Testing for stationarity-ergodicity and for comovements ... · tic comovement, which correspond to...

Journal of Econometrics 96 (2000) 39}73

Testing for stationarity-ergodicity and forcomovements between nonlinear discrete time

Markov processes

Valentina Corradi!, Norman R. Swanson", Halbert White#,*!University of Pennsylvania, Philadelphia, PA 19104, USA

"Pennsylvania State University, University Park, PA 16802, USA#University of California-San Diego, Department of Economics, 9500 Gilman Driver,

La Jolla CA 92093-0508, USA

Abstract

In this paper we introduce a class of nonlinear data generating processes (DGPs) thatare "rst order Markov and can be represented as the sum of a linear plus a boundednonlinear component. We use the concepts of geometric ergodicity and of linear stochas-tic comovement, which correspond to the linear concepts of integratedness and cointeg-ratedness, to characterize the DGPs. We show that the stationarity test due toKwiatowski et al. (1992, Journal of Econometrics, 54, 159}178) and the cointegration testof Shin (1994, Econometric Theory, 10, 91}115) are applicable in the current context,although the Shin test has a di!erent limiting distribution. We also propose a consistenttest which has a null of linear cointegration (comovement), and an alternative of&non-linear cointegration'. Monte Carlo evidence is presented which suggests that the testhas useful "nite sample power against a variety of nonlinear alternatives. An empiricalillustration is also provided. ( 2000 Elsevier Science S.A. All rights reserved.

JEL classixcation: C12; C22

Keywords: Cointegration; Linear stochastic comovement; Markov processes; Nonlin-earities

*Corresponding author. Fax: (858)-534-7040.E-mail address: [email protected] (H. White)

0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 5 0 - 0

1. Introduction

In most econometric applications there is little theoretical justi"cation forbelieving in the correctness of linear speci"cations when modeling economicvariables. In consequence, nonlinear time series models have received increasingattention during the last few years (for example, see Tong (1990), Granger andTeraK svirta (1993), Granger (1995), Granger et al. (1997), Anderson and Vahid(1998), and the references contained therein). Nevertheless, when nonlinearmodels are speci"ed, correct inference requires some knowledge as to whetherthe underlying data generating processes (DGPs) are stationary and ergodic insome appropriate sense, or instead have trajectories that explode with positiveprobability as the time span approaches in"nity. In the linear case it is commonto test for unit roots in order to check whether a series is integrated of order 1,denoted I(1), or integrated of order 0, denoted I(0), where a process is said to beI(d) if the scaled partial sum of its dth di!erence satis"es a functional centrallimit theorem (FCLT). If one has two or more I(1) series, it is common to test forcointegration in order to determine whether there exists a linear combination ofthe variables which is I(0). However, the concepts of integratedness and cointeg-ratedness typically apply to linear DGPs, in the sense that the conditional meanis assumed to be a linear function of a set of conditioning variables. In contrast,strictly convex or concave transformations of random walks have a unit rootcomponent, but they are not I(1), in the sense that their "rst di!erences need notbe short memory processes (Corradi, 1995).

In this paper we examine nonlinear DGPs that are "rst-order Markov andcan be represented as the sum of a linear plus a bounded nonlinear component.For such DGPs, we exploit results by Chan (see Appendix to Tong, 1990) toobtain simple conditions for distinguishing between processes that are geomet-ric ergodic (and thus strong mixing) and processes having trajectories thatexplode with positive probability as ¹PR. Using these conditions, we replacethe concept of cointegratedness with concept of linear stochastic comovement.Speci"cally, if X"(X

i,t, i"1, 2,2, p, t"1, 2,2, ¹) is a nonergodic Markov

process in Rp, in the sense that the trajectories explode with positive probabilityas ¹PR, but there exists an r dimensional linear combination, say h@

0X

t, with

h0a full column rank p]r matrix (r(p) that is ergodic in Rr, then there is linear

stochastic comovement among the components of X. We use the term &Markovprocess' to mean a process in which the state space is continuous and time isdiscrete. Note that our approach di!ers from that of Granger and Hallman(1991). According to their terminology, two long memory series, say X

tand

>thave an attractor if there exists a linear combination of nonlinear functions of

Xtand >

t, say g(X

t)!h(>

t) that is short memory. In contrast, we consider the

case of linear combinations among the components of nonlinear and nonergodicMarkov processes that form nonlinear and ergodic Markov processes. Forour class of nonlinear DGPs, the null hypothesis of ergodicity and the null

40 V. Corradi et al. / Journal of Econometrics 96 (2000) 39}73

hypothesis of no unit root can be formulated in the same way. Similarly, the nullhypothesis of linear stochastic comovement and the null hypothesis of cointeg-ration can be formulated in the same way. Thus, the presence of stochasticcomovement implies the presence of cointegration among the linear compo-nents of our nonlinear models, and vice versa. Given this framework, one of ourmain goals is to propose a &nonlinear cointegration' test, for which the nullhypothesis is linear cointegration, and the alternative is nonlinear cointegration.The test which we propose is consistent against a wide variety of nonlinearalternative, including neural network models with sigmoidal activation func-tions (e.g. logistic cumulative distribution functions (cdfs)). We show using aseries of Monte Carlo experiments that our nonlinearity test has the ability todistinguish between a variety of linear and nonlinear models for moderatesample sizes.

As we typically do not have information concerning the precise form of thenonlinear component, we examine the e!ect that neglected nonlinearities haveon tests for the null of stationarity (unit root) and for the null of cointegration(no cointegration). We note that in the presence of neglected nonlinearities, testswith a null hypothesis of integratedness, as well as tests with a null hypothesis ofno cointegration, do not have easily determined limiting distributions. This isbecause in the presence of neglected non-linearities the innovation terms are nolonger strong mixing and in general do not satisfy standard invariance prin-ciples. Consequently, standard unit root asymptotics no longer necessarilyapply. Along these lines, we "rst examine the stationarity test proposed byKwiatkowski et al. (1992). We show that this test has a well-de"ned limitingdistribution under the null hypothesis of general nonlinear stationary-ergodicDGPs and has power not only against the alternative of integratedness, but alsoagainst alternatives involving a range of nonlinear-nonergodic processes. Sec-ond, we show that the Shin (1994) test for the null hypothesis of cointegrationcan be used to test for stochastic comovement, although the limiting distributionof the test is di!erent. Interestingly, the ADF unit root and Johansen cointegra-tion tests no longer have straightforward limiting distributions in general.Nevertheless, we show using a series of Monte Carlo experiments that these testsmay still be reliable in practice, in the sense that they exhibit moderate bias andreasonable power (e.g. the empirical power is more than 0.5 for samples as small as250 observations when the nonlinear component in our model is a logistic cdf ).

The rest of the paper is organized as follows. In Section 2 we describe ourset-up. In Section 3, we examine stationary-ergodicity and cointegration(comovement) tests. In Section 4 we propose a test for distinguishing betweenlinear and nonlinear cointegration. In Section 5 we summarize the results froma series of Monte Carlo experiments, while Section 6 contains an empiricalillustration. Section 7 concludes. All the proofs are collected in an Appendix.In the sequel, N denotes weak convergence;= denotes a standard Brownianmotion.

V. Corradi et al. / Journal of Econometrics 96 (2000) 39}73 41

2. Assumption and preliminary results

We start by considering the following DGP:

Xt"AX

t~1#g

0(h1@

0X

t~1,2, hj@

0X

t~1)#e

t, (2.1)

where Xt: XPRp, t"1, 2,2, ¹ with (X, F, P) an underlying probability

space, and hi0

denotes the ith-column of h0, a full column rank p]r matrix, with

r)p, and 1)j)r(p. Assume also that

A1. etis identically and independently distributed (iid), has a distribution which

is absolutely continuous with respect to the Lebesgue measure in Rp, andhas positive density everywhere.

A2. E(et)"0 and E(e

te@t)"R where R is positive de"nite and E((e@

tet)2)(R.

A3. g0: RjPRp is bounded, Lipschitz continous, and di!erentiable in the

neighborhood of the origin. Furthermore, g0

is not everywhere equal tozero.

Although A3 is a somewhat strong assumption, it should be noted that a widevariety of nonlinearities are contained within the class of DGPs which weexamine. For example various neural network models with sigmoidal activationfunctions satisfy A3. Examples include feedforward arti"cial neural networkswith a single &hidden unit' and either logistic or normal cdfs as activationfunctions (see, e.g. Kuan and White, 1994). Other examples of functional formsfor g

0include modi"ed exponential autoregressive models where g

0(x)"xe~x

2

(Tong, 1990, p. 129), and symmetric smooth transition autoregressive (STAR)type models where g

0(x)"x/(1#ex2). On the other hand, the logistic STAR

g0(x)"x/(1#e~x) and the exponential STAR (g

0(x)"x(1!ex2)) (see, e.g.

TeraK svirta and Anderson, 1992) are ruled out. Furthermore g0

may containa constant term. Higher-order lag structures are allowed by a variant of (2.1), asa p-dimensional k-order Markov process can be written as a kp-dimensional"rst-order Markov process with a positive semi-de"nite R of rank p.

Proposition 2.1. For DGP (2.1), suppose that A1}A3 hold. If all of the eigenvaluesof the matrix A are strictly less than one in absolute value, then X"(X

i,t;

i"1,2, p, t"1, 2,2, ¹) is a geometric ergodic Markov process in Rp, with aninvariant probability measure which is absolutely continuous with respect to theLebesgue measure in Rp.

Dexnition 2.2. (Linear stochastic comovement). Assume that X is a nonergodicMarkov process in Rp, in the sense that each component of X approachesin"nity with positive probability as ¹PR. Assume also that there exists somefull column rank p]r (r(p) matrix, h

0"(h1

0,2, hr

0), such that h@

0X

tis an

ergodic Markov process, in the sense that it has an invariant probabilitymeasure which is absolutely continuous with respect to the Lebesgue measure in


Rr. Then, there exists linear stochastic comovement among the components of X,and hi

0is the ith comovement vector, i"1,2, r.

In the sequel we specialize De"nition 2.2. to the DGP given in (2.1). For thisreason, we use h

0in (2.1). Now let U,A!I, where I is the p]p identity matrix.

We assume either of the following:

A4(i). U"ah@0, where a and h

0are full column rank p]r matrices, r(p. The

eigenvalues of U, say ji, are such that !2(j

1)j

22)j

r(0.

A4(ii). A"I.

Proposition 2.3. Assume that (2.1), and A1}A4(i) hold. Then: (i) X is a nonergodicMarkov process in Rp and P(EX

TEPR),g'0 as ¹PR.

(ii) h@0X

tis a geometric ergodic Markov process in Rr which has an invariant

probability measure, n, that is absolutely continuous with respect to the Lebesguemeasure (k) in Rr, and which has density l"dn/dk. Further,

h@0*X

t"h@

0UX

t~1#h@

0g0(h1@

0X

t~1,2, hj@

0X

t~1)#h@

0et.

It follows that there is stochastic comovement among the components ofX (from De"nition 2.2).

Proposition 2.4. Assume that (2.1), A1}A3, and A4(ii) hold. Then both X and h@0X

are nonergodic Markov processes in Rp and Rr, respectively. It also follows that

P[EXTEPR]'0 and P[Eh@

0X

TEPR]'0.

