Amemiya A Ronald Takeshi

John F Gallant

Cheng Geweke

Peter Hsiao

Arnold Robinson

Zellner

Aït-Sahalia Yacine

H Badi

M W Baltagi

Marcus J Brandt

Songnian Chambers

Manfred Chen

Miguel A Deistler

Jean-Marie Delgado

Sylvia Dufour

Eric Fruhwirth- -Schnatter

John C Ghysels

Javier Ham

Han Hidalgo

Yongmiao Hong

Bo E Hong

Maxwell L Honoré

Yuichi King

G M Kitamura

Chung-Ming Koop

Naoto Kuan

Kajal Kunitomo

Q Lahiri

Tong Li

Oliver Li

James G Linton

Robert Mackinnon

Rosa L Mcculloch

Franz C Matzkin

Dale J Palm

Nicholas Poirier

B M Polson

Ingmar Pötscher

Peter C Prucha

Eric Reiss

Frank Renault

Robin Schorfheide

Fallaw Sickles

G J Sowell

Herman Van Den Berg

Dijk Van

H Quang

Edward Vuong

Tom Vytlacil

Andrew Wansbeek

Weiss

Christian Francq
email: christian.francq@univ-lille3.fr

Svetlana Makarova
email: makarova@eu.spb.ru

Jean-Michel Zakoïan
email: zakoian@ensae.fr

A class of stochastic unit-root bilinear processes: Mixing properties and unit-root test

Keywords: JEL classification: C22, C12, C52 Augmented Dickey-Fuller test, Bilinear processes, GARCH, Mixing, Phillips-Perron test, Stochastic unit-roots

A class of stochastic unit-root bilinear processes, allowing for GARCH-type effects with asymmetries, is studied. Necessary and sufficient conditions for the strict and second-order stationarity of the error process are given. The strictly stationary solution is shown to be strongly mixing under mild additional assumptions. It follows that, in this model, the standard (non-stochastic) unit-root tests of Phillips-Perron and Dickey-Fuller are asymptotically valid to detect the presence of a (stochastic) unit-root. The finite sample properties of these tests are studied via Monte Carlo experiments.

A c c e p t e d m a n u s c r i p t 1 Introduction

It is now recognized that many economic series display nonstationarities and nonlinearities. Empirical researchers often find standard linear models, i.e. with iid (independent and identically distributed) innovations, inappropriate for differenced series. For this reason, recent papers dealing with unit roots have been concerned with modeling the error term of the linear dynamics as a non-iid process. Results on estimating and testing unit roots with non-iid errors can be found in [START_REF] Phillips | Time Series Regression with a Unit Root[END_REF], [START_REF] Kim | Some evidence on the accuracy of Phillips-Perron tests using alternative estimates of nuisance parameters[END_REF], [START_REF] Seo | Distribution theory for unit root tests with conditional heteroskedasticity[END_REF], [START_REF] Ling | Asymptotic inference for unit root with GARCH (1,1) errors[END_REF], [START_REF] Ling | Estimation and testing of stationarity for double-autoregressive models[END_REF], [START_REF] Rodrigues | The performance of unit root tests under level-dependent heteroskedasticity[END_REF] and the references therein. [START_REF] Charemza | Conditional testing for unit-root bilinearity in financial time series: some theoretical and empirical results[END_REF] showed that unit-roots models with bilinear errors have interesting economic interpretations, and are empirically relevant. Following this paper, we also consider a unit-root model with bilinear errors, but our specification is different. Our model allows for stationary increments, contrary to the model by [START_REF] Charemza | Conditional testing for unit-root bilinearity in financial time series: some theoretical and empirical results[END_REF]. A natural practice, followed by [START_REF] Charemza | Conditional testing for unit-root bilinearity in financial time series: some theoretical and empirical results[END_REF], is to test for the presence of unit roots in a first step, and then to perform specifications tests on the noise dynamics in a second step. Caution is needed, however, in the blind application of standard unit root tests in the framework of non-iid errors. [START_REF] Rodrigues | The performance of unit root tests under level-dependent heteroskedasticity[END_REF] present numerical experiments showing that non-iid errors may cause severe distortions in conventional unit-root tests. [START_REF] Ling | Estimation and testing of stationarity for double-autoregressive models[END_REF] provided an example of a unit-root model with non-iid errors, namely the so-called double-autoregressive model, in which the LS estimator of the AR coefficient does not converge in law to the standard Dickey-Fuller (DF) distribution. For such models, the most commonly used unit-root tests, i.e. the Phillips-Perron and augmented DF tests, may not have the correct asymptotic size.

An important issue for linear models with non-iid errors thus concerns the validity of those unitroot tests. [START_REF] Phillips | Time Series Regression with a Unit Root[END_REF] and [START_REF] Phillips | Testing for a Unit Root in Time Series Regression[END_REF] showed that, under moment and mixing conditions on the noise process, the unit-root hypothesis can be tested using the standard DF asymptotic distribution. The main goal of this paper is to establish the validity of those standard unit-root tests for the bilinear model under consideration. This requires analyzing in detail the probability structure of the model, in particular its mixing properties. 1 Apart from the unit-root testing problem, these properties have of course independent interest.

The paper is organized as follows. The general model is presented in Section 2, and the existence of stationary solutions is studied. Under a mild additional assumption on the distribution of the 1 Mixing is one way to characterize the decrease of dependence when the variables become sufficiently far apart (see e.g. [START_REF] Davidson | Stochastic Limit Theory[END_REF].

A c c e p t e d m a n u s c r i p t

iid process, the strictly stationary solution is shown to be strongly mixing in Section 3. Section 4 is devoted to examining the validity of the Phillips-Perron and augmented DF unit-root tests in our framework. Monte Carlo experiments are presented in Section 5. Concluding remarks are given in Section 6.

