.4 5.2 4.7 5.1 5.21 10.1 10.7 27.2 27.8 20.1 17.1 21.5 18.95 11.7 11.8 31.5 28.1 23.0 18.5 24.0 20.2

10 11.8 11.8 32.7 27.6 23.1 19.0 25.0 20.720 12.5 12.1 32.6 27.9 23.8 19.5 25.1 20.4

10 0 5.2 5.0 5.4 5.2 5.0 4.8 5.1 4.91 21.0 22.0 63.9 56.8 42.6 34.7 45.0 36.55 29.0 24.6 74.9 56.6 52.9 39.1 55.3 40.3

10 29.2 24.6 75.0 55.9 54.4 39.4 56.8 40.320 31.4 25.2 76.1 56.9 56.1 40.8 58.3 41.2

20 0 5.3 5.2 5.4 5.6 5.3 4.7 5.3 4.91 58.5 55.3 98.4 89.3 86.7 70.1 88.9 72.75 81.4 60.3 99.7 88.1 96.1 77.0 96.9 75.6

10 84.0 59.4 99.6 86.8 96.7 77.8 97.5 74.620 84.8 59.7 99.7 87.1 97.0 77.9 97.7 74.7

50 0 5.1 5.0 5.3 5.3 5.0 5.1 5.1 4.61 99.9 97.1 100.0 99.9 100.0 98.9 100.0 98.95 100.0 97.9 100.0 99.8 100.0 99.6 100.0 99.3

10 100.0 97.2 100.0 99.7 100.0 99.4 100.0 98.820 100.0 97.3 100.0 99.7 100.0 99.5 100.0 98.9

Data generating process is given by: ∆yt = ϕyt−1(1 − exp(−γy2t−1)) + εt, withϕ = −c/T , γ = g2/T , εt ∼ N(0, σ2). OLS, GLS, REC and MAX stand for OLSdemeaning, GLS demeaning, recursive mean adjustment and the maximum procedureby Leybourne (1995). DF and KSS denote the linear Dickey-Fuller test and the non-linear Kapetanios et al. (2003) test, respectively. Size of tests is considered wheneverg2 = 0, power of tests is considered whenever g2 > 0.

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the power of unit root tests against nonlinear local …...pecially when the dgp is close to the...

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