The ergodicity of the process de"ned in (2.1) is implied by the stability of theassociated deterministic dynamical system. This allows us to analyze the er-godicity of the stochastic system by examining the eigenvalues of A or U. Thismeans that the same conditions which ensure the ergodicity of the linear part ofour model also ensure the ergodicity of the entire nonlinear process. Thus, forthe processes which we are considering, the existence of stochastic comovementis equivalent to the existence of cointegration among the linear components inthe model. In particular, the existence of r comovement vectors implies theexistence of r cointegrating vectors in the linear part of the model. For brevity,we refer to this subsequently as just &cointegration'.

Observe that in the case where A!I"U"ah@0there is a clear interpretation

of the argument of g0

in terms of cointegrating vectors. On the other hand whenA"I, g

0depends on some generic linear combination of the X 's. Thus under

A4(i), the nonlinear component is a geometric ergodic process, while underA4(ii), the nonlinear component is a nonlinear nonergodic process. In order to


keep a &continuous' relationship between the arguments of g0

and A!I, wecould consider the following variation of (2.1):

*Xt"ah@

0X

t~1#g

0((h

0, 0)@X

t~1)#e

t,

where h0

is a p]r (r(p) matrix, 0 is a p](p!r) matrix of zeroes, and h0"0

when r"0. Thus, when r"0, Xtis an I(1) process with a deterministic trend

component. It should be noted that all the theorems below hold for this specialcase. (This point was kindly communicated to us by Herman Bierens.)

The following facts will be frequently used in the paper.

Fact 2.5 (From Athreya and Pantula, 1986, Theorem 1). Geometric ergodicdiscrete time Markov processes are strong mixing. Further, the speed at which themixing coezcient declines to zero is proportional to the speed at which thetransition distribution converges to the invariant probability measure. Thus, whenthe transition distribution approaches the invariant probability measure at a geo-metric rate, the mixing coezcients also decay at a geometric rate.

To ensure the next fact, we add another assumption.

A5. X0

is a random p-vector and h@0X

0is drawn from a density l, where l is the

density associated with the invariant probability measure, n, as de"ned inProposition 2.3(ii).

Fact 2.6 (from Meyn, 1989). For (2.1), if A1}A4(i) and A5 hold, then h@0X

thas

density l for all t"1, 2,2, ¹. Thus X is strictly stationary, in addition to beinga geometric ergodic process (and thus strong mixing).

3. Testing for stationarity}ergodicity and for linear stochastic comovement

We begin by considering the one-dimensional case, (i.e. p"1), and the testproposed by Kwiatkowski et al. (1992). Without loss of generality assume thath0"1, so that (2.1) can be written as:

Xt"aX

t~1#g

0(X

t~1)#e

t(3.1)

The null hypothesis considered by Kwiatkowski et al. (1992) is rather generaland includes (3.1) when a(1, and A5 holds. However, the alternative issomewhat restrictive, as X

tis assumed to be an integrated time series character-

ized by the sum of a random walk component, a stationary (short memory)component, and possibly a time trend component. This alternative does notinclude nonlinear nonergodic DGPs such as (3.1), with a"1. For this case the"rst di!erence of X

tis not a strong mixing process, in general, as it displays &too

much' memory. Nevertheless, we show below that the statistic proposed byKwiatkowski et al. (1992) does have power against (3.1) with a"1.


Theorem 3.1. Assume that (3.1), and A1}A3 hold.

(i) If DaD(1, and A5 holds, then

ST"

1

s2lT

1

¹2

T+t/1A

t+j/1

(Xj!XM )B

2NP

1

0

<2r

dr,

where <r"=

r!r=

1, =

r"W(r), 0)r)1, XM "¹~1+T

t/1X

t, and

s2lT"

1

¹

T+t/1

(Xt!XM )2#

2

¹

lT+t/1

lT+j/1A1!

t

lT#1B

]T+

j/t`1

(Xj!XM )(X

j~t!XM ),

with lT"o(¹1@2).

(ii) If a"1, then

P[ST'C

T]P1 as ¹PR,

where CTPR and

CTlT

¹

P0, as ¹PR.

Part (i) of Theorem 3.1 ensures that the distribution under our null is exactlythe same as the distribution of the Kwiatkowski et al. (1992) test statistic.Further, part (ii) of Theorem 3.1 ensures that under our alternative, S

Tdiverges

at the same rate as does the Kwiatkowski et al. (1992) statistic under theiralternative. However, note that our alternative is more general than theiralternative of integratedness, as it includes DGPs consisting of a unit rootcomponent plus a possibly long memory component. Assumption A5, whichensures strict stationarity, can be relaxed and replaced by the weaker property ofconstancy of the "rst moment (i.e. E(X

t)"E(X) for all t). If E(X

t) depends on t,

though, the partial sums of Xt!XM do not necessarily satisfy a FCLT and so the

numerator of the test statistic may diverge. Recently, Domowitz and El-Gamal(1993,1997) have proposed a test for the null hypothesis of ergodicity. Their testis based on the convergence of Cesaro averages of the iterates from di!erentinitial densities, has the correct size under the null, regardless of whether theprocess is stationary or not, and has power against the alternative of noner-godicity given a maintained assumption of stationarity.

Now we turn to the case where p'1, and examine the cointegration test ofShin (1994). Although our testing framework holds for arbitrary "nite p, forsimplicity we limit ourselves to the case of p"2, r"0, 1. The extension to thecase of p'2 gives no further insight into the e!ect of neglected nonlinearities.As mentioned in Section 2, the existence of stochastic comovement is equivalent


to the existence of cointegration, in the current context. Thus, tests that have(no) stochastic comovement under the null hypothesis are equivalent to teststhat have (no) cointegration under the null. In this section we consider thefollowing two DGPs:

*Xt"g

0(h@

0X

t~1)#e

t(3.2)

and

*Xt"UX

t~1#g

0(h@

0X

t~1)#e

t, (3.3)

where h0"(!h

1, h

2)@ and U"ah@

0, such that !2(!a

1h1#a

2h2(0.

From (3.3) we have that

h@0*X

t"(!a

1h1#a

2h2)h@

0X

t~1#h@

0g0(h@

0X

t~1)#h@

0et. (3.4)

Let

h~12

h@0"c@

0"(!h~1

2h1, 1)"(!b

0, 1)

and

d"!a1h1#a

2h2,

so that

c@0*X

t"dc@

0X

t~1#c@

0g0(h@

0X

t~1)#c@

0et. (3.5)

As h@0X

tis a geometric ergodic process, c@

0X

tis also a geometric ergodic process.

It is convenient to use a triangular representation of (3.3):

X1, t"

t+j/1

l1, j

and X2, t"b

0X

1, t#l

2, t, (3.6)

where

l1, t"a

1h2c@0X

t~1#g

0,1(h@

0X

t~1)#e

1, t, (3.7)

l2, t"(d#1)c@

0X

t~1#c@

0g0(h@

0X

t~1)#c@

0et, (3.8)

and g0,i

, i"1, 2, denotes the ith component of g0.

In order to test the null hypothesis of stochastic comovement, we use thestatistic proposed by Shin (1994) for testing the null of cointegration. Althoughstochastic comovement and cointegration are equivalent concepts, we showthat, unless E(l

1, t)"0 for all t, the distribution that we obtain under our null

hypothesis di!ers from that obtained by Shin. The underlying intuition is that


unless E(l1, t

)"0, X1, t

displays a deterministic trend component. In fact, we canwrite X

1, tin (3.6) as

X1, t"nt#

t+j/1

(l1,j

!n),

where E(l1, t

)"n for all t, as under A5, l1t

is strictly stationary. X2, t

can bewritten in an analogous way. The role of the comovement vector is to &cancelout' both stochastic and deterministic trends when the linear combination,X

2, t!b

0X

1, t, is formed. Furthermore, X

2, t!b

0X

1, tis an ergodic process. As

X1, t

is dominated by its trend component, the estimator of the comovementvector is ¹3@2-consistent. Also, the asymptotic distribution of ¹3@2(bK

T!b

0),

where bKT

is the coe$cient from the regression of X2, t

on X1, t

and a constantterm (as in (3.6)), di!ers from the asymptotic distribution for the driftless case.Note that the only di!erence between X

2, tin (3.6) and Eq. (2.1) in Hansen

(1992a) is that E(l2, t

)"/O0, so that we need to introduce a constant term intoour cointegrating regression.

Theorem 3.2. Assume that (3.3), and A1}A4(i), A5 hold, and that E(l1, t

)O0. Then

¹3@2(bKT!b

0)N

12

npl2AP

1

0

s d=s!1/2=

1B,where p2l2"<ar(l

2, t), as dexned in (3.8), and =

s"W(s), 0)s)1.

As in Theorem 3.1(i), we require only that the "rst moment of Xtis constant,

and A5 ensures this. The same is also true for Theorem 3.3(i) below.Note that :1

0s d=

s!1/2=

1"1/2=

1!:1

0=

sds&N(0, 1/12), as :1

0=

sds&

N(0, 1/13) and Cov(1/2=1, :1

0=

sds"1/4. Thus, the limiting distribution in

Theorem 3.2 is normal. The representation of the limiting distribution given inTheorem 3.2 is more convenient for computing the limiting distribution of thetest for the null of stochastic comovement (cointegration). If instead E(l

1, t)"0

for all t, then

¹(bKT!b

0)NAP

1

0

=21,s

dsB~1

AP1

0

=1,s

d=2, sB#*

12,

where *12"0 and E(=

1, t,=

2, t)"0 for all t, only if E(l

1, t, l

2, s)"0 for all t, s.

Nevertheless, for a wide class of nonlinearities E(l1, t

)O0, for all t (see below).Under the alternative of no stochastic comovement, bK

Tis bounded in probabil-

ity. This holds even though we do not obtain a limiting distribution. (The reasonwhy we do not obtain a limiting distribution is because the partial sum of the


nonlinear component, in general, does not satisfy a standard invariance prin-ciple.)

We now show that the Shin (1994, Eq. (6)) Ck test can be applied to test thenull hypothesis of stochastic comovement, although the limiting distributionunder the null is di!erent. Let mK

tbe the residual from the regression of X

2, ton

X1, t

and a constant term.

Theorem 3.3. (i) Assume that (3.3), and A1}A4(i), A5 hold, and that E(l1, t

)O0.Then,

1

s( 2lT

1

¹2

T+t/1

At+j/1

mKtB

2NP

1

0

Q2s

ds

where

Qs"(=

s!s=

1)!6AP

1

0

s d=s!1/2=

1B(s2!s)

and

s( 2lT"

1

¹

T+t/1

m2t#

2

¹

lT+t/1A1!

t

lT#1B

T+

j/t`1

mKjmKj~t

,

where lT"o(¹1@2).

(ii) Assume that (3.2), and A1}A3, A4(ii) hold. Then,

PC1

s( 2lT

1

¹2

T+t/1

At+j/1

mKtB

2'C

TDP1 as ¹PR,

where CTPR and C

TlT/¹P0, as ¹PR.