ECM with bilinear innovations

We consider the following ECM(p)-BL(q) model

         ∆y t = φy t-1 + ψ 1 ∆y t-1 + • • • + ψ p ∆y t-p + u t , u t = (1 + b 1 u t-1 + • • • + b q u t-q )ǫ t , ǫ t iid (0, σ 2 ǫ ) (1)
where φ, ψ 1 , . . . , ψ p , b 1 , . . . , b q are real coefficients and σ 2 ǫ > 0. The two equations have the following interpretations. The first one is a standard error correction form of an autoregressive model, capturing the linear behaviour of the possibly I(1) (when φ = 0) series y t . Nonlinear effects are introduced in the second equation through a bilinear specification. Bilinear models have been studied by [START_REF] Granger | An introduction to bilinear time series models[END_REF], who introduced them in the time series literature, by Subba [START_REF] Subba Rao | An introduction to bispectral analysis and bilinear time series models[END_REF] and by many others. As will be seen, under appropriate conditions on the coefficients b i , (u t ) is a white noise, i.e. a centered non-correlated process. It is an iid white noise when all the coefficients b i are equal to zero (i.e. when no nonlinear effects are present in the data).

Bilinearity is a very general way to introduce nonlinearity in economic series. Its main advantage over other types of nonlinearities (e.g. Threshold AR, SETAR, Exponential AR) is that it is compatible with the properties of a noise. These properties are of course shared by ARCH models. The ECM(p)-BL(q) can be seen as an alternative to the extensively studied ECM(p)-ARCH(q). Remark 2.1 The first two conditional moments of u t are given by

E(u t | u t-1 , . . .) = 0, Var(u t | u t-1 , . . .) = (1 + b 1 u t-1 + • • • + b q u t-q ) 2 σ 2 ǫ .
This form of conditional variance is a particular case of the quadratic ARCH, introduced by Sentana (1995), and also a particular case of Linear ARCH (LARCH), introduced by [START_REF] Robinson | Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression[END_REF] and recently studied by [START_REF] Giraitis | A model for long memory conditional heteroscedasticity[END_REF], [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF]. It is seen that the conditional variance is asymmetric: a negative shock u t-i may increase the conditional variance by a larger amount than a positive shock of the same magnitude. This so-called leverage-effect property is often described as one of the main stylized facts of financial time series,

A c c e p t e d m a n u s c r i p t

and constitutes the main motivation of numerous extensions of the standard GARCH models, in particular the EGARCH (see [START_REF] Nelson | Conditional Heteroskedasticity in Asset Returns : a New Approach[END_REF], the GJR (Glosten, Jagannathan and Runkle, 1994), the TARCH [START_REF] Zakoïan | Threshold Heteroskedastic Models[END_REF] and the APARCH models [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF]. Many nonlinear models, for instance the Markov-switching models introduced by [START_REF] Hamilton | A new approach to the economic analysis of nonstationary time series and the business cycle[END_REF], or nonparametric methods (see [START_REF] Pagan | Alternative Models for Conditional Stock Volatility[END_REF] can also be employed to take into account asymmetric conditional heteroscedasticity. An interesting feature of the ECM-BL model, which is transparent on the news impact curve displayed in Figure 1, is that the volatility is not minimal at zero. In other words, an increase of small positive returns may lower volatility. One can imagine that the volatility is minimal when the returns correspond to the free-risk return (-1/b 1 on the figure). This interpretation, as well as the leverage effect, of course require b 1 < 0. Finally, the volatility is not bounded away from 0, contrary to most GARCH models. (1) with q = 1, b 1 = -0.2 and σ ǫ = 1 (full line) compared with the news impact curve of the ARCH(1) model

Var(u t | u t-1 ) u t-1 -
u t = 1 + b 2 1 u 2 t-1 ǫ t (dotted line).
u t = (1 + b 1 u t-1 + b 2 u t-2 )ǫ t , ǫ t iid N (0, 1)
A: second order stationarity, A ∪ B: strict stationarity, and C: non stationary.

Remark 2.2 This model allows for stochastic unit-root interpretations (see [START_REF] Granger | An introduction to stochastic unit-root processes[END_REF], [START_REF] Leybourne | Can Economic Time Series Be Differenced to Stationarity[END_REF] for details on stochastic unit-root models). Taking p = 0 and q = 1, we have y t = ρ t y t-1 + v t , where ρ t = 1 + φ + b 1 ǫ t has mean 1 (justifying the name stochastic unit-root) when φ = 0, and v t = {1b(1 + φ)y t-2 }ǫ t is an error term which is uncorrelated with the y t-i for i > 0.

Remark 2.3 For the dynamics of u t , Charemza, Lifshits and Makarova (2005) used a slightly different bilinear specification given by

u t = ǫ t + bǫ t-1 y t-1 , ǫ t iid (0, σ 2 ǫ ), y 0 = ǫ 0 = 0.

A c c e p t e d m a n u s c r i p t

This model has received an economic interpretation as being derived from a model of speculative behavior. In their paper [START_REF] Charemza | Conditional testing for unit-root bilinearity in financial time series: some theoretical and empirical results[END_REF] were mostly concerned by testing the assumption that b = 0, giving rise to the so-called "b-test". When b = 0 the error term is not stochastically stable (in particular, as demonstrated by the authors, the variance of u t tends to infinity). Therefore this model is not suitable for I(1) series.

Strict stationarity

We first give a condition for the existence of a strictly stationary white noise solution (u t ). Because, strictly speaking, (u t ) does not belong to the standard class of bilinear models2 , existing results, e.g. those established by [START_REF] Liu | On the general bilinear time series model[END_REF], cannot be directly applied. Let u t = (u t , . . . , u t-q+1 ) ′ ∈ R q and c t = (ǫ t , 0, . . . , 0) ′ ∈ R q . Then, the second equation in ( 1) is equivalently written as

u t = c t + A t u t-1 :=   ǫ t 0 q-1   +   b 1:q-1 ǫ t b q ǫ t I q-1 0 q-1   u t-1 , (2)
where b 1:q-1 = (b 1 , . . . , b q-1 ) and I k is the k × k identity matrix. Notice that (c t , A t ) is an iid sequence of matrices. Let A = |a ij | for any matrix A = (a ij ) and let log + x = max(log x, 0)

for any positive number x. Since E(log

+ |ǫ t |) ≤ E|ǫ t | < ∞ we have E(log + A t ) < ∞,

and thus

we can define the top-Lyapunov exponent γ(A) of the sequence A = (A t ):

γ(A) := inf t∈N * 1 t E(log A t A t-1 . . . A 1 ) = lim t→∞ a.s. 1 t log A t A t-1 . . . A 1 . (3)
If γ(A) < 0, the unique strictly stationary solution to (2), in view of [START_REF] Bougerol | Strict stationarity of generalized autoregressive processes[END_REF] Theorem 1.1), is

u t = c t + ∞ k=1 A t A t-1 . . . A t-k+1 c t-k . (4)
It is straightforward that the strict stationarity of (u t ) is equivalent to the strict stationarity of (u t ).

