Adaptive Consistent Unit Root Tests Based on Autoregressive Threshold Model

Transcription

1 Adaptive Consistent Unit Root ests Based on Autoregressive hreshold Model Frédérique Bec Alain Guay Emmanuel Guerre January 2004 first version: February 2002) Abstract his paper aims to understand how threshold unit-root tests can improve linear ones. We consider SupWald tests from a threshold autoregressive model computed with an adaptive set of thresholds. he claimed theoretical benefits of adaptation are consistency against any stationary ergodic alternative as an ADF test and an asymptotic power improvement compared to a linear Wald unit-root test. A second contribution of the paper is a general asymptotic null limit theory when the threshold variable is a level variable. We obtain a pivotal null limiting distribution under some simple conditions for bounded or unbounded thresholds. A third contribution is examples of adaptive threshold sets. he choice of the auxiliary threshold models as well as the calibration of the threshold sets are specifically designed for nonlinear stationary alternatives relevant for macroeconomic and financial topics involving arbitrage in presence of transaction costs. A Monte-Carlo study and an application to the interest rates spread for French, German, New-Zealander and US post-980 monthly data illustrate the ability of the adaptive SupWald tests to reject unit-root when the ADF does not. JEL Classification: C2, C22, C32, E43 Keywords:Unit root test, threshold autoregressive model, term structure of interest rates. Financial supports from FCAR, CRES and LSA are gratefully acknowledged. Previous versions were presented at the SCSE 2002 conference, macroeconomic seminar of CRES, ESEM 2002, EMM first annual conference and Rencontre d économétrie et statistique Lille3 Littoral We benefited from discussions with D. Andrews, M. Carrasco, C. Francq, R.M de Jong, G. Laroque, A. Rahbek and J.M. Zakoian. CRES ENSAE, imbre J20 3, av. Pierre Larousse, Malakoff Cedex, France. bec@ensae.fr CIRPEE, Université du Québec à Montréal, Canada. guay.alain@uqam.ca LSA Université Paris 6 and CRES-LS, France. eguerre@ccr.jussieu.fr

2 Introduction A debate on the capability of linear unit-root tests to detect nonlinear stationary alternatives has recently grown in the econometric literature. Indeed, the presence of fixed adjustment costs, transaction costs or arbitrage boundaries can create nonlinear adjustments in economic variables quite close to nonstationarity. Economic policy characterized by discrete intervention to manage exchange rate, target zone or inflation output targets could also induce such nonlinear dynamics. Empirical studies as Anderson [997], Michael, Nobay and Peel [997], Obstfeld and aylor [997] or Sollis, Leybourne and Newbold [2002] also argued for nonlinear dynamics. On the other hand, the simulation studies of Balke and Fomby [997], Pippenger and Goering [993] and aylor [200] have risen doubts about the power of standard linear unit-root tests against nonlinear stationary alternatives. As a consequence, a fast developing branch of the econometric literature has proposed as a remedy to use an auxiliary nonlinear dynamic model in place of a linear autoregression to build a unit-root test. his includes among others a hreshold Autoregressive AR) specification as in Bec, Ben Salem and Carrasco [forthcoming], Berben and van Dijk [999], Caner and Hansen [200], Enders and Granger [998], Gonzalez and Gonzalo [998], Kapetanios and Shin [2002], Seo [2003], Shin and Lee [200] and Shin and Lee [2003], or a smooth transition autoregressive specification as in Kapetanios, Shin and Snell [2003]. A substantial difficulty is then that the threshold parameter is not identified under the null. Consequently, much attention has been focused on the null limiting distribution of a threshold unit-root test. But the intuition which has led to consider such a nonlinear specification remains unclear: how threshold unit-root tests can improve on linear ones? Answering this fundamental question requires to consider carefully the alternative so as to compare the competing linear and nonlinear tests. A first contribution of the present paper is to reorient the construction of threshold unit root tests towards consistency and power comparison issues. In place of the linear autoregression of the augmented Dickey-Fuller ADF) statistic, a general threshold specification is considered to serve as an auxiliary model to build a unit-root test. We first show that, in these threshold models, a growing regime gives as a limit a linear augmented autoregression with a non zero autoregressive coefficient under the alternative. As a consequence a Wald statistic for testing vanishing autoregressive coefficients will diverge if computed at such thresholds. However, this threshold is unknown so that, as many of the references above, we propose a SupWald test 2

3 SupWald Λ ) which maximizes the Wald statistic over a set of thresholds Λ, being the sample size. In previous works, a quantile choice of Λ ensuring a minimal percentage of observations in each regime was considered, see e.g. Caner and Hansen [200]. But, due to this restrictive quantile choice, nothing ensures consistency since such Λ does not necessarily contain a threshold associated with a diverging Wald statistic. herefore, a more general construction of Λ should be considered to achieve consistency. Under the alternative, the set Λ should include diverging thresholds corresponding to a growing regime. But, under the null, a large regime is a source of nonidentification and may produce a divergent null limiting distribution. As a consequence, we argue that the behavior of the thresholds set Λ must be sensitive to the hypothesis verified by the data at hand, and we refer to this property as adaptation. he claimed theoretical benefits of adaptation are twofold. First, this gives consistency against nonlinear) stationary ergodic alternatives. his finding clarifies in particular early critics on the possible inconsistency of threshold unit-root tests. Second, under the alternative an adaptive SupWald test has a larger critical region than a linear unit-root Wald test using the same critical value, so that the SupWald test rejects the null more often asymptotically. his finding indicates a possible power improvement when using adaptive threshold unit-root tests compared to linear ones, and therefore justifies the consideration of a more complex threshold model to base a unit-root test. A second contribution is a general asymptotic theory under the null. Such a theory must cope in particular with random thresholds sets Λ and give conditions ensuring a finite null limiting distribution. We consider a general 3-regime AR specification as a baseline model. Following Bec et al. [forthcoming], Berben and van Dijk [999], Enders and Granger [998], Kapetanios and Shin [2002] and Seo [2003], the lagged level variable is chosen as the threshold variable, which is therefore nonstationary under the null. his differs from the choice of Caner and Hansen [200], Gonzalez and Gonzalo [998] and Shin and Lee [2003] who consider an ad hoc stationary threshold. By contrast, our approach is in line with many macroeconomic or financial toy models involving arbitrage behaviour in presence of transaction costs. Moreover, it yields a pivotal null limit distribution which simplifies the implementation of the test. We consider general unbounded thresholds sets Λ covering the quantile choice of Bec et al. [forthcoming] and Berben and van Dijk [999], as well as bounded ones which can improve the power of the test by lowering critical values. Finding the null limiting distribution of such a SupWald test requires 3