From part (i) of Theorem 3.3 note that the asymptotic distribution under thenull is a functional of only one Brownian motion=, where= is the weak limit,property rescaled, of the partial sums of l

2, t!E(l

2, t)"c@

0X

t!E(c@

0X

t). Thus,

the asymptotic behavior of the statistic is not a!ected by whether l1, t

and l2, t

in(3.7) and (3.8) are correlated or not. As mentioned above, this is due to the factthat the asymptotic behavior of X

1, tis dominated by its trend component. This

di!ers from the linear case, g0"0, in which the non-zero correlation between

l1, t

and l2, t

results in a nuisance parameter in the limiting distribution(when b

0is estimated by OLS). The asymptotic critical values for the distribu-

tion given in Theorem 3.3(i) are reported below. The simulated critical values arebased on sample size n"2000 and 20,000 replications.


Linear stochastic comovement test critical values

Nominal size

1% 5% 10% 90% 95% 99%

0.219 0.150 0.121 0.028 0.023 0.018

Note that our critical values are smaller than those reported in Shin (1994) for hisCk test statistic. This di!erence arises because of the nonlinear component in (3.3).

In practice we do not know whether E(l1, t

)"0 or not. If E(l1, t

)O0, then thecritical values tabulated above should be used. On the other hand, if E(l

1, t)"0,

then the asymptotic distribution of our test is the same as that of Ck in Shin (1994,Theorem 1), provided E(l

1, tl2,t

)"0. Given these facts, we suggest applying thecomovement test in the following way. If the test statistic is less than our criticalvalue, accept the null of (cointegration) comovement. On the other hand if the teststatistic is above Shin's critical value for Ck, we have evidence of the absence ofcointegration (comovement). If the test statistic falls in the region between the twocritical values (say the &intermediate region'), construct the following statistic:dT"¹~1@2+T

t/1p( ~1T

*X1, t

, where p( 2T

is a consistent estimator of the long runvariance of ¹~1@2+*X

1, t. Under cointegration (comovement), d

TNN(0, 1) when

E(l1, t

)"0, and diverges at rate ¹1@2 when E(l1, t

)O0. On the other hand, whenthere is no cointegration (comovement), we must distinguish between three cases.First, assume that E(l

1, t)"0 . Here, there are two cases: (i) E(l

1, t)"0 because

there is no nonlinear component; or (ii) E(l1, t

)"0 because there is zero meannonlinear component. Under (i), d

Tis normally distributed (as l

1, tis a zero mean

mixing process). Under (ii), the statistic is not normally distributed in general.Finally, assume that E(l

1,t)O0. In this case, the estimator of the variance term in

dT

diverges at a rate less than or equal to l1@2T

, while the numerator of dT

divergesat a faster rate. These facts suggest a procedure for testing for cointegration(comovement) when the test statistic is in the &intermediate region'. In particular, ifwe reject E(l

1, t)"0, reject the null of cointegration (comovement). If we fail to

reject E(l1, t

)"0, then use Shin's critical values. (In order to use Shin's criticalvalues in this case, an e$cient estimator of the cointegrating vector shouldbe constructed, as in Shin (1994).) Note that the use of d

1may result in the false

rejection of E(l1,t

)"0 when there is no cointegration (comovement) and thenonlinear component has zero mean. However, in this case we still reject the nullof cointegration (comovement), which is the correct inference.

4. Testing for nonlinear cointegration

In this section we propose a test for the null of linear cointegration (comove-ment) against the alternative of nonlinear cointegration (comovement).


A number of papers which examine nonlinear cointegration have recentlyappeared. For example, Balke and Fomby (1997) propose a test for thresholdcointegration. Also, Granger (1995) suggests testing for the null of linear cointeg-ration by regressing the residuals from a standard cointegrating regression ontheir lagged values and a nonlinear function, and then performing a LagrangeMultiplier (LM) type test. A similar approach is examined by Swanson (1999)who regresses the "rst di!erences of the data on their lagged values and ona polynomial function of the cointegrating vector. He shows, using Monte Carloexperiments, that such tests have good power when g

0is a logistic cdf. Neverthe-

less, this class of tests does not have unit asymptotic power against generalnonlinear alternatives. One reason for this is the LM tests are implementedusing polynomial test functions, and the use of polynomials does not ensure testconsistency.

In our test, cointegration is maintained under both the null and the alterna-tive hypothesis. Let g(

tbe the residual and tK be the slope coe$cient from the

least squares regression of c( @TX

ton c( @

TX

t~1and a constant, where c( @

T"(!bK

T, 1)

and bKT

is the coe$cient from the regression on X2,t

on X1,t

and a constant.Thus,

g(t"Ac( @TX

t!

1

¹

T+t/1

c( @TX

tB!tKTAc( @TXt~1

!

1

¹

T+t/1

c( @TX

t~1B. (4.1)

Under the null of no nonlinearity, we have J¹(tKT!t

0)"O

p(1) and

¹(bKT!b

0)"O

p(1). Let

gt"Ac@0Xt

!

1

¹

T+t/1

c@0X

tB!t0Ac@0Xt~1

!

1

¹

T+t/1

c@0X

tB, (4.2)

so that gtis uncorrelated with any function of c@

0X

t~1. Under the alternative of

nonlinearity of the type described in Section 2, J¹(tKT!tH)"O

p(1), where

tHOt0, ¹3@2(bK

T!b

0)"O

p(1), and

gt"Ac@0Xt

!

1

¹

T+q/1

c@0XqB!tHAc@0Xt~1

!

1

¹

T+q/1

c@0XqB.

Under the alternative, gt

includes the neglected nonlinear term(g

0(h@

0X

t~1)!E(g

0(h@

0X

t~1))), where h~1

2h@0,c@

0"(!h~1

2h1, 1). Thus, g

tis cor-

related with some function of c@0X

t~1. If we use as a test function, call it g, an

exponential, as in Bierens (1990), or any other generically comprehensive testfunction, as described in Stinchcombe and White (1998, Section 3, hereafter SW),then under the alternative

E(gt(g

0(c@

0X

t~1q)!E(g

0(c@

0X

t~1q))))O0


for all q3T, a subset of R whose complement T# has Lebesgue measure zero

and is not dense in R. Given the J¹ consistency of tKT

and the ¹3@2 consistencyof bK

Tunder the alternative, we also have

1

¹

T+q/1Ag( tAg(c( @

TX

t~1q)!

1

¹

T+q/1

g(c( @TX

t~1q)BBPMq in Prob.

where MqO0, for all q3T.According to Theorem 3.10 in Stinchcombe and White (1998), if g is a real

analytic function, then g delivers a consistent test, regardless of g0, provided that

g is not a polynomial. One natural choice for g is the logistic cdf, as it is anon-polynomial real analytic function.

As the parameter q is not identi"ed under the null hypothesis, our tests fallsinto the class of tests with nuisance parameters present only under the alterna-tive. Thus, although the asymptotic size of the test statistic is not a!ected by theactual value of q, the "nite sample size will be a!ected, while the power will bea!ected both in "nite samples and asymptotically. Consider the following twoDGPs:

*Xt"UX

t~1#e

t(4.3)

and

*Xt"UX

t~1#g

0(h@

0X

t~1)#e

t, (4.4)

where h0"h

2c0. Assume that U satis"es A4(i), g

0satis"es A3, and e

tsatis"es A1

and A2. From (4.3) note that

*c@0X

t"dc@

0X

t~1#c@

0et,

and from (4.4) note that

*c@0X

t"dc@

0X

t~1#c@

0g0(h@

0X

t~1)#c@

0et,

where d"!a1h1#a

2h2. The proposed test is based on the statistic:

1

J¹

T+t/2

(g(c( @TX

t~1q)!g(6 )g(

t, (4.5)

where g(6 "(1/¹)+Tt/2

g(c( @TX

t~1q). It is shown in the proof of Theorem 4.1 that

(4.5) can be written as

1

J¹

0+t/2

((g(c@TX

t~1q)!g6 )!M~1

(c@0X)2Mc@0Xg

(c@0X

t~1!c@

0XM ))g

t#o

p(1), (4.6)


where M(c@0X)2

"E((c@0X

t!c@

0XM )2), Mc@0Xg

"E((c@0X

t!c@

0XM )(g(c@

0X

t~1q)g6 )),

g6 "(1/¹)+Tt/2

g(c@0X

t~1q): From the central limit theorem for strong mixing

processes (e.g. White, 1984, p. 124) and the asymptotic equivalence lemma, notethat the limiting distribution of (4.6), when scaled by p2

Tis a zero mean normal,

where

p2T"VarA¹~1@2

T+t/2

((g(c@0X

t~1q)!E(g))!M~1

(c@0X)2Mc@0Xg

(c@0X

t~1!c@

0XM ))g

tB,(4.7)

and p2TPp2

0as ¹PR. A convenient estimator for p2

0is given by

p( 2lT"

1

¹

T+t/2

(/K qt)2#

2

¹

lT+t/2A1!

t

lT#1B

T+

j/t`1

/K qj/K q

j~t, (4.8)

where

/K qt"((g(c( @

TX

t~1q)!g(6 ) (4.9)

!A1

¹

T+t/2

(c( @TX

t~1!c( @

TXM )2B

~1

A1

¹

T+t/2

(c( @TX

t~1!c( @

TXM )(g(c( @

TX

t~1q)!g(6 )B

](c( @TX

t~1!c( @XM ))g(

t,

with lT"o(¹1@2). Assume also that

A6. The test function g is a real nonpolynomial analytic function, with bounded"rst two derivatives.

Note that various sigmoidal functions (e.g. the logistic cdf) satisfy A6.

Theorem 4.1. (i) Assume that (4.3), A1}A2, A4(i), and A6 hold. Dexne

mqT,

1

J¹

1

p(lT

T+t/2

g(t(g(c( @

tX

t~1q)!g(6 ).

Then

(mqT)2Ns2

1

for each q3T, and p( 2lT

dexned in (4.8). The same result follows when g0

isa constant.


(ii) For DGP (4.4), suppose that A1}A4(ii), A5}A6 hold, and that g0

is nota constant. Then for each q3T,

P[DmqTD2'C

T]P1 as ¹PR,

where CTPR and C

TlT/¹P0, as ¹PR.

Note that strict stationarity is required only under the alternative, as the "rstmoment of (4.3) is constant. In fact, in Theorem 4.1(ii), as in Theorems 3.1(i), 3.2and 3.3(i) above, we impose A5 simply because it implies the constancy of the"rst moment. Also, note that the case where g

0is constant is covered by the null

hypothesis.In the current context we do not provide a &sup' type result, as in Bierens

(1990) and Stinchcombe and White (1998), for example. The intuitive reason forthis is that the o

p(1) term in (4.6) holds pointwise in q, but not necessarily

uniformly in q. As will become clear in the proof of Theorem 5.1, this is due tothe fact that in the nonstationary case, we cannot invoke the usual uniform lawof large numbers. We appeal instead to invariance principles and to results onconvergence to stochastic integrals that hold pointwise in q, but not necessarilyuniformly.