It is also seen that the strictly stationary solution is nonanticipative (i.e. with u t function of the ǫ t-i , i ≥ 0) and ergodic, as a function of the iid process (ǫ t ). By Lemma 2 given in the appendix, and Theorem 2.5 in [START_REF] Bougerol | Strict stationarity of generalized autoregressive processes[END_REF], the sufficient condition γ(A) < 0 is also necessary for the existence of a nonanticipative strictly stationary solution.

When q = 1 we have γ

(A) = E log |b 1 ǫ t | = log |b 1 | + E log |ǫ t |
, and the strict stationarity condition γ(A) < 0 takes the simpler form:

|b 1 | < e -E log |ǫt| .
(5)

A c c e p t e d m a n u s c r i p t

When ǫ t is Gaussian, the necessary and sufficient condition is |b 1 |σ ǫ < 1.88736.

When q > 1, the strict stationarity region can not be given explicitly. In Figure 2, the strict stationarity region has been evaluated using (3) and simulations of the sequence (A t ) in the case q = 2 and ǫ t ∼ N (0, 1) (see [START_REF] Cline | Evaluating the Lyapounov exponent and existence of moments for threshold AR-ARCH models[END_REF] for methods to delineate stationarity regions in a more general framework). The strict stationarity curve passes at the points

(b 1 , b 2 ) = ±e -E log |ǫt| , 0 ,
as can be seen from ( 5), and at the points (b 1 , b 2 ) = 0, ±e -E log |ǫt| , as can be shown by algebraic computations. It is interesting to note that the stationarity region is not symmetric with respect to the diagonal b 1 = b 2 .

Second-order stationarity

Results concerning the existence of second-order stationary solutions of bilinear models are wellknown, and they can be straightforwardly extended to our model. Let (u t ) be a solution to the 2nd equation in (1). Then it is easily seen that E(u t ) = 0 and E(u

t u t-h ) = E(ǫ t )E(1 + b 1 u t-1 + • • • + b q u t-q )u t-h = 0 for any h > 0. Moreover, we have 1 - q i=1 b 2 i σ 2 ǫ Eu 2 t = σ 2 ǫ > 0. It follows that q i=1 b 2 i σ 2 ǫ < 1 (6)
is a necessary condition for second-order stationarity. It is shown in the appendix that the condition is also sufficient. Note that Giraitis, [START_REF] Giraitis | A model for long memory conditional heteroscedasticity[END_REF] show that when q = ∞, condition ( 6) is necessary and sufficient for the existence of a strictly stationary and covariance stationary solution. However, as in GARCH models, strict stationarity is less restrictive than second-order stationarity when q is finite, since our condition for strict stationarity is both necessary and sufficient.

Whether necessary and sufficient conditions for strict stationarity can be obtained for infinite-order

models is an open issue, to our knowledge. The results of this section are summarized in the next theorem.

Theorem 2.1 The second equation of (1) admits a strictly stationary solution (u t ) if and only if γ(A) < 0, where A = (A t ) is defined in (2). Under this condition, the strictly stationary solution is unique, nonanticipative and ergodic. This solution admits a second order moment if and only if

q i=1 b 2 i σ 2 ǫ < 1.
In this case, the solution is a conditionally heteroskedastic white noise.

As illustrated in Figure 2, the second order stationarity region is generally much more restrictive than the strict stationarity region.

A c c e p t e d m a n u s c r i p t 3 Mixing properties

The result of this section is stated in Theorem 3.1 below. It concerns mixing properties of the process (u t ), which will be crucial for applying unit-root tests to Model (1). The proof relies on Markov chain techniques, and consists in showing geometric ergodicity. General conditions for ergodicity and mixing of Markov chains are provided in the book by [START_REF] Meyn | Markov Chains and Stochastic stability[END_REF]. References dealing with mixing properties of various classes of processes can be found in [START_REF] Francq | Mixing properties of a general class of GARCH(1,1) models without moment assumptions on the observed process[END_REF].

Theorem 3.1 Let f be the density of ǫ t and assume that f > 0. If γ(A) < 0, where A = (A t ) is defined in (2), then the strictly stationary solution (u t ) is strongly mixing with geometric rate.

It is worth noting that the mixing property of Theorem 3.1 is shown without moment assumption on u t . The proof of is given in the appendix, and relies on the following lemma.

Lemma 1 Assume that (i) (X t ) is a µ-irreducible Feller chain, for some measure µ on (E, E) whose support has nonempty interior, (ii) (X t ) is an aperiodic chain, (iii) there exists a compact set C ⊂ E, an integer m ≥ 1, and a nonnegative continuous function

(test function) g : E → [0, +∞) such that E[g(X t+m )|X t = x] ≤ (1 -β)g(x) -β, x ∈ C c , E[g(X t+m )|X t = x] ≤ b, x ∈ C,
for some strictly positive constants β and b. Then (X t ) is geometrically ergodic.

Lemma 1 is a criterion for geometric ergocity, which has the particularity of being based on mstep transitions, instead of 1-step transitions as is usually the case. The proof is obtained from a straightforward adaptation of Meyn and Tweedie (1993, Theorem 19.1.3).

Unit-root testing

The Phillips-Perron and augmented DF tests are arguably the most popular unit-root tests. Both of them have been derived under precise assumptions, the validity of which is questionable for the model of this paper.