4 to establish a new functional version of the limit results of Park and Phillips [200] which can be used in other contexts, as for instance the smooth transition autoregressive specification of Kapetanios et al. [2003]. A third contribution is to give examples of adaptive, unbounded or bounded, sets of thresholds Λ such that the resulting SupWald statistics have a pivotal null limiting distribution. he examples are based on a construction of Λ using the consistent ADF statistic. he unbounded example is a modification of the quantile-based Λ while the bounded example is new. In view of specific macroeconomic applications, we design in the simulation experiment some SupWald tests powerful against stationary alternatives characterized by reversion to a central regime and local nonstationarity, as considered in e.g. Balke and Fomby [997] and aylor [200]. his in particular reveals the interest of adaptive SupWald tests compared to the linear ADF. An application to the yield spread dynamics illustrates the ability of adaptive SupWald tests to detect stationarity when the ADF does not. he remainder of the paper is as follows. Section 2 introduces and motivates adaptation. Section 3 is devoted to the null limiting distribution of the SupWald tests. Section 4 gives the examples of adaptive Λ for the threshold specification used in the simulation experiments of Section 5. Section 6 applies the SupWald tests to the yield spread dynamics. Section 7 gives some final remarks and proofs are gathered in Section 8 and four Appendices. 2 Construction of the test, adaptation and consistency Consider the following null unit-root hypothesis H 0 : y t = al) y t + ε t for t, y 0 = = y p = 0, where {ε t } is a strong) white noise sequence with variance σ 2 and al) is of known order p 0 with roots outside the unit circle, to be tested against any strictly stationary alternatives in H : {y t } is a stationary ergodic process with finite second order moments, such that the matrix Var[y t,..., y t p ] ) has an inverse. 4

5 on the basis of observations y t, t = p + ),...,. he ADF test is based on the auxiliary augmented autoregressive model y t = µ + ρy t + al) y t + v t. 2.) We consider instead a 3-regime hreshold AutoRegression AR) specification. Let {s t } be a threshold variable and λ = [λ, λ 2 ] a threshold vector with λ λ 2 +. Our baseline general AR model of order p is µ + ρ y t + a L) y t if s t, λ ] = I λ), y t = u t + µ 2 + ρ 2 y t + a 2 L) y t if s t λ, λ 2 ) = I 2 λ), µ 3 + ρ 3 y t + a 3 L) y t if s t [λ 2, + ) = I 3 λ). Bec et al. [forthcoming], Berben and van Dijk [999], Enders and Granger [998], Kapetanios and Shin [2002] and Seo [2003] consider a threshold variable s t = y t which is integrated under H 0 but stationary under H, while Caner and Hansen [200] and Shin and Lee [2003] use a stationary s t under H 0 and H as for instance s t = y t, see also Gonzalez and Gonzalo [998]. Our approach assumes that {y t, s t } is stationary under H, hence allowing for all the choices considered in the references above. Various restrictions of 2.2) have been considered in the literature. In addition to the general specification 2.2), we will refer here to the three following examples. Example : 3-regime AR with common dynamics. Kapetanios and Shin [2002] and Seo [2003] considered the restricted AR specification a L) = a 2 L) = a 3 L) imposing furthermore µ 2 = ρ 2 = 0 to build a test of H 0 ), i.e. µ + ρ y t if s t, λ ] = I λ), y t = u t + al) y t + µ 2 + ρ 2 y t if s t λ, λ 2 ) = I 2 λ), µ 3 + ρ 3 y t if s t [λ 2, + ) = I 3 λ). 2.2) Example 2: symmetric mirroring 3-regime AR. Consider Example 2 with λ = λ 2 = λ > 0, µ 3 = µ, ρ 3 = ρ, i.e. µ + ρ y t if s t, λ] = I λ), y t = u t + al) y t + µ 2 + ρ 2 y t if s t λ, λ) = I 2 λ), µ + ρ y t if s t [λ, + ) = I 3 λ). 5

6 his AR model is, for s t = y t, an extended version of the model considered by aylor [200] for the Purchasing Power Parity analysis with transaction costs, see also Balke and Fomby [997]. he central area may be well characterized by a unit-root since proportional transaction costs as measured by λ are likely to prevent arbitrage for small price differentials. he lower and upper regimes mirror each other indicating symmetry of arbitrage. Suitable restrictions on µ and ρ produce a mean-reverting effect towards the central band so that stationarity may hold and PPP is maintained. Besides, such a specification is parsimonious thanks to a symmetric central regime defined with a unique threshold and constrained lower and upper regimes. Example 3: unrestricted 2-regime AR. Caner and Hansen [200] and Shin and Lee [2003] consider a 2-regime specification µ + ρ y t + a L) y t if s t, λ] = I λ), y t = u t + µ 2 + ρ 2 y t + a 2 L) y t if s t λ, + ) = I 2 λ), see also Berben and van Dijk [999] and Enders and Granger [998] under additional parameter restrictions. We introduce now some suitable compact notations to deal with the Examples. All the AR models considered above can be represented as a linear regression model y t = x t λ)β + u t, λ, β) Θ λ R k, 2.3) for a suitable choice of the row vector of covariates x t λ) among the linear combinations of Is t I j λ)), y t Is t I j λ)), y t Is t I j λ)),..., y t p Is t I j λ)), j =, 2, 3. As well, the coefficient β of 2.3) stacks some linear combinations of the primary AR coefficients µ j, ρ j and a j L). As possible in 2.2) and Examples -3, we assume that the choice of the covariates x t λ) in 2.3) is such that the corresponding β is unconstrained, and then can be estimated with Ordinary Least Squares OLS). 2 In 2.2) and Example, Θ λ = {λ R 2 ; λ < λ 2 }, while Θ λ = R + in Example 2 and Θ λ = R in Example 3. he OLS estimators of β and Varu t ) are, Although we limit our analysis to 2 or 3-regime models, many of our results easily extend to a larger number of regimes. 2 For instance in Example 2, such an x tλ) stacks [, y t ]Is t I 2λ)) for the central regime, [Is t I λ)) Is t I 3λ)), y t Is t I λ) I 3λ))] for the outer regimes and 3, and y t,..., y t p. 6