As mentioned above, although the asymptotic size of the test statistic is nota!ected by the choice of a particular q, the "nite sample size and power area!ected. There are at least two ways of addressing this issue. First, we canconstruct the statistic for di!erent q's and apply Bonferroni type bounds as inLee et al. (1993), for example. Second, let q

1,2, q

pbe chosen according to

a particular design (e.g. randomly), and let GK be a consistent estimator ofCov(/qi

l, /qk

l), so that GK is the matrix whose i, k element is given by

1

¹

T+t/1

/K qit/K qkt#

2

¹

lT+t/1A1!

t

lT#1B

T+

j/t`1

/K qij/K qkj~t

,

where /K qtis de"ned in (4.9), and /q

tis de"ned as in (4.9), but with c( @

Treplaced with

c@0. Then, (mq1

T,2, mqp

T)@GK ~1(mq1

T,2, mqp

T)Ns2

p, for arbitrary and "nite p. Whether

the above result also holds for p"pT, with p

TPR at an appropriate rate as

¹PR, is left for future research.

5. Monte Carlo results

In this section, a summary of Monte Carlo experiments based on the abovetest, and for samples of 100, 250, and 500 observations, is given. For the sake ofbrevity, much of the discussion focuses on the nonlinear cointegration (NLCI)test.

Before turning to our "nite sample NLCI test results, it is worth reiteratingthat unit root and cointegration tests are not generally robust to the inclusion of


nonlinearities. As noted above, however, it turns out that the Kwiatkowski et al.(1992) stationarity and the Shin (1994) cointegration tests have well-de"nedlimiting distributions and unit asymptotic power, in our context. The samecannot be said for the augmented Dickey}Fuller (ADF) unit root test. Inparticular, note that under the null hypothesis of a unit root, if we neglect toaccount for the nonlinear component, the error term is given by l

t"e

t#g

0( ) ),

where g0( ) ) is generally not a strong mixing process. Thus, standard unit root

asymptotics no longer apply, and the usual limiting distribution of the ADF teststatistic is typically no longer valid (see, e.g. Ermini and Granger, 1993). On theother hand, under the alternative of no unit root, the error term is a strongmixing process, so that we expect ADF tests to have reasonable power in largesamples.

Table 1 reports the results from a Monte Carlo experiment based on the ADFtest, using data generated according to

Xt"a#bX

t~1#cg

0(X

t~1)#e

t,

where Xtis a scalar, e

tis a scalar IN(0, 1) random variable, g

0( ) ) is the logistic

cdf, and a"0 (results for aO0 and for di!erent g0( ) ) are qualitatively similar,

and are available upon request from the authors). Notice that in the table,b varies from !0.9 to 1.0, so that empirical power of a variety of di!erentparameterizations, and empirical size (b"1) is reported. Also, note that theparameter c is alternately !0.5, !0.1, 0.1, and 0.5. Here and below, resultsbased on 5% nominal size tests are reported (results for 10% size tests aresimilar, and are not included for the sake of brevity). The "nite sample power ofthe ADF test is good (power is always close to or equal to unity, except whenDbD"0.9), as expected. The "nite sample size of the ADF test is between 0.084and 0.056, even for samples of 100 observations, and improves as we move from100 to 500 observations, when q( q is used. This suggests that within our context,the ADF test can still be used to signal the presence of a unit root, even whena bounded nonlinear component is added to the DGP, as long as q( q is used. Thisis perhaps not surprising, as the mean of g

0( ) ) is not generally zero. In summary,

one might argue in favour of using q( q in our context, as the "nite sample size iscloser to the nominal size than when q( and q( k are used. Further, the "nite samplepower of the q( q test is comparable to the power associated with the use of q( andq( k, except when b"0.9.

Table 2 reports the results from a Monte Carlo experiment based on theJohansen cointegration test, using data generated according to

*Xt"d#eZ

t~1#fg

0(Z

t~1)#e

t, (5.1)

where Xt"(X

1, t, X

2, t)@ is a 2]1 vector, e

tis a 2]1 vector whose components

are distributed IN(0,1), Zt"!X

2, tif e

1"e

2"0, otherwise Z

t"X

1, t!X

2, t,

g0(x)"(2/[1#e~x])!1, d"(d

1, d

2)@, d

1"d

2"0.2, e"(e

1, e

2)@, and


Tab

le1

Augm

ente

dD

ickey}Fulle

rte

stper

form

ance

und

erneg

lect

ednonlin

earity

!

bc

T"10

0T"

250

T"50

0

q(q( k

q( qq(

q( kq( q

q(q( k

q( q1.

0!

0.5

0.05

60.

086

0.08

40.

054

0.10

30.

086

0.05

00.

115

0.09

81.

0!

0.1

0.08

90.

067

0.07

10.

109

0.06

10.

061

0.12

10.

066

0.06

71.

0!

0.5

0.01

60.

053

0.06

70.

009

0.03

20.

057

0.00

30.

027

0.05

41.

0!

0.5

0.00

10.

015

0.06

70.

000

0.00

60.

056

0.00

00.

005

0.05

7

!0.

9!

0.5

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

0!

0.6

!0.

51.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

!0.

3!

0.5

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

00.

3!

0.5

1.00

00.

999

0.99

61.

000

1.00

01.

000

1.00

01.

000

1.00

00.

6!

0.5

0.99

60.

994

0.98

51.

000

1.00

01.

000

1.00

01.

000

1.00

00.

9!

0.5

0.73

70.

727

0.52

90.

998

0.99

90.

994

1.00

01.

000

1.00

0!

0.9

!0.

11.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

!0.

6!

0.1

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

0!

0.3

!0.

11.

000

1.00

00.

999

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.3

!0.

11.

000

0.99

90.

995

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.6

!0.

10.

999

0.98

90.

979

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.9

!0.

10.

821

0.44

30.

286

1.00

00.

983

0.91

21.

000

1.00

01.

000

!0.

90.

11.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

!0.

60.

11.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

!0.

30.

11.

000

1.00

00.

999

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.3

0.1

1.00

00.

999

0.99

31.

000

1.00

01.

000

1.00

01.

000

1.00

00.

60.

11.

000

0.98

50.

973

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.9

0.1

0.63

30.

286

0.18

50.

997

0.90

60.

712

1.00

01.

000

0.99

7!

0.9

0.5

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

0!

0.6

0.5

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

0!

0.3

0.5

1.00

01.

000

0.99

91.

000

1.00

01.

000

1.00

01.

000

1.00

00.

30.

50.

999

0.99

80.

990

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.6

0.5

0.97

20.

973

0.94

81.

000

1.00

01.

000

1.00

01.

000

1.00

00.

90.

50.

022

0.28

00.

184

0.08

00.

783

0.60

60.

690

0.99

50.

978

!Bas

edon

the

augm

ente

dD

ickey}Fulle

r(A

DF)t

est,

entr

iesco

rres

pond

toth

eem

pirica

lfre

quen

cyofr

ejec

tion

ofth

enull

hyp

oth

esis

ofa

unit

root

.Thr

eeve

rsio

nsofth

ete

stre

gres

sion

sar

eru

n:w

ith

no

const

ant

or

linea

rde

term

inistic

tren

d(q(

),w

ith

aco

nsta

nt

only

(q(k),

and

with

aco

nst

ant

and

alinea

rdet

erm

inisitic

tren

d(q(

q).D

ata

arege

nera

ted

acco

rdin

gto

thefo

llow

ing

pro

cess

:Xt"

a#bX

t~1#

cg0(X

t~1)#

e t,wher

eX

tisa

scal

ar,e

tisa

scal

arIN

(0,1

)ra

ndo

mva

riab

le,g

0())i

sth

elo

gist

iccd

f,an

da"

0.0.

The"rs

tfo

urro

wsof

entr

iesin

the

tabl

ere

port

the

empiric

alsize

oft

hete

stbas

edon

a5%

nom

inal

size

,while

the

rem

aini

ngro

wsre

port

the

empiric

alpow

er,a

lso

fora

5%nom

inal

size

test

.All

exper

imen

tsar

ere

peat

edfo

rsa

mple

sof¹

"10

0,25

0,an

d50

0ob

serv

atio

ns.R

esul

tsar

eba

sed

on50

00M

onte

Car

lore

plic

atio

ns.


Table 2Johansen cointegration test performance under neglected nonlinearity!

e1

e2

f1

f2

T"100 T"250 T"500

Trace 1 Trace 2 Trace 1 Trace 2 Trace 1 Trace 2

0.0 0.0 !2.0 0.0 0.720 0.086 0.748 0.084 0.743 0.0740.0 0.0 !5.0 0.0 0.683 0.080 0.744 0.076 0.764 0.0750.0 0.0 !2.0 2.0 0.287 0.957 0.016 0.935 0.000 0.9470.0 0.0 !5.0 5.0 0.282 0.955 0.014 0.936 0.000 0.949

!0.2 0.2 !2.0 0.0 0.449 0.967 0.074 0.938 0.003 0.949!0.2 0.4 !2.0 0.0 0.344 0.959 0.032 0.936 0.000 0.949!0.2 0.6 !2.0 0.0 0.300 0.956 0.020 0.937 0.000 0.948!0.2 0.2 !5.0 0.0 0.471 0.969 0.093 0.941 0.004 0.944!0.2 0.4 !5.0 0.0 0.355 0.956 0.039 0.933 0.001 0.946!0.2 0.6 !5.0 0.0 0.342 0.960 0.031 0.940 0.000 0.953!0.2 0.2 !2.0 2.0 0.280 0.955 0.014 0.937 0.000 0.949!0.2 0.4 !2.0 2.0 0.286 0.955 0.015 0.934 0.000 0.950!0.2 0.6 !2.0 2.0 0.325 0.961 0.023 0.939 0.001 0.950!0.2 0.2 !5.0 5.0 0.285 0.895 0.014 0.937 0.000 0.950!0.2 0.4 !5.0 5.0 0.330 0.957 0.023 0.936 0.000 0.945!0.2 0.6 !5.0 5.0 0.374 0.958 0.042 0.933 0.001 0.943

!Based on the Johansen trace test statistic, entries correspond to the empirical frequency of rejectionof the null hypothesis no cointegration, in favor of a cointegrating space rank of unity. Two versionsof the test statistic are constructed: with no constant or linear deterministic trend (Trace 1), and witha drift and linear deterministic trend in the levels, and a drift in the di!erences (Trace 2: this versionof the test corresponds to Case 1 in Osterwald-Lenum (1992)). Data are generated according to thefollowing process: *X

t"d#eZ

t`1#fg

0(Z

t~1#e

t), where X

t"(X

1, t, X

2, t)@ is a 2]1 vector, e

tis

a 2]1 vector whose components are distributed IN(0, 1), and Zt"!X

2, tif e

1"e

2"0, otherwise

Zt"X

1, t!X

2, t. Also, g

0(x)"(2/[1#e~x])!1, d"(d

1, d

2)@, d

1"d

2"0.2, e"(e

1, e

2)@, and

f"( f1, f

2)@. The "rst four rows of entries in the table report the empirical size of the test based on

a 5% nominal size, while the remaining rows report the empirical power, also for a 5% nominal sizetest. All experiments are repeated for samples of ¹"100, 250, and 500 observations. Results arebased on 5000 Monte Carlo replications.

f"( f1, f

2)@. The values used for e are e

1"e

2"0 (empirical size), and

e1"!0.2, e

2"M0.2, 0.4, 0.6N (empirical power). This DGP is the same as that

used in Park and Ogaki (1991), except that we also include a nonlinear compon-ent. Note that in Table 2 it is clear that the empirical power of the Johansen testis quite good (always above 0.895, even for samples of only 100 observations)only for the Trace 2 test, which includes an intercept in the di!erenced vectorautoregression (the intercept in the DGPs is nonzero). However, even for theTrace 2 test, the empirical size is only relatively close to the nominal size (e.g.0.086 for ¹"100, and lower for higher values of ¹) when the nonlinearcomponent enters only one of the equations in the system (i.e. f

2"0). Thus, the

Johansen test performs more poorly when the complexity of the nonlinearity in


(5.1) is increased. This is perhaps not too surprising, given that the Johansen testis not valid in our context.