A c c e p t e d m a n u s c r i p t

Indeed, a number of researchers (e.g. Gonzalo and Lee (1998), [START_REF] Yoon | A re-evaluation of the performance of the unit root and cointegration tests under STUR: no more pitfalls[END_REF] or Rodrigues and Rubia ( 2005)) have found, by means of numerical experiments or with theoretical arguments, that the standard unit-root tests may be in failure, or may suffer from severe size distortion or inconsistency, when the errors are not iid. The result obtained by [START_REF] Ling | Estimation and testing of stationarity for double-autoregressive models[END_REF] is particularly interesting: he shows that for the double autoregressive model

   y t = y t-1 + φy t-1 + ǫ t ǫ t = ω + αy 2 t-1 η t , (7)
the LSE has not the usual DF asymptotic distribution. Thus the standard unit-root tests are not valid (even asymptotically) to test φ = 0 in this model, which has however an ergodic and stationary solution (under appropriate assumptions).

We start by presenting the conditions given by [START_REF] Phillips | Time Series Regression with a Unit Root[END_REF] for the validity of the standard unit-root test, before verifying them on our model.

Phillips-Perron tests

In his seminal paper, [START_REF] Phillips | Time Series Regression with a Unit Root[END_REF] studied the random walk

x t = ax t-1 + v t , a = 1, t = 1, 2, . . . ,
where the initial value x 0 may be any random variable whose distribution is fixed. He showed that the standard least squares estimator ân := n t=2 x t x t-1 / n t=2 x 2 t-1 consistently estimates a = 1, under very general assumptions on the error terms v t . More precisely, denoting by α v (k) the strong mixing coefficients of the process (v t ), Phillips found that under the assumptions i)

Ev t = 0 for all t, ii) ∞ k=1 {α v (k)} ν 2+ν < ∞, for some ν > 0, iii) sup t E |v t | 2+ν < ∞, iv) ϑ 2 v := lim n→∞ Var n -1/2 n t=1 v t exists and ϑ 2 v > 0,
the standardized least squares estimator satisfies

Z φ := n (â n -1) - n 2 σ2 ân 2ŝ 2 v θ2 v -ŝ2 v ⇒ (1/2) W 2 (1) -1 1 0 W 2 (t)dt , (8)

A c c e p t e d m a n u s c r i p t

where {W (t), t ∈ [0, 1]} denotes a standard Brownian motion, θ2 v is a weakly consistent estimator of ϑ 2 v defined in iv) above, σ2 ân = ŝ2 v / n t=2 x 2 t-1 , and

ŝ2 v = 1 n -1 n t=1 (x t -ân x t-1 ) 2 (9)
is a weakly consistent estimator of s 2 v := Ev 2 t . Note that ân , σ2 ân and ŝ2 v are available in any standard regression software. For the estimation of ϑ 2 v , a HAC-type estimator can be used, as proposed by [START_REF] Phillips | Time Series Regression with a Unit Root[END_REF]. Phillips also found the asymptotic distribution of the associated regression t statistics:

Z t := ŝv n θv σân Z φ ⇒ (1/2) W 2 (1) -1 1 0 W 2 (t)dt 1/2 . ( 10
)
Note that the momemt condition (iii) is not satisfied for the double autoregressive model (7). As [START_REF] Ling | Estimation and testing of stationarity for double-autoregressive models[END_REF] showed, the convergences ( 8) and ( 10) do not hold for this model.

Validity of the Phillips-Perron test for the bilinear process

We are interested in testing the unit-root assumption

H 0 : φ = 0
in Model (1). We keep the notation of the previous section, with x t replaced by y t (and thus v t = y ty t-1 ). The next theorem states that ( 8) and ( 10) hold under H 0 . A drift term and/or a deterministic time trend could be added to our model, leading to the limiting distributions obtained by [START_REF] Phillips | Testing for a Unit Root in Time Series Regression[END_REF]. The stochastic unit-root hypothesis can then be tested by the standard Phillips-Perron tests, in exactly the same way as when the unit root is not stochastic.

Theorem 4.1 Let the assumptions of Theorem 3.1 be satisfied. Assume that the zeroes of the polynomial ψ(z) := 1 -p i=1 ψ i z i are outside the unit disk, and the stationary solution of the second equation in (1) satisfies E|u t | 2+ν < ∞ for some ν > 0. Under H 0 the weak convergences ( 8) and (10) hold.

The proof is given in the appendix. The estimator ŝ2

v can be replaced by the simpler estimator n -1 n t=1 (x tx t-1 ) 2 . Phillips (1987, Theorem 4.2) shows that there exists a consistent HAC estimator θ2

v under the addition moment assumption E|u t | 4+ν < ∞. As stated in Theorem 4.1, other estimators than the HAC may be employed. The choice of the estimators of s 2 v and ϑ 2 v may however be important for the behavior of the statistics Z φ and Z t in finite samples and/or under the alternative φ = 0.

A c c e p t e d m a n u s c r i p t

For α ∈ (0, 1), let df φ (α) and df t (α) be the α-quantiles of the distributions defined in the right-hand sides of ( 8) and ( 10). These quantiles are given in Fuller (1976, p. 371). In particular df φ (5%) = -8.1 and df t (5%) = -1.95. The alternative we consider is

H 1 : (1 -z)ψ(z) -φz = 0 when |z| ≤ 1.
Under H 1 we assume that (y t ) is the nonanticipative stationary solution of (1).

lim n→∞ P Z φ ≤ df φ (α) = 1 and lim n→∞ P {Z t ≤ df t (α)} = 1.
The last limit is obtained with the restrictions lim sup n→∞ θ2 v < ∞ a.s and θ2 v > 0 a.s for all n. The consistency of the Z φ -based test is obtained whatever the nonnegative estimator θ2 v .