7 for each threshold λ ) β λ) = x tλ)x t λ) x tλ) y t, σ 2 λ) = y t x t λ) β λ)) 2, 2.4) where M denotes the Moore-Penrose inverse of M when M has no inverse. Let us stack the autoregressive coefficients ρ j of the AR model under consideration into a vector ρ. Under H and stationarity of {y t, s t }, the limit ρ j λ) of ρ j λ) writes, for each λ [ ρ λ) = R E x t λ)x t λ ]) [ ] E x t λ) y t where R is a selection matrix with Rβ = ρ. 2.5) he Wald statistic to test that ρ λ) = 0 for a given λ is Wald λ) = ) ) R β λ) R x tλ)x t λ) R R β λ)/ σ 2 λ), 2.6) where the selection matrix R is as in 2.5). Under H, Wald λ) diverges if λ is such that ρ j λ) 0, for at least a regime j, or ρ λ) 0 equivalently. he SupWald statistic studied in this paper is then SupWald Λ ) = sup λ Λ Wald λ) 2.7) where Λ is a random set of admissible threshold vector λ = [λ, λ 2 ] in Θ λ. Consistency of the SupWald test is equivalent to the divergence of the SupWald statistic. his necessitates to show that there is, asymptotically, a threshold λ in Λ such that ρ λ) does not vanish. As explained below, the study of the pseudo-true autoregressive coefficients ρ λ) for all values of λ is a difficult issue, see Remarks 2.2 and 2.3. o show that ρ λ) 0 for some λ, we relate our AR specification to the autoregressive linear model 2.). Indeed, the pseudo-true autoregressive coefficient ρ of 2.) differs from 0 under the alternative, see also Remark 2.. Considering constant coefficients across regimes into 2.2), Examples and 3 directly yield 2.) at the price of an unidentified threshold parameter λ. But this is not true in Example 2 due to the coefficient µ which appears with a different sign in the upper and lower regimes. However, in Example 2, 2.) is a limit model of the AR specification when the regime interval I 2 λ) grows to R, as in 2.2), Examples and 3. he next definition makes more precise such a situation where the autoregressive linear model 2.) is a limit case of a AR when a regime grows. 7

8 Definition Consider a restriction 2.3) of the 3-regime threshold autoregressive model 2.2) such that the parameters µ j, ρ j and a j L) are constant across a subset J of regime indices. Let S be the support of the stationary threshold variable s t and I J λ) = j J I j λ). his restricted autoregressive threshold model nests the augmented linear autoregressive model 2.) through regimes J if and only if, i. here exists a sequence λ n Θ λ such that I J λ n ) S when n goes to infinity i.e. lim n Ps t / I J λ n )) = 0)); ii. he covariates admit a partition x t λ) = [x J t λ), x J t λ)] where x J t λ) groups the variables appearing in regimes J with lim IJ λ) S x J t λ) = [, y t, y t,..., y t p ], x J t λ) = x J t λ)is t / I J λ)) and lim IJ λ) S x J t λ) = 0. All our Examples verify Definition and nest 2.) through regime 2. 3 he introduction of the set of regimes J allows to consider restricted AR models, as for instance assuming identical upper and lower regimes in 2.2) so that the resulting AR nests 2.) through regimes 2 or J = {, 3}. he next lemma studies the pseudo-true autoregressive coefficients ρ j λ) under H, when a regime grows. It is a corner stone result for consistency. Indeed, Lemma is essential to show that the Wald statistic Wald λ) diverges for a well-chosen λ. Lemma Assume that the threshold variable s t is chosen such that {y t, s t } is stationary for any alternative {y t } in H. Let S be the support of the stationary s t. Consider a restriction 2.3) of the AR model 2.2) which nests the linear augmented autoregressive model through regimes J. hen, for any {y t } in H and j J, lim IJ λ) S ρ j λ) < 0. As a consequence, there exists a threshold parameter λ j such that ρ j λ j ) < 0. Remark 2.. he proof of Lemma shows that, under the alternatives of H, the autoregressive coefficient ρ in 2.) is strictly negative. his yields consistency of the ADF or Wald test of ρ = 0 in the augmented linear autoregressive model 2.). 3 Consider for instance Example with J = {2}. Set x J tλ) = [Is t I 2λ)), y t Is t I 2λ)), y t,..., y t p] and x J tλ) = [Is t I λ)), y t Is t I λ)), Is t I 2λ)), y t Is t I 2λ))], Θ λ = R 2 which are such that Definition holds. 8