We now turn to a discussion of our results based on the NLCI test. In order toillustrate the performance of our test statistic under various scenarios, the resultsof three di!erent experiments are reported. In all cases, the nonlinear functionused in the construction of the NLCI test is g(x)"(2/(1#e~x))!1. The dataare generated according to (5.1), with

Table 3: g0(x)"g(x), d

1"d

2"0.2, f

1"M0,!2.0,!5.0N, f

2"M0, 2.0, 5.0N,

Table 4: g0(x)"sin(x), d

1"d

2"0.2, f

1"M0,!1.0,!2.0N, f

2"M0, 1.0, 2.0N,

Table 5: g0(x)"sin(x), if DxD)p/2, g

0(x)"g(x) if DxD)p/2,

d1"d

2"0.1, f

1"M0, !2.0, !5.0N, f

2"M0, 2.0, 5.0N.

Note that the experiment reported in Table 5 uses data which are generatedaccording to two di!erent forms of nonlinear error correction, depending onhow far x is from the origin, and hence the DGP used is a type of threshold errorcorrection model. However, note that in this case the nonlinear function isdiscontinuous at n/2, so that assumption A3 is not satis"ed. Thus, the results inTable 5 can be interpreted as yielding evidence of the usefulness of the NLCI testfor &modest' departures from A3. Finally, various other parameterizations of theabove DGPs were also examined and are omitted because the Monte Carloresults are similar. Also, the overall results did not change when q was varied.Thus, all reported results use q"1. The results presented in Tables 3}5 arestraightforward to interpret. For example, the "nite sample power of theNLCI test is rather low for samples of 100 observations, and is lowerwhen nonlinearity enters through only one equation (compare the lastsix rows of entries with the previous six rows, in each table). In particular, the"nite sample power ranges from 0.105 to 0.487 across all DGPs, when f

2"0 and

lT"0. The power of the test increases, though, as the sample size increases, and

for samples of 500 observations, the rejection frequency of the NLCI test hasa lower bound of 0.836, across all parametrizations and DGPs, when l

T"0. The

empirical size of the test is reported in the "rst four rows of entries in Table 3.For l

T"0, the empirical size ranges from 0.038 to 0.050 for 100 observations,

and from 0.051 to 0.053 for 500 observations. Note also that for lT3, the

empirical size is low when 100 observations are used (the range is 0.018}0.024),but is much closer to the nominal size (the range of 0.045}0.047) when 500observations are used.

6. Empirical illustration

Nonlinear models have been used in empirical studies with varying degrees ofsuccess in recent years. Examples of such models include smooth transitionautoregressive models (TeraK svirta and Anderson, 1992), threshold autoregres-sive models (Pesaran and Potter, 1997) and Altissimo and Violante (1995),


Tab

le3

Nonl

inea

rity

test

per

form

ance

:!g 0(x

)"(2

/(1#

e~x))!

1

ef 1

f 2T"

100

T"25

0T"

500

l T"

l T1

l T"

l T2

l T"

l T3

l T"

l T1

l T"

l T2

l T"

l T3

l T"

l T1

l T"

l T2

l T"

l T3

0.2

0.0

0.0

0.03

80.

034

0.01

80.

053

0.05

20.

044

0.05

50.

054

0.04

70.

40.

00.

00.

046

0.03

80.

024

0.05

20.

045

0.03

70.

053

0.05

10.

045

0.6

0.0

0.0

0.05

00.

038

0.02

30.

048

0.04

20.

037

0.05

10.

048

0.04

5

0.2

!2.

00.

00.

105

0.08

50.

051

0.44

10.

392

0.34

00.

836

0.81

60.

783

0.4

!2.

00.

00.

108

0.08

40.

050

0.51

40.

460

0.39

60.

891

0.87

90.

847

0.6

!2.

00.

00.

123

0.09

20.

058

0.62

60.

581

0.50

90.

954

0.94

60.

932

0.2

!5.

00.

00.

192

0.25

40.

176

0.82

20.

826

0.79

50.

998

0.99

80.

998

0.4

!5.

00.

00.

278

0.39

30.

269

0.78

40.

797

0.75

60.

999

0.99

90.

999

0.6

!5.

00.

00.

465

0.56

80.

408

0.74

40.

770

0.71

10.

999

0.99

90.

998

0.2

!2.

02.

00.

313

0.28

30.

197

0.97

80.

976

0.96

41.

000

1.00

01.

000

0.4

!2.

02.

00.

315

0.30

70.

221

0.98

00.

978

0.97

01.

000

1.00

01.

000

0.6

!2.

02.

00.

331

0.35

20.

259

0.97

20.

971

0.96

71.

000

1.00

01.

000

0.2

!5.

05.

00.

321

0.30

40.

220

0.97

10.

969

0.96

01.

000

1.00

01.

000

0.4

!5.

05.

00.

635

0.66

90.

470

0.76

70.

778

0.68

00.

994

0.99

40.

992

0.6

!5.

05.

00.

615

0.63

40.

423

0.80

50.

806

0.71

00.

992

0.99

20.

988

!Bas

edon

the

nonlin

ear

coin

tegr

atio

nte

stdiscu

ssed

abov

e,en

trie

sco

rres

pond

toth

eem

pirica

lfreq

uency

ofre

ject

ion

ofth

enul

lhyp

oth

esis

oflinea

rco

inte

grat

ion,

infa

vorof

a"ndi

ng

ofn

onlin

earco

inte

grat

ion.T

estst

atistics

are

cons

truct

edfo

rva

rious

sam

ple

size

s(T"

100,

250,

and

500

obse

rvat

ions),

for

3di!

eren

tva

lues

ofth

ela

gtr

unca

tion

par

amet

er:l t1"

0,l T

2"in

tege

r[4(¹

/100

)1@4],

and

l T3"

inte

ger[

12(¹

/100

)1@4],

and

bas

edon

the

nonl

inea

rfu

nct

ion:g

(x)"

[2/(1#

e~x)]!

1.D

ata

are

gene

rate

dac

cord

ing

toth

efo

llow

ing

pro

cess

:*X

t"d#

eZt~

1#fg

0(Zt~

1)#e t

wher

eX

t"(X

1,t,X

2,t)@

isa

2]1

vect

or,

e tis

a2]

1ve

ctor

whose

com

pone

nts

are

distr

ibute

dIN

(0,1

)an

dZ

t"X

1,t!

X2,t.

Also,d"

(d1,d

@ 2),d 1"

d 2"0.

2,e"

(e1,e

2)@,an

df"

(f1,f

2)@.The"rs

tth

ree

row

sof

entr

iesin

the

tabl

ere

por

tth

eem

pirica

lsiz

eoft

he

test

base

don

a5%

nom

inal

size

,while

the

rem

ainin

gro

wsre

port

the

empiric

alpow

er,a

lso

for

a5%

nom

inal

size

test

.Res

ults

are

bas

edon

5000

Mon

teC

arlo

replic

atio

ns.


Tab

le4

Nonl

inea

rity

test

per

form

ance

:!g 0(x

)"sin(x

)

e 2f 1

f 2T"

100

T"25

0T"

500

l T"

l T1

l T"

l T2

l T"

l T3

l T"

l T1

l T"

l T2

l T"

l T3

l T"

l T1

l T"

l T2

l T"

l T3

0.2

!2.

00.

00.

341

0.28

70.

184

0.80

80.

749

0.69

90.

984

0.97

60.

970

0.4

!2.

00.

00.

305

0.25

50.

165

0.83

30.

799

0.75

60.

989

0.98

70.

985

0.6

!2.

00.

00.

311

0.26

90.

171

0.86

70.

850

0.81

90.

993

0.99

30.

992

0.2

!5.

00.

00.

467

0.39

80.

288

0.82

10.

753

0.70

40.

975

0.96

30.

954

0.4

!5.

00.

00.

483

0.42

30.

321

0.87

40.

841

0.81

70.

988

0.98

50.

983

0.6

!5.

00.

00.

487

0.46

10.

373

0.89

30.

881

0.86

00.

992

0.99

10.

990

0.2

!2.

02.

00.

510

0.42

30.

313

0.85

30.

795

0.74

70.

986

0.97

70.

971

0.4

!2.

02.

00.

523

0.46

00.

354

0.90

80.

881

0.85

80.

995

0.99

20.

991

0.6

!2.

02.

00.

555

0.52

20.

430

0.93

30.

924

0.91

00.

995

0.99

40.

994

0.2

!5.

05.

00.

078

0.07

90.

059

0.30

20.

327

0.28

60.

671

0.68

60.

651

0.4

!5.

05.

00.

179

0.17

90.

134

0.52

40.

532

0.49

30.

864

0.87

00.

851

0.6

!5.

05.

00.

426

0.42

20.

353

0.78

20.

784

0.75

90.

995

0.95

60.

950

!See

note

sto

Tab

le3.

All

row

sre

port

empiric

alpow

erofth

ete

st.


Tab

le5

Nonl

inea

rity

test

per

form

ance

:!g 0(x

)"sin(x

)if

DxD)

p/2,

oth

erw

ise

g 0(x)"

[2/(1#

e~x)]!

1

e 2f 1

f 2T"

100

T"25

0T"

500

l T"

l T1

l T"

l T2

l T"

l T3

l T"

l T1

l T"

l T2

l T"

l T3

l T"

l T1

l T"

l T2

l T"

l T3

0.2

!2.

00.

00.

146

0.12

10.

080

0.52

10.

493

0.44

20.

923

0.91

90.

903

0.4

!2.

00.

00.

137

0.11

60.

076

0.55

10.

527

0.47

60.

938

0.93

50.

924

0.6

!2.

00.

00.

142

0.12

00.

076

0.58

90.

561

0.50

40.

953

0.94

90.

939

0.2

!5.

00.

00.

165

0.14

30.

088

0.62

90.

602

0.55

20.

962

0.95

80.

951

0.4

!5.

00.

00.

235

0.22

30.

160

0.77

00.

768

0.74

40.

981

0.98

10.

979

0.6

!5.

00.

00.

223

0.22

40.

162

0.77

30.

776

0.75

30.

983

0.98

40.

984

0.2

!2.

02.

00.

334

0.31

40.

223

0.86

00.

856

0.83

80.

993

0.99

30.

992

0.4

!2.

02.

00.

327

0.31

20.

227

0.87

60.

874

0.86

40.

993

0.99

40.

994

0.6

!2.

02.

00.

339

0.33

60.

249

0.87

90.

879

0.86

40.

995

0.99

50.

995

0.2

!5.

05.

00.

241

0.25

40.

207

0.69

20.

707

0.68

20.

963

0.96

50.

961

0.4

!5.

05.

00.

304

0.40

40.

302

0.80

50.

832

0.80

70.

983

0.98

70.

985

0.6

!5.

05.

00.

370

0.47

70.

356

0.80

40.

839

0.80

80.

978

0.98

20.