Augmented DF tests

The approach followed by [START_REF] Dickey | Distribution of the estimators for autoregressive time series with a unit root[END_REF] is based on the pth-order autoregression defined by the first equation of (1):

∆y t = (φ, ψ ′ )X t + u t , where X t = (y t-1 , V ′ t ) ′ ,
V t = (∆y t-1 , . . . ∆y t-p ) ′ and ψ = (ψ 1 , . . . , ψ p ) ′ . The least-squares estimator of (φ, ψ ′ ) ′ is defined by

( φ, ψ′ ) ′ = n t=1 X t X ′ t -1 n t=1 ∆y t X t , ψ = ( ψ1 , . . . , ψp ) ′ .
The following theorem is similar to Theorem 4.1-Corollary 4.1. For the sake of conciseness we only consider the test based on φ, and we omit the studentized version.

Theorem 4.2 Assume Model (1) satisfies the assumptions of Theorem 4.1. Under H 0

DF φ := n φ 1 -ψ1 -• • • -ψp ⇒ (1/2) W 2 (1) -1 1 0 W 2 (t)dt (11)

A c c e p t e d m a n u s c r i p t and, under the additional moment assumption

Eu 4 t < ∞, √ n ψ -ψ ⇒ N 0, Σ ψ := EV t V ′ t -1 Eu 2 t V t V ′ t EV t V ′ t -1 . ( 12
)
We have lim n→∞ P DF φ ≤ df φ (α) = α under the unit-root assumption H 0 , and

lim n→∞ P DF φ ≤ df φ (α) = 1 under the stationarity assumption H 1 .
As in the case of an independent noise, the asymptotic null-distribution of √ n ψψ is the same whether the variable y t-1 is included or not in the regression (of course only in the case φ = 0). However, the asymptotic variance Σ ψ depends on the noise distribution through the b i coefficients and the moments of ǫ (see the example below). This is not surprising because the asymptotic variance of the LS estimator in stationary ARMA models is modified when, in the noise assumptions, independence is replaced by uncorrelatedness (see [START_REF] Francq | Estimating Linear Representations of Nonlinear Processes[END_REF].

Interestingly, this is not the case for the distribution of φ which turns out to be the same as for an independent noise.

In the simple case p = 1 with φ = ψ 1 = 0, b 4 1 < 1/3 and ǫ t ∼ N (0, 1), straightforward computations show that

Σ ψ = (1 -b 2 1 )(1 + 3b 2 1 + 12b 4 1 ) 1 -3b 4 1 .
It is seen that this asymptotic variance can be arbitrarily bigger (for b 1 close to 1/3) than for an iid noise.

Small sample properties of the standard unit-root tests

This section investigates the finite-sample properties of the tests. Partial results of Monte-Carlo experiments are presented in Tables 1-2 below. Complementary results are available from the authors. In the two tables, the relative frequencies of rejection, denoted α, are computed over N = 10, 000 independent replications. Table 1 displays the behavior of the tests under the unit-root hypothesis H 0 . To estimate the long-run variance ϑ 2 v , HAC-type estimators with different kernels have been used. QS stands for the Quadratic-Spectral kernel, TH for the Turkey-Hanning kernel and Tr for the Triangular kernel (see [START_REF] Newey | A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix[END_REF] and [START_REF] Andrews | Heteroskedasticity and autocorrelation consistent covariance matrix estimation[END_REF] for definitions).

To gauge if the difference between α and the nominal level α is significant or not, the statistic 12, and all the other Monte Carlo experiments we performed, can be summarized as follows. The presence of bilinear terms is sensible in finite samples, however the size distortion is tiny for moderate and large sample sizes. Another teaching from our experiments is that the Phillips-Perron test performs slightly better than the augmented DF test.

z = (α -α)(α(1 -α)/N ) -1/2 is computed. Since N is

Conclusion

In this paper we considered a class of AR models with bilinear innovations, in the spirit of [START_REF] Charemza | Conditional testing for unit-root bilinearity in financial time series: some theoretical and empirical results[END_REF] but suitable for I(1) series. This specification can be seen as a stochastic unit-root model. From another viewpoint this model is also of the GARCH type and displays asymmetries. We established necessary and sufficient strict and second-order stationarity conditions. We showed that the strict stationary solution is geometrically ergodic. Testing for unit roots in the presence of conditional heteroscedasticity is clearly important in financial applications, in particular to know if the economic shocks are persistent or not. The ergodicity results were used to demonstrate that the standard Phillips-Perron and augmented DF tests are asymptotically valid in this framework, which is not the case for other stochastic unit-roots models recently considered in the literature. Indeed, [START_REF] Gonzalo | Pitfalls in testing for long run relationships[END_REF] and [START_REF] Yoon | A re-evaluation of the performance of the unit root and cointegration tests under STUR: no more pitfalls[END_REF] showed that the standard unit-root tests do not work for detecting the stochastic unit-root hypothesis H 0 : φ = 0 in the model y t = (1 + φ + α t )y t-1 + ǫ t with α t = ρα t-1 + η t , and [START_REF] Ling | Estimation and testing of stationarity for double-autoregressive models[END_REF] formally showed that standard unit-root tests are not asymptotically valid for the DAR model which, like the model considered in the present paper, is an AR model with GARCH-type innovations. Monte Carlo experiments have also shown that the standard Phillips-Perron and augmented DF tests have good finite sample properties for testing the stochastic unit-root hypothesis in Model (1). From these numerical experiments and the asymptotic study, we draw the conclusion that the range of application of the conventional unit-root tests is broader than the sole detection of deterministic unit-roots.

A c c e p t e d m a n u s c r i p t APPENDIX

We first establish a lemma, which allows to apply Theorem 2.5 in [START_REF] Bougerol | Strict stationarity of generalized autoregressive processes[END_REF]. An affine subspace H of R q is said to be invariant under (2) if it satisfies

∀x ∈ H, A 1 x + c 1 ∈ H a.s. (13)
Model ( 2) is said to be irreducible if R q is the unique invariant affine subspace. Note that this notion of irreducibility is different from the one used in Section 3.

Lemma 2 Model (2) is irreducible.