9 Remark 2.2. Note that Lemma holds for any alternatives of H and then in particular for stationary threshold autoregressive processes driven by 2.2). his allows us to clarify an important issue raised by Balke and Fomby [997] on the problematic use of AR specification for unit-root testing. Chan, Petruccelli, ong and Woolford [985] showed that, in the case p = 0, the solution of µ if y t, λ 0 ] y t = ε t + µ 2 if y t λ 0, λ0 2 ) 2.8) µ 3 if y t [λ 0 2, + ) is stationary provided µ 3 < 0 < µ, although each regime has a unit root since ρ j = 0 for j =, 2, 3. Balke and Fomby [997] consequently argued that his suggests that, in general, just examining the autoregressive parameters... is not enough to determine whether the series is stationary. his is true for 2.8) when limiting to the true value λ 0 of the threshold vector, but Lemma indicates that this does not hold for all λ. herefore a Wald test based on a threshold which, as in a SupWald test, maximizes the Wald statistic for null autoregressive coefficients ρ j λ) can be consistent provided Λ is large enough under the alternative. By contrast, restricting to an estimate of the true threshold vector λ 0, as in Berben and van Dijk [999], may not detect stationary AR alternatives as 2.8). herefore choosing a threshold to perform a threshold unit root test must be interpreted as a selection procedure and not as an estimation issue. Remark 2.3. Studying the pseudo-true values ρ j λ) for any stationary alternatives is known to be a difficult issue. 4 Lemma cannot hold for any threshold vector λ as seen from the threshold model 2.8). he proof of Lemma uses the fact that the restricted AR model nests the linear model 2.) through a regime. his approach also allows to understand how a threshold specification can outperform a linear one in the context of unit-root testing, see heorem 2. As mentioned above, consistency of the SupWald test requires to show that there is, asymptotically, a λ in the threshold set Λ of 2.7) such that the Wald statistic Wald λ) diverges 4 his would give as a by product necessary conditions for stationarity of a general AR while such conditions are only known for p = 0, see Chan et al. [985]. 9

10 under H. We referer to this property of Λ as adaptation, as formally introduced, in connection with Lemma, in the next Definition. Definition 2 Assume that the threshold variable s t is chosen such that {y t, s t } is stationary for any alternative {y t } in H. Let S be the support of the stationary s t. Consider a restriction 2.3) of the AR model 2.2) which nests the linear augmented autoregressive model 2.) through regimes J. hen a random set Λ of admissible thresholds is J -adaptive if and only if, for any alternatives {y t } of H, there exists a deterministic sequence λ in Θ λ with lim I J λ ) = S, and λ is in Λ with a probability tending to. he statistic SupWald Λ ) is J -adaptive if and only if Λ is. Examples of adaptive Λ, based on a preliminary consistent test statistic as the ADF, will be given in Section 4. hese examples are such that Λ goes to the set Θ λ of all admissible under the alternatives. Applying Lemma for an adaptive Λ yields the following consistency-type result. heorem Assume that the threshold variable s t is chosen such that {y t, s t } is stationary for any alternative {y t } in H. Consider a restriction 2.3) of the AR model 2.2) which nests the linear augmented autoregressive model 2.) through regimes J. hen, if Λ is J -adaptive, SupWald Λ ) diverges to + in probability for any {y t } in H. As done in Section 3, heorem must be completed by showing that the SupWald statistic admits a finite null limiting distribution to propose a test. heorem then shows consistency of the asymptotic α-level test which rejects H 0 if SupWald Λ ) z α, z α being the α quantile of the null limiting distribution. By contrast with previous works, consistency is shown to hold against any stationary alternatives of H. 5 heorem therefore shows that adaptive SupWald tests can compete with linear unit-root tests from the consistency view point. he next theorem completes heorem by indicating a possible power improvement of nonlinear unit-root tests compared to linear ones, therefore justifying the introduction of nonlinear models to build a unit-root test. Indeed, under the alternative, an adaptive SupWald test has, at first-order, a 5 See for instance Kapetanios and Shin [2002] which are among the rare papers which tackle the consistency issue. hese authors consider specific threshold stationary alternatives dictated by the auxiliary nonlinear model used to build a test, and hence excludes stationary threshold models as 2.8) among others. See also Kapetanios et al. [2003]. 0

11 larger critical region than a linear Wald test based on the augmented Dickey-Fuller statistic τ using the same critical value, and can therefore reject the null more often asymptotically. heorem 2 Assume that the threshold variable s t is chosen such that {y t, s t } is stationary for any alternative {y t } in H. Consider a restriction 2.3) of the AR model 2.2) which nests the linear augmented autoregressive model through regimes J. Let SupWald Λ ) be a J -adaptive test with an asymptotic level α critical value z α and τ the augmented Dickey-Fuller statistic. hen, for any alternative {y t } in H, { τ 2 z α + o P )) } {SupWald Λ ) z α } so that P SupWald Λ ) z α ) P τ 2 z α + o P )) ). Proving heorem 2 crucially uses adaptation of Λ and the fact that the auxiliary AR model nests the linear autoregressive model 2.), so that SupWald Λ ) τ 2 + o P)) under H. However, the message of heorem 2 must be taken with a word of caution since the asymptotic level of the test which rejects H 0 if τ 2 z α is not controlled. But this indicates that SupWald tests with small critical values should be preferred. An important factor affecting z α is the size of Λ under H 0, i.e. a Λ Λ will produce smaller critical values than Λ. he choice of the auxiliary model is also important, since using a parsimonious AR model gives smaller critical values as well as more precisely estimated ρ j λ) coefficients under H ), or because a specific AR model may be more appropriate on an economic grounds. 3 Asymptotic distributions under the null Let us introduce some notations and a preliminary limit result. We study the null limiting distribution of the SupWald statistic computed from the unrestricted threshold model 2.2) and the restricted AR of Example. We set the threshold variable s t to y t. Let λ, λ ν 0 be three random variables which are assumed to be O P ) under the null. We study the two following types of sets Λ of admissible thresholds: Unbounded Λ : Λ = and {λ = [λ, λ 2 ] ; λ λ λ 2 λ, λ 2 λ ν }, 3.) Bounded Λ : Λ = { λ = [λ, λ 2 ] ; λ λ λ 2 λ, λ 2 λ ν }. 3.2)