980

!See

note

sto

Tab

le3.


nonlinear error correction models (Granger and Swanson, 1996), thresholderror correction models (Balke and Fomby, 1997), and the references containedtherein. In this section, we do not estimate new varieties of nonlinear models, butrather, we illustrate the use of the nonlinear error-correction test discussedabove. In order to do this, we examine data on the term structure of interestrates, which have been kindly provided to us by Heather Anderson. The dataconsist of monthly nominal yield to maturity "gures from the Fama TwelveMonth Treasury Bill Term Structure File, for the period January 1970}Decem-ber 1988. Six variables, denoted R1}R6, are examined, and correspond toTreasury bills with one month to maturity, Treasury bills with two months tomaturity, and so on, up to bills with 6 months to maturity. A detailed discussionof the data is given in Hall et al. (1992), as well as in Anderson (1997).

We consider three types of tests: (i) For the one-dimensional case, we con-struct both the ADF statistic for the null hypothesis of a unit root and theKwiatkowski et al. (1992) test, described in Section 3, for the null of stationar-ity/ergodicity. (ii) For the two-dimensional case, we construct the Johansen&trace' test statistic (1988, 1991) for the null of no cointegration, and the Shin(1994) test for the null of cointegration (comovement). For the latter cointegra-tion test, we compare the results using both the critical values in Shin (1994) andthe critical values reported above. (iii) Also for the two-dimensional case, weperform the test for nonlinear cointegration described in Section 5.

Test results are reported in Table 6. Panel A contains ADF and Kwiatkowskiet al. (1992) test results, where l

Tdenotes the number of lags used in the

computation of the estimated variance (see above). For the ADF test, q( k isreported, although test regressions without a constant were also run for allvariables, and our "ndings did not di!er. Note that the outcomes of the ADFand the Kwiatkowski et al. (1992) tests agree. In particular, for all maturities, theunit root null hypothesis is not rejected (using the ADF test) while the null ofstationarity ergodicity is consistently rejected (using the Kwiatkowski et al.(1992) test), regardless of the value of l

T.

Panel B reports results based on cointegration and comovement tests. For thesake of brevity, only bivariate combinations which include R1 are reported on.Complete results are available from the authors. As mentioned above, thelimiting distribution from Shin (1994) does not apply in the presence of neglectednonlinearity. Interestingly, even using the smaller critical values reported inSection 3 above, we fail to reject the null of stochastic comovement for 3 of 5bivariate combinations, based on statistics constructed using l

T"4 and 8

(columns 5 and 6 of the table). Furthermore, for the other two bivariatecombinations, use of the standard Shin (1994) critical values leads to a failureto reject at a 5% level (and in some cases a 1% level). These results agree withthe theory posited by Hall et al. (1992) which suggests that any bivariatecombination of our nominal interest rate series is cointegrated. Given these"ndings, it may be of interest of test the bivariate combinations for nonlinear


Table 6Empirical illustration: the term structure of interest rates!

Panel A: Unit root and stationarity ergodicity tests

Variable Kwiatkowski et al. (1992)ADF

q( k (lags) lT"0 1 2 3 4 5 6 7 8

R1 !1.60 (7)H 4.95H 2.55H 1.73H 1.33H 1.08H 0.92H 0.80H 0.71H 0.64HR2 !1.61 (6)H 5.25H 2.68H 1.82H 1.39H 1.13H 0.95H 0.83H 0.73H 0.66HR3 !1.64 (6)H 5.30H 2.71H 1.84H 1.40H 1.14H 0.96H 0.83H 0.74H 0.67HR4 !1.67 (8)H 5.34H 2.72H 1.85H 1.41H 1.14H 0.97H 0.84H 0.74H 0.67HR5 !1.68 (7)H 5.42H 2.76H 1.87H 1.43H 1.16H 0.98H 0.85H 0.75H 0.68HR6 !1.43 (7)H 5.54H 2.82H 1.91H 1.46H 1.18H 1.00H 0.87H 0.77H 0.69H

Panel B: Cointegration and comovement tests

Variables Johansen Shin NLCI

lT"0 1 4 8 l

T"0 1 4 8

R1, R2 20.3H 0.44% 0.41% 0.38% 0.34% 4.16HH 3.78HH 3.52H 2.88H(0.041) (0.051) (0.060) (0.089)

R1, R3 24.6H 0.37% 0.31$ 0.26% 0.25$ 3.99HH 2.67H 1.90 1.85(0.045) (0.102) (0.168) (0.173)

R1, R4 25.6H 0.25$ 0.21$ 0.17# 0.16# 3.55H 3.00H 2.61H 2.45(0.059) (0.083) (0.106) (0.117)

R1, R5 27.8H 0.27$ 0.22# 0.16# 0.15" 2.60H 1.95 1.63 1.68(0.106) (0.162) (0.201) (0.194)

R1, R6 26.1H 0.40% 0.30$ 0.20# 0.18# 2.50 1.85 1.52 1.62(0.113) (0.173) (0.217) (0.203)

!The data are monthly Treasury-Bill nominal yield to maturity "gures for the period 1970 : 1}1988 : 12. R1 is the series for billswith one month to maturity, R2 is the series for bills with two months to maturity, and so on up until R6 which is the series forbills with 6 months to maturity. Panel A contains augmented Dickey}Fuller (ADF) and Kwiatkowski et al. (1992) test statistics(as discussed above). For the ADF tests, the &lag augmentations' used is in brackets, chosen based on an examination of residualautocorrelations. All starred entires in Panel A correspond to evidence of a unit root (nonstationary-ergodicity) at the 5% levelusing critical values from Kwiatkowski et al. (1992) or MacKinnon (1991). In Panel B, the second column contains theJohansen (1988,1991) trace test statistics, where the associated vector autoregressions are estimated with a constant in thecointegrating relation, a linear determinstic trend in the data (results were the same without the deterministic trend), and 6 lagsof each variable (similar results were found for the 12 lag case). Starred entries indicate rejection of the null hypothesis of nocointegration (in favor of cointegrating space rank of unity) using the 5% level critical value. The last 8 columns of PanelB contain Shin (1994) cointegration (comovement) and nonlinear error correction test statistics. For each of these two statistics,values are tabulated for l

T"0, 1, 4, 8.

"For the Shin-type tests superscrips &b' and &c' denote failure to reject the null hypothesis of cointegration (comovement) usingthe 5% and 1% (respectively) critical values in Section 3 of the paper.#is the same as footnote &b' above.$is the same as footnotes &b' and &c', but use the critical values of Shin (1994). For the nonlinear cointegration test (last4 columns), values of the statistics, (mq

T)2, which is used in the modi"ed Bonferroni bound of Hochberg (1988) de"ned as

a"minj/1,2,m

(m!j#1)P(j)

, where P(j)

is the p-value of the test statistic, is reported. Here the values used for q are q"M2.0,5.0, 8.0, 10.0N, so that m"4. Modi"ed Bonferroni bounds are given in brackets below statistic values. Rejection of the null oflinear cointegration in favor of the alternative of nonlinear cointegration at a 5% and 10% size are denoted by superscipts Hand HH, respectively.%is the same as footnote &d' above.


error correction. The appropriate test statistics are reported in the last 4 col-umns of the table, with modi"ed Bonferroni bounds given in brackets belowstatistic values (see footnote to Table 6). Entries superscripted H(HH) denoterejection of the null hypothesis of linear cointegration (comovement) in favor ofnonlinear cointegration at a 5% (10%) level. Thus, for the pair of seriesconsisting of (R1, R2), some evidence of nonlinear cointegration is found, regard-less of the value of l

T. Weaker evidence (i.e. rejections for some of l

T) of nonlinear

cointegration is also found for (R1, R3), (R1, R4) and (R1, R5). Based on a com-parison of linear and nonlinear 1-step ahead forecast errors, Anderson (1997)"nds evidence of nonlinear error correction among the variables consideredhere, consistent with our "ndings.

7. Conclusions

In this paper we introduce a class of nonlinear Markov processes character-ized by the sum of a linear component plus a bounded nonlinear component. Inthe one-dimensional case, the ergodicity of the process is equivalent to theabsence of a unitary or explosive root, and in the multidimensional case theexistence of linear stochastic comovement is equivalent to the existence of coin-tegration.

We show that the statistic proposed by Kwiatkowski et al. (1992) has a well-de"ned limiting distribution under the null of general stationary-ergodic nonlin-ear processes, and has power not only against the alternative of integratedness,but also against the alternative of a more general nonlinear nonergodic process.We also show that the cointegration test statistic proposed by Shin (1994) isconsistent, in our context, although the critical values of the test are quitedi!erent from those tabulated by Shin (1994) for the linear case. Finally, wepropose a consistent test for the null hypothesis of linear cointegration againstthe alternative of nonlinear cointegration (NLCI). In a series of Monte Carloexperiments, we "nd that the NLCI test has good "nite sample size and power.Further, in an illustration of the NLCI test in which we examine the termstructure of interest rate, we "nd some evidence that bivariate models of interestrates of di!erent maturities may be nonlinearly cointegrated.

Acknowledgements

We are grateful to two anonymous referees, Filippo Altissimo, FrankDiebold, Rob Engle, Clive Granger, Jin Hahn, Ross Starr, Shinichi Sakata,Chor-Yiu Sin, Ruth Williams and to seminar participants at the University ofPennsylvania, Pennsylvania State University and the 1995 Winter Meetings ofthe Econometric Society for helpful comments on an earlier version of this


paper. We also owe special thanks to Herman Bierens for very useful sugges-tions, and to Heather Anderson for providing us with the data used in theillustration of our nonlinearity test. Swanson thanks the National ScienceFoundation and the Research and Graduate Studies O$ce at PennsylvaniaState University for "nancial assistance. White's participation was supported byNSF grant SBR 9511253.

Appendix A

Proof of Proposition 2.1. Note "rst that

Xt"AX

t~1#(g

0(h1@

0X

t~12, hj@

0X

t~1)!g

0(0))#(e

t#g

0(0))

"H(Xt~1

)#(et#g

0(0)).

Assume that the associated deterministic system, say xt, is given by x

t"H(x

t~1).

The proposition follows from Theroem 4.3 of Tong (1990), once we have shownthat his assumptions B1}B3 are satis"ed. First, note that A1}A3 imply B2}B3. Itremains to show that B1 is satis"ed. As lim

,x,?=(g

0(x)!g

0(0))/ExE"0 from

Theorem 1.3.5(a) in Kocic and Ladas (1993), it follows that H is asymptoticallystable at large, so that B1 is satis"ed. h

Proof of Proposition 2.3. (i) Let Xt"+t

j/0At~je

j#+t

j/1At~jg

0(h1@

0X

j~1,2,

hj@0X

j~1)"XI

t#g8

0,t. As XI

tis a linear component, from Johansen (1988) and

from the Granger representation theorem (Engle and Granger, 1987), it followsthat the Wold representation for XI

tis *XI

t"C(¸)e

t. Using the Beveridge and

Nelson (1981) decomposition it follows that XIt"C(1)+t

j/0ej#CH(¸)e

t, where

CH(¸)"+=j/0

(+=i/j`1

Ci)¸j. Note also that g8

0,T~1)O

p(¹), as it is the sum of

¹ bounded components. Now, XIT/¹1@2NB

1, where B

1is a p-dimensional mean

zero normal with covariance matrix equal to C(1)RC(1)@, and so is a non-degenerate random variable. If g8

0,T/¹1@2PR or g8

0,T/¹1@2P0, then X

TPR

at rate ¹1@2 (if XIT

is the component of higher order of probability) or at a ratefaster than ¹1@2 (if the nonlinear component is of higher order than the linearcomponent). Finally, consider the case in which g8

0,T/¹1@2NG, where G is either

a nondegenerate or a degenerate random variable. As B1

is a continuouslydistributed nondegenerate random variable, P(u: B

1(u)"!G(u))"0. The

result follows directly.(ii) The result follows from the fact that h@

0X

tsatis"es the assumptions of

Proposition 2.1. h

Proof of Proposition 2.4. Using the arguments from the proof ofProposition 2.3, and by setting A"I, it follows that EX

TE diverges.