Proof. For simplicity, we only give the proof for q = 2. The arguments are the same for q > 2, but the proof requires tedious notations in the general case. Let H be an affine subspace of R 2 satisfying (13). By stationarity, we have,

∀x = (x 1 , x 2 ) ′ ∈ H, A 2 (A 1 x + c 1 ) + c 2 = ǫ 2 (b 2 1 x 1 ǫ 1 + b 1 b 2 x 2 ǫ 1 + b 2 x 1 + b 1 ǫ 1 + 1) ǫ 1 (b 1 x 1 + b 2 x 2 + 1) ∈ H a.s. ( 14
)
Taking the expectation of the vector defined in ( 14), we obtain 0 ∈ H. Taking x = 0 in ( 13) and ( 14), we obtain

0 0 ∈ H, ǫ 1 0 ∈ H a.s., ǫ 2 (ǫ 1 b 1 + 1) ǫ 1 ∈ H a.s. ( 15
)
Since σ ǫ > 0, ǫ 1 is not almost surely equal to 0. Thus (15) entails that the linear subspace H = R 2 .

2

Proof that ( 6) is sufficient for the existence of a second-order stationary solution. Let the vector norm X 2 = {EX ′ X} 1/2 , where X ′ denotes the transpose of a vector X belonging to the space L 2 of the square-integrable random variables. We will show that u t defined in (4) is the mean-square limit of the sequence (u tN ) N defined by

u tN = c t + N k=1 A t A t-1 . . . A t-k+1 c t-k .
It suffices to show that (u tN ) N is a Cauchy sequence in L 2 , i.e. that for N ′ > N , u tN ′u tN 2 tends to zero when N goes to infinity. Denote by ⊗ the Kronecker product of matrices and by vec the operator stacking the columns of a matrix (see e.g. [START_REF] Harville | Matrix Algebra from a Statistician's Perspective[END_REF] for details about these matrix operators). We have vec(ABC) = (C ′ ⊗ A)vec(B) and thus

vec(A 1 . . . A n BC n . . . C 1 ) = (C ′ 1 ⊗ A 1 ) . . . (C ′ n ⊗ A n )vec(B) (16)
for any conformable matrices A i , C i and B. It follows that, for N ′ > N ,

u tN ′ -u tN 2 ≤ N ′ k=N +1 A t . . . A t-k+1 c t-k 2 = N ′ k=N +1 {E(A t . . . A t-k+1 c t-k ) ′ (A t . . . A t-k+1 c t-k )} 1/2 = N ′ k=N +1 [E{vec(c ′ t-k A ′ t-k+1 . . . A ′ t A t . . . A t-k+1 c t-k )}] 1/2 = N ′ k=N +1 Ec ′ t ⊗ c ′ t (EA ′ t ⊗ A ′ t ) k vecI q 2 , (17)

A c c e p t e d m a n u s c r i p t

where the inequality follows from the Minkowski inequality, and the last equality follows from ( 16) and the independence of the matrices A t-i and c t-k . Denote by ρ(A) the spectral radius of a square matrix A. Using ρ(A) = lim k→∞ A k 1/k , it can be shown that the right-hand side of (17) tends to 0 as N → ∞ if ρ {E (A t ⊗ A t )} < 1. We have

E (A t ⊗ A t ) = σ 2 ǫ B ⊗ B + J ⊗ J,
where

B = b 1:q-1 b q 0 q-1×q-1 0 q-1 , J = 0 ′ q-1 0 I q-1 0 q-1
.

By induction, it can be shown that

det σ 2 ǫ B ⊗ B + J ⊗ J -λI q 2 = (-λ) q 2 B 1 λ ,
where B(z) = 1 -q i=1 b 2 i σ 2 ǫ z i . It is well-known that the roots of the polynomial B(z) are outside the unit disk if and only if ( 6) holds (see e.g. [START_REF] Francq | Maximum Likelihood Estimation of Pure GARCH and ARMA-GARCH Processes[END_REF], Proposition 1). Thus (6) entails that the spectral radius of EA t ⊗ A t is strictly less than 1, which allows to conclude that (u tN ) N is a Cauchy sequence in L 2 . Therefore u t is in L 2 . Proof of Theorem 3.1. To establish the geometric ergodicity of (u t ) defined by (4) we verify the three conditions of Lemma 1. Let for x = (x 1 , . . . , x q ) ′ ∈ R q , ψ(x) = 1 + q i=1 b i x i . We have

u t = ψ(u t-1 )ǫ t .
Let λ denote the Lebesgue measure on R. For any bounded continuous function h, E(h(u t )|u t-1 = x) = h(ψ(x)ǫ, x 1 , . . . , x q-1 )f (ǫ)λ(dǫ)

is a continuous function of x = (x 1 , . . . , x q ), by continuity of ψ and h and by application of the Lebesgue theorem. It follows that (u t ) is a Feller chain. Now we will check that (u t ) is λ q -irreducible, where λ q is the Lebesgue measure on (R q , B(R q )). To avoid cumbersome notations we will only establish this result when q = 2, the extension to higher dimensions being straightforward. For B ∈ B(R 2 ) and x = (x 1 , x 2 ) ∈ R 2 we have

P 2 (x, B) = P {(u 2 , u 1 ) ∈ B} , where u 1 = ǫ 1 ψ(x), u 2 = ǫ 2 ψ(u 1 , x 1 ). (18)
First consider x such that ψ(x) = 0. Let T x : (ǫ 1 , ǫ 2 ) → (u 1 , u 2 ). Let ǫ 0 1 be the point such that ψ(u 0 1 , x 1 ) = 0 where u 0

1 = ψ(x)ǫ 0 1 . The mapping T x is one-to-one from R \ ǫ 0 1 × R to R \ u 0 1
× R, and admits continuous derivatives. Since (ǫ 1 , ǫ 2 ) admits a density, the changeof-variables theorem shows that (u 1 , u 2 ) also admits a density. In view of (18), it follows that P 2 (x, B) > 0 whenever λ 2 (B) > 0.