12 hese restrictions on λ aim to ensure identification of the AR, i.e. a sufficient number of observations in each regime so that the threshold model can be estimated. he unbounded case 3.) covers as a special case the quantile-based choices of Λ used in Bec et al. [forthcoming] or Berben and van Dijk [999] among others. Before commenting the qualitative difference between the unbounded and bounded cases and introducing our main assumptions, let us introduce a limit theorem for sums of transformations of the y t due to Park and Phillips [200], see also Park and Phillips [999]. he next definitions are from Park and Phillips [200]. A map f ) from R to R is regular if it is continuous in a neighborhood of infinity, and, for any compact subset C of R, there exist some continuous functions f ɛ ) and f ɛ ) with lim ɛ 0 f ɛ f ɛ )w)dw = 0, and δ ɛ > 0 such that f ɛ w ) fw) f ɛ w ) for all w w δ ɛ on C. A finite-dimensional vector of functions is regular if each entry is regular. ypical examples of such functions are the indicators Iw I j λ)) j =, 3, of the lower and upper regimes of the AR model 2.2). A map f ) from R to R is I-regular if it is integrable, square integrable, and satisfies the Lipschitz condition fw) fw ) K w w on its support. A finite-dimensional vector of functions is I-regular if each entry is I-regular. A typical example of such functions is the indicator Iw I 2 λ)) of the central regime of 2.2). he study of such transformations requires the introduction of the local time L W, ) of a standard Brownian motion {W v)} v [0,]. Let A be any Borel subset of R. he integral s 0 I AW v))dv defines a measure called the occupation time which admits a density L W, s) with respect to the Lebesgue measure, so that s 0 I AW v))dv = + I Aw)L W w, s), the so called Occupation ime Formula. he limit distribution of Park and Phillips [200] depends on L W 0, ) which is distributed like the absolute value of a standard normal. More details on the local time can be found in Revuz and Yor [999]. Park and Phillips [200] use the two following assumptions: Assumption Es). he i.i.d. ε t s are such that Eε t = 0 and E ε t 4+s <. he ε t s have a bounded density and lim y y γ E expiyε ) = 0 for some γ > 0. Assumption L. For t y t y t = i=0 π iε t i with y 0 = 0, and where π 0 =, i=0 π i 0 and i= i π i <. Note that H 0 is a special case of Assumption L. he next theorem is a straightforward consequence of heorems 3. and 3.2 in Park and Phillips [200]. 2

13 heorem 3 Park and Phillips [200]) Let σ 2 = Varε t ), σ 2 y = σ 2 i=0 π i) 2, {W v)} v [0,] and {Bw)} w R be two independent standard Brownian motions. 6 Let F and F 2 be collections of regular maps and I-regular maps respectively. hen, under Assumptions L and Es), s > 4, the finite dimensional marginal distributions of { ) yt f, ) yt f ε t, f 2 y t ), /4 } f 2 y t ) ε t, f, f 2 ) F F 2, converge to the ones of { f σ y W v)) dv, σ f σ y W v)) dw v), L W 0, ) σ y 0 0 f 2 w)dw, σl/2 W 0, ) σy /2 f 2 w)dbw) }. he Brownian motion {W v)} v [0,] is the limit in distribution of {y [ v] / } v [0,] as given by the Donsker heorem. he sums of non linear transformations f y t / ) in heorem 3 therefore refer to Donsker asymptotics while we refer to limits of sums of nonlinear transformations f 2 y t ) as local time asymptotics. Finding the null limiting distribution of SupWald statistics requires a functional version of heorem 3 given by heorem B. in Appendix B, see also Lemma B.. 7 As explained after heorem 2, the critical values of the SupWald test are influenced by the size of Λ. Since the Wald statistic is piecewise constant, the SupWald statistic is a maximum over pairs y t, y t ), t, t in Λ so that an interesting indicator of the size of Λ is the number #{y t, y t ) Λ } of such pairs. As seen from heorem 3, we have for λ, λ replaced by deterministic numbers l < l, I l yt ) l = I l y t l ) = O P ), I l y ) t l = O P ), 6 Recall that a Brownian motion over R is defined as Bw) = B +w) for w 0, Bw) = B w) for w < 0, where {B +w)} w R + and {B w)} w R + are two independent Brownian motions over R +. 7 Such a functional result was derived in Wang [2002] for the special case of the random walk i.e. p = 0 in H 0), Donsker asymptotics and indicator functions. heorem B. is more general and is proven with different arguments. heorems 3 and B. also allow to consider smooth transition autoregressive model as in Kapetanios et al. [2003]. 3

14 so that #{y t, y t ) Λ } = O P 2 ) in the unbounded case while #{y t, y t ) Λ } = O P ) in the bounded case. Although a direct comparison of the critical values generated by unbounded or bounded Λ is difficult because such Λ are not nested, one can expect that a bounded thresholds set gives lower critical values. We now give our main assumption on the boundaries λ, λ and ν of the definitions 3.[]0dΩ009f4.242

15 3. Unbounded thresholds he next theorem studies the SupWald statistic for unbounded thresholds under a condition which applies to standard quantile choices of Λ. heorem 4 Consider the unrestricted threshold model 2.2) with s t = y t and let Λ be a set of unbounded thresholds as in 3.). Assume that Assumption Es) holds with s > 4 and that Assumption Λ holds with, almost surely inf W v) < λ < λ < sup W v) and ν > ) v [0,] σ y σ y v [0,] hen, under H 0, SupWald Λ ) converges in distribution to [ ) ) )] λ λ λ ξu 2 + ξ2u 2 + ξ3u 2 sup λ Λ σ y which is finite almost surely. Moreover, if [λ, λ, ν] /σ y has a pivotal distribution i.e. independent of the parameters al) and σ of H 0 ), then SupWald Λ ) is asymptotically pivotal. he key condition of heorem 4 is 3.6) which yields asymptotic identification of each regime since the limit Brownian motion {W v)} v [0,] of {y [ v] / } v [0,] will visit all the regimes I j λ), for any λ in the limit threshold set Λ defined in 3.3). 8 Condition 3.6) holds in particular for usual quantile choice of λ, λ and ν as used in Bec et al. [forthcoming], Berben and van Dijk [999], see also heorem 6 below. Bec et al. [forthcoming] used a modified version of heorem 4 to account for a symmetric central regime. Remark 3.. Note that the SupWald statistic has a pivotal null limiting distribution under a rather simple condition so that the test can be implemented through comparison with asymptotic critical values. his contrasts with Caner and Hansen [200] approach which uses a bootstrap procedure to correct the influence of some nuisance parameters under the null. his is due to the choice of y t as the threshold variable while Caner and Hansen [200] analysis retains a stationary variable. Remark 3.2. he null limiting distribution of heorem 4 accounts for the presence of the three autoregressive coefficients ρ j λ) in 2.2) through the variables ξ ju λ), j =, 2, 3, and for the 8 In the case of a simplified AR model of order and for random walks, Wang [2002] noted that a condition as 3.6) was needed to avoid diverging test statistics, but did not formally introduced it. σ y σ y 5