Furthermore, h@0X

t"h@

0+t

j/1ej#h@

0+t

j/1g0(h1@

0,2, hj@

0X

j~1). Thus, noting that

¹~1@2(h@0+T

j/1ej)Nh@

0B2, where B

2is a p-dimensional mean zero normal with

covariance matrix equal to R, the result follows by the same argument used inthe Proof of Proposition 2.3(i). h

Proof of Theorem 3.1. (i) First, note that

1

J¹

*Tr++j/1

(Xj!XM )"

1

J¹

*Tr++j/1

(Xj!a)!

1

J¹

*Tr++j/1

(XM !a)

"

1

J¹

*Tr++j/1

(Xj!a)!

1

¹

*Tr++j/1A

1

J¹

T+t/1

(Xt!a)BNp

X=

r!p

Xr=

1,

where a"E(Xt) for all t, p2

X"lim

T?=Var(¹~1@2+T

t/1(X

t!a)2), and ="

(=s, s3[0, 1]) is a standard Brownian motion. Also, by Fact 2.6, X

tis a strictly

stationary mixing process, with mixing coe$cients decaying at a geometric rate.Furthermore, given A2, X

tis fourth-order stationary. Thus, by Lemma 1 in

Andrews (1991), Xt!E(X

t) satis"es his Assumption A. Given that XM is ¹1@2-

consistent for the true "rst moment, from Theorem 1(a) in Andrews (1991), itfollows that s( 2

lTPp2

Xin Prob. The result then follows by the continuous mapping

theorem.(ii) We need to show that the numerator of S

T, say ¹~2+T

t/1(+t

j/1(X

j!XM ))2

explodes at a faster rate than does s( 2lT. First, consider the numerator. Under the

alternative, Xt"+t

j/1ej#+t

j/1g0(X

j~1) Thus,

1

¹2

T+t/1A

t+j/1

(Xj!XM )B

2"

1

¹2

T+t/1A

t+j/1A

j+i/1

(ei!e6 )

#

j+i/1

(g0(X

i~1)!g6

0)BB

2

"

1

¹2

T+t/1A

t+j/1

j+i/1

(ei!e6 )B

2#

1

¹2

T+t/1A

t+j/1

j+i/1

(g0(X

i~1)!g6

0)B

2

#

2

¹2

T+t/1A

t+j/1A

j+i/1

(ei!e6 )BA

i+i/1

(g0(X

i~1)!g6

0)BB,

where e6 and g60denote sample means. From Kwiatkowski et al. (1992, p. 168), we

know that

1

¹4

T+t/1A

t+j/1

j+i/1

(ei!e6 )B

2Np2eP

1

0AP

u

0

=ksdsB

2du,


where =ks"=

s!:1

0=qdq, and = is a standard Brownian motion. Now, as

g0

is bounded, it follows that (1/¹4)+Tt/1

(+tj/1

+ji/1

(g0(X

i~1)!g6

0))2"O

p(¹).

Turning now to the denominator, note "rst from Kwiatkowski et al. (1992,p. 168), that it follows that

1

¹lTA

1

¹

T+t/1A

t+j/1

(ei!e6 )B

2

B#1

¹lTA

2

¹

lT+t/1A1!

t

lT#1B

T+

j/t`1

]AAj+i/1

(ei!e6 )B A

j~t+s/1

(es!e6 )BBBNp2eP

1

0

(=ks)2 ds.

In order to prove part (ii), it thus su$ces to show that

T+t/1A

t+j/1

j+i/1

(g0(X

i~1)!g6

0)B

2&¹2

T+t/1A

t+j/1

(g0(X

j~1)!g6

0)B

2, (A.1)

where & means &of the same order of probability'. Let M be the LHSof (A.1) and Q be the RHS of (A.1). Also, let g

i,c"g

0(X

i)!(1/¹)+T

i/1g0(X

i).

Note that

M"

T+i/1

i2g21, c

#

T~1+i/1

i2g22,c

#2#

T~k+i/1

i2g2k,c#2#g2

T

#2T~1+i/1

i2g1, c

g2, c

#2T~2+i/1

i2g1,c

g3,c

#2#2T~k+i/1

i2g1,c

gk`1, c

#2#2g1,c

gT, c

#2#2T~k+i/1

i2gk, c

gk`1,c

#2T~k~1

+i/1

i2gk,c

gk`2, c

#2#2gk,c

gT,c

#2#2gT~1,c

gT, c

.

Also,

Q"¹g21,c

#(¹!1)g22, c

#2#(¹!k)g2k,c#2#g2

T, c

#2(¹!1)g1, c

g2, c

#2(¹!2)g1,c

g3,c

#2#2(¹!k)g1, c

gk`1,c

#2#2g1,c

gT, c

#2#2(¹!k)gk,c

gk`1, c

#2(¹!k!1)gk, c

gk`2,c

#2#2gk, c

gT,c

#2#2gT~1,c

gT, c

.


By comparing M and Q term by term, observe that M&¹2Q. By the sameargument,

1

¹2

T+t/1C

t+j/1

j+i/1

gi,cD

2&l~1

T C1

¹ CT+t/1A

t+j/1

gj,cB

2

D#

2

¹AlT+t/1A1!

t

lT#1B

T+

j/t`1A

j+i/1

gi,cBA

j~t+i/1

gi,cBBD.

Thus, if the denominator explodes at rate ¹1`glT, g3[0, 1], the numerator

will explode at rate ¹2`g, and the ratio of the two will then explodeat rate ¹l~1

T. Note that g"0 is the case in which either the linear component is

dominant or the two components, linear and nonlinear, are of the same order ofprobability. We have g'0 in the case in which the nonlinear component isdominant. h

Proof of Theorem 3.2. First, note that

¹3@2(bKT!b

0)"

(1/¹3@2)+Tt/1

(X1, t!XM

1)u

2, t(1/¹3)+T

t/1(X

1, t!XM

1)2

where u2, t

"l2, t!/, and E(l

1, t)"/, for all t. Now,

1

¹3

T+t/1

(X1, t!XM

1)2"

1

¹3

T+t/1A

t+j/1

u1,j

!

1

¹

T+t/1A

t+j/1

u1, tB#nAt!

¹#1

2 BB2

"

1

¹3

T+t/1AnAt!

¹#1

2 BB2#o

p(1),

where u1, t"l

1, t!n, and E(l

1, t)"n, for all t. Thus (1/¹3)+T

t/1(X

1, t!XM

1)2P

n2/12 in Prob. Further,

1

¹3@2

T+t/1

(X1, t!XM

1)u

2, t"

1

¹3@2

T+t/1A

t+j/1

u1, j

!

1

¹

T+t/1A

t+j/1

u1, jBBu2, t

#

1

¹3@2

T+t/1

nAt!¹#1

2 Bu2, t"

1

¹3@2

T+t/1

nAt!¹#1

2 Bu2, t#op(1)

Nnpl2P1

0

s d=s!

n2

pl2=1.


Thus,

¹3@2(bKT!b

0)N

12

npl2AP

1

0

s d=s!(1/2)=

1B.

Proof of Theorem 3.3 (i) Let

mKt"(X

2, t!XM

2)!bK

T(X

1, t!XM

1)"(X

2, t!XM

2)!b

0(X

1, t!XM

1)

#(bKT!b

0)(X

1, t!XM

1).

Then,

1

J¹

t+j/1

mKj"

1

J¹

t+j/1

((X2, j

!XM2)!b

0(X

1,j!XM

1))

#¹3@2(bKT!b

0)

1

¹2

t+j/1

(X1,j

!XM1)

"

1

J¹

t+j/1

((X2, t!XM

2)!b

0(X

1, t!XM

1))

#¹3@2(bKT!b

0)

1

¹2

t+j/1Anj!n

¹#1

2 B#op(1).

Given Theorem 3.3(i), it follows that

1

¹2

T+t/1A

t+j/1

mKjB

2Np2l2P

1

0

Q2s

ds,

where

Qs"(=

s!s=

1)!6AP

1

0

s d=s!(1/2)=

1B(s2!s).

As ¹3@2(bKT!b

0)"O

p(1), and using the same argument used in the proof of

Theorem 3.1(i), it follows that s2lTPp2l2 in Prob. The result follows.

(ii) Recall that

bKT"

+Tt/1

(X1, t

!XM1)(X

2, t!XM

2)

+Tt/1

(X1, t!XM

1)2

.

Given that X1, t

and X2, t

are of the same order of probability (note that thenonlinear components, g

0,1, g

0,2, depend on the same argument, h@

0X

t),

bKT"O

p(1). Thus, s~2

lT(1/¹2)+T

t/1(+t

j/1mKj)2 will explode at rate ¹l~1

T, by the same

argument used in the proof of Theorem 3.1(ii). h


Proof of Theorem 4.1. (i) Recall that bKT

is the slope coe$cient from the regres-sion of X

2, ton X

1, tand a constant, and c(

T"(!bK

T, 1). Let g(

tbe de"ned as in Eq.

(4.1), and g(6 "(1/¹)+Tt/2

g(c( @TX

t~1q). Finally, let g6 "(1/¹)+T

t/2g(c( @

TX

t~1q). Given

Eq. (4.2), note that

1

J¹

T+t/2

g(t(g(c( @

TX

t~1q!g(6 )"

1

J¹

T+t/2

g(t(g(c( @

TX

t~1q)!g(6 )

!(bKT!b

0)

1

J¹

T+t/2

(X1, t!XM

1)(g(c( @

TX

t~1q)!g(6 )

!t0(bK

T!b

0)

1

J¹

T+t/2

(X1, t!XM

1)(g(c( @

TX

t~1q)!g(6 )

!(tKT!t

0)

1

J¹

T+t/2

(c@0X

t~1!c@

0XM )(g(c( @

TX

t~1q)!g(6

!(tKT!t

0)(bK

T!b

0)

1

J¹

T+t/2

(X1, t~1

!XM1)(g(c( @

TX

t~1q)!g(6 ).

Now write

g(c( @TX

t~1q)!g(6 "(g(c( @

0X

t~1q)!g6 )#(g(c( @

TX

t~1q)!g(c@

0X

t~1q)!(g(6 !g6 )).