Now consider x such that ψ(x) = 0. The previous argument fails because the distribution of (u 1 , u 2 ) = (0, u 2 ) has no density with respect to λ 2 . The problem is easily solved by considering three-steps transition probabilities, and by showing that (u 2 , u 3 ) has a density whenever ψ(0, x 1 ) = 0. When ψ(x) = ψ(0, x 1 ) = 0, four-steps transition probabilities allow to conclude that (u 3 , u 4 ) has a density. Hence for all x, if λ 2 (B) > 0 then P t (x, B) > 0 for some t ∈ {2, 3, 4}. This completes the proof of (i). To prove (ii) we will still limit ourselves to the case q = 2. Let C be a compact subset of R 2 such that λ 2 (C) > 0 and ψ(x) = 0 for any x ∈ C. We have just seen that, for any x ∈ C, P 2 (x, B) > 0 whenever λ 2 (B) > 0. Moreover, by continuity of the function x → P 2 (x, B), the compactness of C entails that inf x∈C P 2 (x, B) = P 2 (x * , B) > 0, for some x * ∈ C. Setting for some x * * ∈ C. By arguments similar to those used in the proof of step (i), we show that P 3 (x, C) > 0 for all x ∈ C, and thus we have P 3 (x * * , C) > 0. Hence C is also ν 5 -small, with ν 5 = P 3 (x * * , C)ν 2 . In view of Meyn and Tweedie (1993, p. 116-118), we can conclude that (u t ) is aperiodic.

Finally, we will verify condition (iii). We will use the following result which, under a slightly different form, is contained in the proof of Lemma 2.3 by [START_REF] Berkes | GARCH processes: structure and estimation[END_REF].

Lemma 3 Let X be an almost surely positive random variable. If EX r < ∞ for some r > 0 and if E log X < 0, then there exists s > 0 such that EX s < 1.

Since γ(A) < 0, there exists an integer k > 0 such that E(log A t A t-1 . . . A t-k ) < 0 (see the first definition of γ(A) given in (3) and use the strict stationarity of the sequence (A t )). On the other hand, we have

E( A t A t-1 . . . A t-k ) ≤ E A t E A t-1 . . . E A t-k ≤ (E A t ) k+1 < ∞ (19)
using the facts that the norm is multiplicative and that the matrices A t are iid. Lemma 3 entails the existence of some s ∈]0, 1[ such that

ρ := E( A t A t-1 . . . A t-k s ) < 1. (20)
By a recursive expansion of the first equality in (2) we get

u t = c t + A t c t-1 + • • • + A t . . . A t-k+1 c t-k + A t . . . A t-k u t-k-1
and thus, the norm being multiplicative,

u t ≤ k i=0 A t . . . A t-i+1 c t-i + A t . . . A t-k u t-k-1 ,
the first term in the sum, for i = 0, being equal to c t by convention. Because s ∈ [0, 1), it follows from the elementary inequality (a + b) s ≤ a s + b s , for a ≥ 0 and b ≥ 0, that

u t s ≤ k i=0 A t . . . A t-i+1 s c t-i s + A t . . . A t-k s u t-k-1 s .
Taking the expectations in both sides, conditionally on u t-k-1 = x, yields

E( u t s | u t-k-1 = x) ≤ k i=0 E A t . . . A t-i+1 s E c t-i s + ρ x s ≤ K + ρ x s . (21)
The first inequality uses the independence between the A t-j and c t-i for i > j, and the independence between these matrices and u t-k-1 for k ≥ i. The latter independence is a consequence of the fact that the stationary solution is nonanticipative. The second inequality in (21) follows from arguments

= {x | (1 -β -ρ) x s ≤ K + β}. Clearly C = ∅ since K + β > 0. Moreover C is compact because 1 -β -ρ > 0.
Thus the righthand side of ( 21) is bounded by a constant over C, and it is bounded by (1β) x sβ over the complement of C. It follows that condition (iii) in Lemma 1 is verified, with g(x) = x s , m = k +1, and β and C chosen as indicated above.

2

Proof of Theorem 4.1. Note that the existence of E|u t | 2+ν entails (6). First consider the case p = 0. Then v t = u t , and i)-iv) are straightforwardly satisfied with ϑ

2 v = s 2 v = σ 2 ǫ /(1 -q i=1 b 2 i σ 2 ǫ )
. Thus, when the DGP does not contain augmented variables, the result directly follows from [START_REF] Phillips | Time Series Regression with a Unit Root[END_REF]. In the case p > 0, it is not obvious to know whether v t = ψ -1 (B)u t := ∞ i=0 c i u t-i inherits the mixing property of (u t ) or not. Fortunately, conditions i)-iv) are not necessary for ( 8) and (10). Conditions i)-iv) are those given by [START_REF] Herrndorf | A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables[END_REF] to establish the functional central limit theorem (FCLT) for (v t ). Other conditions ensuring the FCLT rely on the concept of near-epoch dependence (NED), see [START_REF] Davidson | Stochastic Limit Theory[END_REF]. The process (v t ) is geometrically L 2 -NED on the process (u t ) because the sequence

v t -E (v t |u t-m , . . . , u t+m ) 2 = ∞ i=m+1 |c i | u t-i -E (u t-i |u t-m , . . . , u t+m ) 2 ≤ 2 u t 2 ∞ i=m+1 |c i |
tends to zero at an exponential rate as m → ∞. In view of this property, the exponential decrease of the α-mixing coefficients of (u t ), and the fact that iv) holds with

ϑ 2 v = σ 2 ǫ (1 -p i=1 b 2 i σ 2 ǫ )ψ 2 (1) > 0,
we can conclude from Corollary 29.7 in [START_REF] Davidson | Stochastic Limit Theory[END_REF], that

1 √ nϑ v S [nt] t∈[0,1] ⇒ (W (t)) t∈[0,1] , (22)
where S k = v 1 + . . . + v k and [•] denotes the integer part. As shown by [START_REF] Phillips | Time Series Regression with a Unit Root[END_REF], ( 8) and ( 10) are direct consequences of the FCLT ( 22) and of the continuous mapping theorem, which completes the proof. where the inequality follows from the Cauchy-Schwarz inequality and the fact that the innovations of (y t ) are non degenerated. Let v * t = y tân y t-1 and v * t = y ta * y t-1 . The ergodic theorem also shows that

ŝ2 v = 1 n -1 n t=1 v * 2 t = 1 n -1 n t=1
Λ φ ψ -ψ = Λ -1 n t=1 X t X ′ t Λ -1 -1 Λ -1 n t=1 u t X t ( 23
)
where Λ = Diag(n, √ n, . . . , √ n). We have seen that the functional CLT (22) applies to v t := ∆y t = ψ -1 (B)u t . Therefore the analogue of the results (a) and (b) of Theorem 3.1 in [START_REF] Phillips | Time Series Regression with a Unit Root[END_REF] holds.