16 non identification of the threshold parameters λ through a supremum over Λ. By contrast with the limit distribution of the squared ADF statistic τ 2, this a major drawback of the SupWald approach, since it may generate high critical values so that the resulting test may have a low finite-sample power, see the discussion following heorem 2. Remark 3.3. he proof of heorem 4 relies on Lemma C. in Appendix C which allows to consider various restrictions of the general AR specification 2.2. Considering the restricted threshold model of Example instead of 2.2) gives the null limiting distribution of heorem 4, but under the weaker moment condition s < 4. Considering the restricted threshold model of Example 2 would give a limiting distribution with two components, see heorem 6 below. Remark 3.4. heorem 4 admits a modification suitable to 2-regime AR model as in Example 3 which will give a two-component null limit distribution. his requires a drastically different choice of Λ since the identification Condition 3.6) requires Pν > 0) = so that λ = λ 2 as needed in Example 3 is impossible. A specific set of thresholds {λ; λ λ = λ 2 λ } must be used to implement a SupWald test with a two-regime model. Specification tests can be proposed to select a threshold model but their outcomes are difficult to predict under H 0 due to nonstationarity. his may complicate the null limiting distribution of the resulting threshold unit-root test so that choosing an ad hoc model could be preferred. 3.2 Bounded thresholds he next theorem investigates the case of bounded thresholds as in 3.2) for the restricted threshold model of Example. heorem 5 Consider the restricted threshold model of Example with s t = y t and let Λ be a set of bounded threshold parameters as in 3.2). Assume that Assumption Es) holds with s > 4, and that Assumption Λ holds with ν > 0 almost surely. Let Λ be as in 3.3), ξ ju ) be as in 3.4), ξ 2B ) be as in 3.5). hen, under H 0, SupWald Λ ) converges in distribution to ξu0) 2 + sup ξ2bλ) 2 + ξ3u0) 2 λ Λ which is finite almost surely and has a pivotal distribution if λ/ν, λ/ν) has a pivotal distribution. 6

17 Compared to heorem 4, heorem 5 holds under the weaker condition that ν > 0 in place of 3.6). Indeed, under this condition, y t will visit infinitely often the central regime I 2 λ) for any λ in Λ since L W 0, ) is strictly positive. Continuity of the local time ensures that the random variables ξ U 0) and ξ 2U 0) are finite since it yields 0 IW v) < 0)dv > 0 and 0 IW v) > 0)dv > 0 as also seen from the Arcsine Law, see Revuz and Yor [999]. Remark 3.5. As in heorem 4, the SupWald statistic has a pivotal null limiting distribution under a rather simple condition. his condition differs from 3.6) in heorem 4 due to the specific dependence with respect to σ y and σ of local time asymptotics as given in heorem 3. Note that the null limiting distribution does not depend on the local time L W 0, ) due to standardization of the Wald statistic. Remark 3.6. he structure of the null limiting distribution of heorem 5 differs from the one of heorem 4 in several aspects. First, the components ξ ju λ), j F909)

18 Remark 3.8. heorem 5 is proven under the restricted threshold model of Example. Estimating a central dynamic term a 2 L) as in 2.2) involves the difficult study of the items /4 y t kiy t I 2 λ))ε t, k p, with the low standardization by /4 of local time asymptotics. 4 Examples of adaptive consistent tests We combine now the results of Section 2 and 3 to build an adaptive consistent threshold unit root test with a pivotal null limiting distribution. We consider a two-step construction of the SupWald test, where the first step uses a preliminary consistent ADF statistic of H 0 to build the threshold set Λ. We propose here examples of adaptive unbounded and unbounded Λ will be compared in our simulation experiment. We consider the AR model of Example 2 µ + ρ y t if y t I λ) =, λ], y t = al) y t + u t + µ 2 + ρ 2 y t if y t I 2 λ) = λ, λ), µ + ρ y t if y t I 3 λ) = [λ, + ), that λ R +. 4.) he corresponding unbounded and bounded Λ are such that λ = λ and simplify to Unbounded Λ : Λ = {λ R; 2 ν λ λ }, 4.2) Bounded Λ : Λ = { λ R; 2ν λ λ }. 4.3) Our examples of adaptive Λ depend on the empirical quantiles. For q in [0, ], let y q) = y q, ) be the q-th order statistic computed from y 0,..., y, i.e. y q) = inf{y; I y t y) q}. For the Brownian motion W and q in 0, ), let W q) be the quantile of order q computed from the path {W v)} v [0,], i.e. W q) = inf{w; 0 I W v) w)dv q}. Let τ be the Augmented Dickey-Fuller statistic for H 0 and τ its limit in distribution under H 0, that is τ = 0 W v)dw v) W ) 0 [ W v)dv ) ]. 4.4) 0 W 2 2 /2 v)dv 0 W v)dv 8

19 As a scaling factor we use an estimator σ 2 0 of the variance Varv t) of the linear autoregressive model 2.), i.e. σ 2 0 = p + 2) y t γ 0 γ y t γ p+ y t p ) 2 4.5) where γ 0, γ,..., γ p+ are the OLS estimates of the regression coefficients of y t on, y t,..., y t p. Due to the constraints across the upper and lower regimes of 4.), the null limiting distributions of our SupWald statistics now depend upon ζ U λ) = [ 0 W 2 v)i W v) λ) dv ) ] 0 I W v) λ) dv 2 /2 0 W v)i W v) λ) dv ) /2 0 I W v) λ) dv + ζ 2B λ) = Bλ3 ) λ 3 0 W v)i W v) λ) dv, [ 0 0 I W v) λ) dv ] sgn λ W v)) dw v) where sgn λ w) = Iw λ) Iw λ) = Iw I 3 λ)) Iw I λ)). 0 W v)sgn λ W v)) dw v), 4.6) 4.7) 4. Unbounded thresholds Our example of adaptive unbounded Λ modifies the popular quantile choice of the threshold set to achieve adaptation. Let l > 0 be a length parameter to be chosen in the simulation experiment. Define, for 0 < q < /2, 2 ν = y q, λ = y q ) with q = min q) + l τ, 2). 4.8) his choice is such that there are respectively q% and q % of the total sample of y t in the inner and outer regimes of 4.) for any λ of Λ, with q 2 so that the parameters of the outer regime can be estimated. Under H 0, τ is bounded away from 0 and infinity. Since I y t y) converges in distribution to 0 Iσ y W v) y)dv by heorem 3, the null limiting distribution of y q) / is σ y W q) so that the null limiting distributions [λ, ν] of [λ, ν ] is 2ν = σ y W q), λ = σ y W q). 4.9) 9

20 It follows that the set Λ U defined through 4.2) 4.8) is an example of unbounded Λ which verifies the identification Condition 3.6) of heorem 4. Under the alternative, the quantiles y q ) converge to the quantiles of order p of y t by the Glivenko-Cantelli heorem, σ 0 remains bounded, but the ADF statistic now diverges, so that λ converges to the upper bound of the support S y of y t. herefore Λ U defined through 4.2) 4.8) is 2-adaptive ensuring consistency of the SupWald Λ U ) test. he properties of this test are summarized in the next theorem. heorem 6 Consider the hreshold model 4.). Let Λ U be as in 4.2), 4.8), ΛU be as in 3.3), 4.9), and ζ U ), ξ 2U ) be as in 4.6) and 3.4). hen: i. Assume that Es) holds with s > 4. Under H 0, SupWald Λ U ) converges in distribution to sup λ Λ U ζ 2 U λ) + ξ2u 2 λ)), which is finite almost surely and has a pivotal distribution. ii. he threshold set Λ U consistent against any alternative of H. is 2-adaptive, so that the asymptotic α-level SupWaldΛU ) test is 4.2 Bounded thresholds Our example of adaptive bounded Λ is new. By contrast with Λ U, it only depends upon a length parameter l. Let Λ B = {λ R; 2ν λ λ } and Λ B = {λ R; 2ν λ λ} with 2ν = y 3) + σ 0 l τ and λ = 2ν + l σ 0 τ, 4.0) 2ν = σ l τ and λ = 2ν + σl τ. 4.) his choice of ν is such that there is at least two observations left in the inner regime which is sufficient for estimation. he choice of λ does not allow for such a simple control of the number of observations in the outer regime, which is O P ) under H 0. Under H 0, y 3) converges in distribution to W 0) = 0 so that [λ, ν ] converges in distribution to [λ, ν]. Under H, ν converges to the lower bound of the support S y of y t and λ diverges to + so that Λ B 2-adaptive. he properties of the SupWald Λ B ) test are summarized in the next theorem. is heorem 7 Consider the hreshold model 4.). Let Λ B be as in 4.3), 4.0), ΛU be as in 3.3), 4.), and ζ U ), ζ 2B ) be as in 4.6), 4.7). hen: 20

21 i. Assume that Es) holds with s > 4. Under H 0, SupWald Λ B ) converges in distribution to ζu 2 0) + sup λ Λ B ζ2 2B λ), which is finite almost surely and has a pivotal distribution. ii. he threshold set Λ B consistent against any alternative of H. is 2-adaptive so that the asymptotic α-level SupWaldΛB ) test is 5 Simulation experiments As addressed in Caner and Hansen [200] in analogy with a discussion in Andrews [993] concerning trimming in tests for a structural change, an ideal choice of a threshold set Λ for a SupWald test should be based on a trade-off between the null and the alternative. Although Caner and Hansen [200] consider quantile choices of Λ, this also applies in a general set up, asymptotically or for finite sample. As noted in Caner and Hansen [200], a large Λ under the null generates large critical values and, under the alternative, rejection of H 0 requires large values of SupWald Λ ), hence decreasing the power of the test. By contrast, a large Λ suitable under the alternative. Although adaptation can give large Λ asymptotically as in our examples, such a trade-off is useful to select an adaptive Λ in finite sample. However, such a general consideration is difficult to apply due to a too vast H over which it is impossible to evaluate the power of a SupWald test for each alternative. It is therefore more relevant in practice to be more directional and to calibrate a threshold set using a relevant set of alternatives for the alternatives in mind, as done when selecting the AR specification 4.) of Example 2. As suggested in Balke and Fomby [997] and aylor [200], we use for calibration a class of processes with reversion to a central band characterized by local stationarity. We specifically consider such 3-regime threshold models. We then study the power of the resulting SupWald test against Autoregressive Conditional Root models as proposed by Gouriéroux and Robert [2002] and Rahbek and Shephard [2002] and linear alternatives. We mostly use the sample size = 200 corresponding to our application. o ease calibration of tests, we change τ into max, τ ) in 4.8) and 4.0). he length parameter l for Λ B 6. For the unbounded threshold set Λ U, we fix q = 0.5 and set l = 0. hese values of l are chosen to achieve a good power against AR alternatives. is is 2