Using also the ergodic theorem, we deduce

Λ -1 n t=1 X t X ′ t Λ -1 ⇒ Eu 2 t ψ 2 (1) 1 0 W 2 (t)dt 0 ′ p 0 p EV t V ′ t .
Using Proposition 17.2 in [START_REF] Hamilton | Time Series Analysis[END_REF] and the functional CLT applied to (u t ),

1 n n t=1 u t y t-1 = 1 n n t=1 u t (y 0 + v 1 + • • • + v t-1 ) = 1 nψ(1) n t=1 u t (u 1 + • • • + u t-1 ) + o P (1) ⇒ Eu 2 t 2ψ(1) W 2 (1) -1 .
Moreover it is easy to show that ψ(1) = 1 -ψ1 -

Figure 1 :

 1 Figure 1: News impact curve of u t in Model

Figure 2 :

 2 Figure 2: Strict and second-order stationarity regions of the bilinear modelu t = (1 + b 1 u t-1 + b 2 u t-2 )ǫ t , ǫ t iid N (0, 1)

  large, this statistic roughly follows a standard gaussian distribution when α is the actual size of the test. A small size distortion is observed, but for very large values of b only. The difference between α and α is less important with

  B) = P 2 (x * , B), we define a non-trivial measure on B(R 2 ). It follows that C is a ν 2 -small set. Now, consider the five-step transitions. We haveP 5 (x, B) ≥ C P 3 (x, dy)P 2 (y, B) ≥ P 2 (x * , B) inf x∈C P 3 (x, C) = P 2 (x * , B)P 3 (x * * , C),

  used to show (19). Let β > 0 such that 1β > ρ and let C the subset of [0, +∞) q defined by C

  The following result shows, as an immediate consequence of Theorem 4.1, that the asymptotic level of the standard Phillips-Perron test remains valid in our framework. The consistency is less trivial, and is shown in the appendix.

	Corollary 4.1 We suppose that the assumptions of Theorem 4.1 are satisfied. the unit-root
	assumption H 0 ,
	lim

n→∞ P Z φ ≤ df φ (α) = α and lim n→∞ P {Z t ≤ df t (α)} = α

and under the stationarity assumption H 1 ,

Table 1 :

 1 Empirical size: rejection relative frequencies of H 0 : φ = 0 when the DGP is the URB ∆y t = φy t-1 + u t with φ = 0 and u t = ǫ t + bǫ t u t-1 . The values of the z statistic are given into brackets (z ∼ N (0, 1) if the nominal level α = 0.05 is correct)

	Statistic b	n = 100	n = 1000
	QS-Z φ	0.25 0.054 (1.652)	0.050 (-0.184)
		0.99 0.075 (11.333) 0.062 (5.644)
	TH-Z φ	0.25 0.052 (1.055)	0.049 (-0.275)
		0.99 0.065 (6.791)	0.053 (1.193)
	Tr-Z t	0.25 0.053 (1.514)	0.051 (0.367)
		0.99 0.072 (10.048) 0.053 (1.606)
	DF t	0.25 0.053 (1.193)	0.051 (0.275)
		0.99 0.086 (16.656) 0.077 (12.388)

Table 2 :

 2 Empirical power: rejection relative frequencies of H 0 : φ = 0 when α = 0.05, n = 100 and the DGP is the AR(1)-BL(1) model y t = ay t-1 + u t with u t = ǫ t + bǫ t u t-1 . Z φ and Tr-Z t tests than for the other versions. Table 2 compares the empirical powers of two tests. The powers are very close and do not vary much with b. The output of Tables

	b	a = 0.90	a = 0.99
		Tr-Z t DF t	Tr-Z t DF t
	0.00 0.756 0.766 0.083 0.081
	0.25 0.744 0.772 0.082 0.081
	0.50 0.771 0.774 0.085 0.086
	0.75 0.775 0.775 0.090 0.103
	0.99 0.773 0.752 0.107 0.129

2

  Proof of Corollary 4.1. Under H 1 we havey t = y t-1 + φy t-1 + p i=1 ψ i ∆y t-i + u t = ψ * -1 (B)u t =

i≥0 π i u t-i , where ψ * (z) = (1z)ψ(z)φz.

The process (y t ) is then stationary, ergodic and centered. Thus with probability one, we have ân → a * := Ey t y t-1 Ey 2 t < 1,

  which shows the consistency of the Z φ -based test. The consistency of the Z t -based test comes from

	y 2 t -t = (1 -a * 2 )Ey 2 v * = Ev * 2 → s 2 t .	2â n n -1	n t=1	y t y t-1 +	â2 n n -1	n t=1	y 2 t-1
	Therefore we have almost surely
	lim sup n→∞	Z φ /n = lim sup n→∞	ân -1 -	2 n	1 n t=1 y 2 t	( θ2 v -ŝ2 v )
				≤ a * -1 +	s 2 v * 2Ey 2 t	= -(1 -a * ) 1 -	1 + a * 2	< 0,
	lim sup n→∞	Z t / √	n ≤	Ey 2 t lim sup n→∞ θv	(a * -1) 1 -	1 + a * 2	< 0.

2

Proof of Theorem 4.2. We have

Standard bilinear models only allow terms of the form ǫt-iut-j with i, j > 0.

Acknowledgement: We are grateful to an Associate Editor and two referees for their very helpful comments.

This work was supported by INTAS: research project number 03-51-3714.