22 5. Critical values able gives the critical values based on 40,000 simulations of different sample sizes. Note that these critical values are much higher than the squared ones of ADF test. For instance, at the 5% level, the squared critical values of the ADF test is 2.88) 2 = which is smaller than the one of our test. Higher critical values is here a price to pay for the introduction of a threshold parameter. As shown later on in the simulation experiments, this will have some consequences on the relative power of our tests with respect to the ADF test for linear or close to linear DGPs. Note also that the critical values of the test based upon unbounded Λ U are larger in small and medium samples than the ones associated to bounded Λ B, suggesting that the length of Λ is larger in mean than the one for the bounded Λ. he last column for SupWaldΛ U ) and SupWaldΛB ) contains the percentage of y t < Λ and confirms this conjecture. For instance, with a sample size of 200, the percentage of observations in Λ U is 20 % greater than the one for Λ B. Moreover, as expected, the percentage of y t in the inner regime for Λ B decreases sharply with the sample size. his comes from the fact that the number of observations in the bounded interval increases by order O P ) and, then, the percentage of observations in the interval converges to zero. he critical values of the two tests become closer when increases, suggesting that the maximum of the Wald λ) statistic is achieved in mean for moderate thresholds λ. able : Critical values 40,000 simulations) SupWaldΛ U ) SupWaldΛB ) Sample size 5 % 0% 5% % % in Λ U 5 % 0% 5% % % in Λ B

23 5.2 AR alternatives In order to investigate the effect of the choice of the threshold values on the power of the test, we consider the AR alternatives with an integrated inner regime µ + ρ y t if y t λ, y t = a y t + ε t + ρ 2 y t if y t < λ, µ + ρ y t if y t λ,, with µ =.3 ρ λ, ρ 2 = 0. 5.) and ε t is an i.i.d. N 0, ). able 2 reports the power of the ADF test, SupWaldΛ U ) and SupWaldΛ B ). he values in parenthesis in ables 2 to 4 are percentages of y t contained in Λ U and ΛB. As expected these values are greater than the ones under the null see table ) especially for Λ B. his illustrates the adaptive behavior of ΛU percentage of data in the stationary regimes. and ΛB. Finally, % denotes the he tests based on SupWaldΛ U ) and SupWaldΛB ) generally outperform the standard ADF except for cases where the percentage in the stationary regimes is more important. However, for these cases the power of adaptive tests is close to the power of the standard ADF especially for SupWaldΛ B ). For processes characterized by a percentage of data in the stationary regimes below 0%, the gain of the adaptative tests compared to the standard ADF can be as high as 67%. Finally, the test based on SupWaldΛ B ) outperforms the one based on SupWaldΛU ) for all the cases. his gain in power by the bounded interval compared to unbounded interval is due to the fact that the critical values of SupWaldΛ B ) are relatively small since the percentage of observations in Λ U and ΛB are very close. 5.3 Other alternatives Let us now proceed to simulation experiments so as to check the consistency of our SupWald test against a broader set of stationary alternatives, either linear or not. More precisely, we will consider below a stationary autoregressive process HAR a ) and an Autoregressive Conditional Root model HACR a ) proposed by Gouriéroux and Robert [2002] and Rahbek and Shephard [2002]. hose alternatives are respectively given by: H a AR : y t = µ + ρy t + a y t + ε t, 5.2) 23

24 able 2: Empirical power of the unit root tests α = 5%, = 200,,000 simulations) a ρ ρ 2 λ % ADF SupWaldΛ U ) SupWaldΛB ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) where a = 0.3, and H a ACR : y t = + ρ) st y t + ε t, 5.3) where s t is binomial given the past with p t = Ps t = y t, ε t ) = [ + exp α + βy 2 t ))], ρ is a real number, β is non-negative and α and β are finite. In the two models considered here, ε t is an i.i.d. N 0, ). he sample size is = 200. able 3 reports the results for HAR a. As expected, the ADF test outperforms the unit root test based on the threshold specification in the linear stationary case. Indeed, the ADF test is specifically designed for this alternative. As mentioned below able, our test implies the 24

25 able 3: Empirical power of the unit root tests for linear model α = 5%, = 200,,000 simulations) µ ρ ADF SupWaldΛ U ) SupWaldΛB ) ) ) ) ) ) ) ) ) ) ) ) ) computation of the supremum over an interval which involves a loss of power compared to the ADF test for a linear alternative. However, the performance of our tests is quite reasonable. Here again, the power of the tests based on bounded threshold is slightly superior to unbounded threshold. he Markov ACR model exhibits local non stationarity when s t = 0, which is more likely to arise if α + βyt 2 is small. When β > 0 as in our simulation experiment, this source of local stationarity corresponds to a central regime, but with a less precise delimitation than for the AR model 2.). Indeed, due to the randomness of s t, local nonstationarity may also hold outside a central zone. Even though the degree of local nonstationarity of the ACR model is related to the parameters α, β), it is worth computing the percentage of time spent in the stationary regime % in able 4) for interpretation s sake. he values of the parameters are motivated by the example given by Rahbek and Shephard [2002]. he results are reported in able 4. he ADF test slightly dominates the SupWald tests in the case where the time spent in the stationary regime s t = ) is important. In the other cases, the unit root tests based on the threshold specification does remarkably well while the ADF test has poor power. In particular, for processes characterized by a time spent in the stationary regime below 0 %, the SupWald tests clearly dominate the ADF test. For instance, with parameter values equal to α = 00, β =. and ρ = 0.30 and a time spent in the stationary regime equal to 2.2%, the rejection 25

26 able 4: Empirical power of the unit root tests for the Markov ACR alternative α = 5%, = 200,,000 simulations) α β ρ % ADF SupWaldΛ U ) SupWaldΛB ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) rate of SupWald tests based on the bounded interval is 80.6 percent compared to 4.5 percent for the ADF test. Finally, the power of SupWaldΛ B ) slightly dominates the power of SupWaldΛU ) for 3 cases out of 8. According to these simulation results, the tests proposed in this paper seem to be able to detect globally stationary process with a relatively important nonstationary component. his simulation study based on three alternative specifications clearly reveals the usefulness of the proposed tests based on an adaptive choice of the threshold set. It then appears important for practitioners to perform our tests in conjunction with the traditional ADF test. 26