Then,

1

J¹

T+t/2

g(t(g(c( @

TX

t~1q)!g(6 )"

1

J¹

T+t/2

g(t(g(c( @

0X

t~1q)!g6 )

#

1

J¹

T+t/2

gt(g(c@

0X

t~1q)!g(c( @

TX

t~1q)!(g(6 !g6 ))

!(bKT!b

0)

1

J¹

T+t/2

(X1, t!XM

1)(g(c@

0X

t~1q)!g6 )

!(bKT!b

0)

1

J¹

T+t/2

(X1, t!XM

1)(g(c( @

TX

t~1q)!g(c@

0X

t~1q)!(g(6 !g6 ))

!t0(bK

T!b

0)

1

J¹

T+t/2

(Xt~1

!XM1)(g(c@

0X

t~1q)!g6 )

#t0(bK

T!b

0)

1

J¹

T+t/2

(X1, t~1

!XM1)(g(c( @

TX

t~1q)!g(c@

0X

t~1q)!(g(6 !g6 ))


!(tKT!t

0)

1

J¹

T+t/2

c@0(X

t~1!XM

1)(g(c@

0X

t~1q)!g(6 )

#(tKT!t

0)

1

J¹

T+t/2

(c@0X

t~1!c@

0XM )(g(c( @

TX

t~1q)!g(c@

0X

t~1q)!(g(6 !g6 ))

!(tKT!t

0)(bK

T!b

0)

1

J¹

T+t/2

(X1, t~1

!XM1)(g(c@

0X

t~1q)!g6 )

#(tKT!t

0)(bK

T!b

0)

1

J¹

T+t/2

(X1, t~1

!XM1)(g(c( @

TX

t~1q)!g(c( @

0X

t~1q)

!(g(6 !g6 )). (A.2)

We want to show that

1

J¹

T+t/2

g(t(g(c( @

TX

t~1q!g(6 )"

1

J¹

T+t/2

gt(g(c@

0X

t~1q)!g6 )

!(tKT!t

0)

1

J¹

T+t/2

(c@0X

t~1!c@

0XM )(g(c( @

TX

t~1q)!g6 )#o

p(1). (A.3)

As

J¹(tKT!t

0)"

(1/J¹)+Tt/2

(c@0X

t~1!c@

0XM )g

t(1/J¹)+T

t/2(c@

0X

t~1!c@

0XM )2

#op(1),

(A.3) implies (4.6) in the text. Recall that under H0, J¹(tK

T!t

0)"O

p(1) and

¹(bKT!b

0)"O

p(1). As the ninth term on the RHS of (A.2) converges to zero

faster than the "fth term, and the tenth term vanishes faster than the sixth, wecan neglect them. Also, under H

0, X

1, t"+t

j/1l1, j

, where E(l1,j

)"0 for all j.Note that when g

0is a constant, then ¹1@2(tK

T!t

0)"O

p(1) and

¹3@2(bKT!b

0"O

p(1) and E(l

1, j)O0 for all j. By twice applying the mean value

theorem, it turns out that

g(c( @TX

t~1q)!g(c@

0X

t~1q)"(bK

T!b

0)X

1, tqDg(c@HX

t~1q)

"(bKT!b

0)X

1, t~1qDg(c@

0X

t~1q)

!(bK !b0)(bH!b

0)X2

1, t~1q2D2g(cHHX

t~1q),


where Dkg denotes the kth derivative of g with respect to its argument, andcH"(!bH, 1)@, with bH3(b

0, bK

T), cHH"(!bHH, 1), and bHH3(bH, b

0). Thus, the

second term on the RHS of (A.2) can be written as

¹(bKT!b

0)

1

¹3@2

T+t/2

(X1, t~1

!XM1)qg

t(Dg(c@

0X

t~1q)!Dgc0)

!¹(bKT!b

0)¹(bH!b

0)

1

¹5@2

T+t/2

gt((X

1, t~1!XM

1)2

]q2(D2g(cHHXt~1

q)!Dg2cHH), (A.4)

where Dgkc0"(1/¹)+Dkg(c@0X

t~1q) and DgkcHH"(1/¹)+Dkg(c@HHX

t~1q). Note

that Dg(c@0X

t~1q)!Dgc0 is a zero mean strong mixing process, and that (from

Fact 2.5) the usual size conditions required by the invariance principle hold.Also, by a similar argument as that used in Hansen (1992b, Proof of Theorem4.1),

1

¹

T+t/2

(X1, t~1

!XM1)qg

t(Dg(c@

0X

t~1!Dgc0"O

p(1),

so that the "rst term in (A.4) is Op(¹~1@2). Also, note that the absolute value of

the second term in (A.4) is majorized by

¹DbKT!b

0D¹DbH!b

0Dsup

tDD2g(c@HHX

t~1q)

!Dg2cHHDq21

J¹

suptDg

tD

1

¹2

T+t/2

(X1, t~1

!XM1)2. (A.5)

Now, ¹DbH!b0D)¹DbK

T!b

0D"O

p(1) and (1/J¹) sup

tDg

tD"o

p(1). It follows

that the expression in (A.5) is op(1), and thus the expression in (A.4) is also o

p(1).

Now, consider the eighth term on the RHS of (A.2), which can be written as

(tKT!t

0)(bK

T!b

0)

1

J¹

T+t/2

qX1, t~1

(c@0X

t~1!c@

0XM )(Dg(c@

0X

t~1q)!Dgc0)

!(tKT!t

0)(bK

T!b

0)(bH!b

0)

1

J¹

+(X1, t~1

!XM1)2q2

](c@0X

t~1!c@

0XM )(D2g(c@HHX

1, t~1q)!Dg2cHH)"o

p(1),

using the same majorization argument which is used above. By an analogousargument, the fourth, "fth and sixth terms are o

p(1). Given that

¹(bKT!b

0)"O

p(1), from the same argument used in the proof of Theorem 3.1(i)


we have that s2lTPp2

0in Prob. The result follows. Note also that when g

0is a nonzero constant, ¹1@2(tK !t

0)"O

p(1), ¹3@2(bK

T!b

0)"O

p(1), and

E(l1,j

)O0 for all j. Thus, the same arguments used above apply when g0

isa nonzero constant.

(ii) Under the alternative, X1, t"nt#+t

j/1(l

1, t!n)"nt#+t

j/1uj. As X

1, tis

dominated by the deterministic component, ¹3@2(bKT!b

0)"O

p(1). Further,

J¹(tKT!tH)"O

p(1), where tHOt

0. Now all of the terms on the RHS of

(A.2), except for the "rst and the seventh are op(1), by an argument analogous to

that used in part (i), and by the law of large numbers for strong mixing processes,(1/¹)+T

t/2(g(c@

0X

t~1q)!g6 )!M~1c@0Xc@0XMc@0gXt~1

)gtPMO0. The result then fol-

lows. h

References

Altissimo, F., Violante, G.L., 1995. Persistence and Nonlinearity in GNP and Unemployment: anEndogeneous Delay Threshold VAR, Mimeo, University of Pennsylvania.

Anderson, H.M., 1997. Transaction costs and nonlinear adjustment towards equilibrium in the UStreasury bill market. Oxford Bulletin of Economics and Statistics 59, 465}480.

Anderson, H.M., Vahid, F., 1988. Testing multiple equation systems for common nonlinear compo-nents. Journal of Econometrics 84, 1}36.

Andrews, D., 1991. Heteroskedasticity and autocorrelation consistent covariance matrix estimation.Econometrica 59, 817}858.

Athreya, K.B., Pantula, S.G., 1986. Mixing properties of Harris chains and autoregressive processes.Journal of Applied Probability 23, 880}892.

Balke, N.S., Fomby, T.B., 1997. Threshold cointegration. International Economic Review 38,627}645.

Beveridge, S., Nelson, C., 1981. A new approach to decomposition of economic time series intopermanent and transitory components with particular attention to measurement of the businesscycle. Journal of Monetary Economics 7, 151}174.

Bierens, H.J., 1990. A consistent conditional moment test of functional form. Econometrica 58,1443}1458.

Corradi, V., 1995. Nonlinear transformations of integrated time series: a reconsideration. Journal ofTimes Series Analysis 16, 539}549.

Domowitz, I., El-Gamal, M.A., 1993. A consistent test for stationary-ergodicity. EconometricTheory 9, 589}601.

Domowitz, I., El-Gamal, M.A., 1997. A consistent nonparametric test of ergodicity for time serieswith applications. Mimeo, Northwestern University.

Engle, R.F., Granger, C.W.J., 1987. Co-integration and error correction: representation, estimationand testing. Econometrica 55, 547}557.

Ermini, L., Granger, C.W.J., 1993. Some generalization of the algebra of I(1) processes. Journal ofEconometrics 58, 369}384.

Granger, C.W.J., 1995. Modelling nonlinear relationships between extended-memory variables.Econometrica 63, 265}279.

Granger, C.W.J., Hallman, J., 1991. Long memory series with attractors. Oxford Bulletin ofEconomics and Statistics 53, 11}26.

Granger, C.W.J., Inoue, T., Morin, N., 1997. Nonlinear stochastic trends. Journal of Econometrics81, 65}92.


Granger, C.W.J., Swanson, N.R., 1996. Further developments in the study of cointegrated economicvariables. Oxford Bulletin of Economics and Statistics 58, 537}553.

Granger, C.W.J., TeraK svirta, T., 1993. Modelling Non-linear Economic Relationships. OxfordUniversity Press, New York.

Hall, A.D., Anderson, H.M., Granger, C.W.J., 1992. A cointegration analysis of treasury bill yields.The Review of Economics and Statistics 74, 116}126.

Hansen, B.E., 1992a. E$cient estimation and testing of cointegrating vectors in the presence ofdeterministic trends. Journal of Econometrics 53, 87}121.

Hansen, B.E., 1992b. Convergence to stochastic integrals for dependent and heterogeneous pro-cesses. Econometric Theory 8, 489}500.

Hochberg, Y., 1988. A sharper Bonferroni procedure for multiple tests of signi"cance. Biometrika 75,800}802.

Johansen, S., 1988. Statistical analysis of cointegration vectors. Journal of Economics and Dynamicsand Control 12, 231}254.

Johansen, S., 1991. Estimation and hypothesis testing of cointegration vectors in Gaussian vectorautoregressive models. Econometrica 59, 1551}1580.

Kocic, V.L., Ladas, G., 1993. Global Behavior of Nonlinear Di!erence Equations of Higher Orderwith Applications. Kluwer, London.

Kuan, C.-M., White, H., 1994. Arti"cial neural networks: an econometric perspective. EconometricReviews 13, 1}91.

Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., Shin, Y., (KPSS), 1992. Testing the null hypothesis ofstationarity against the alternative of unit root: how sure are we that economic series have a unitroot? Journal of Econometrics 54, 159}178.

Lee, T.H., White, H., Granger, C.W.J., 1993. Testing for neglected nonlinearity in time series models.a comparison of neural network methods and alternative tests. Journal of Econometrics 56,269}290.

MacKinnon, J.G., 1991. Critical values for cointegration tests in long-run economic relationships.In: Engle, R.F., Granger, C.W.J. (Eds.), Long-run Economic Relationships: Readings in Cointeg-ration. Oxford University Press, New York.

Meyn, S.P., 1989. Ergodic theorems for discrete stochastic systems using a stochastic Lyapunovfunction. SIAM Journal of Control and Optimization 27, 1409}1439.

Park, J.Y., Ogaki, M., 1991. Inference in cointegrated models using VAR prewhitening to estimateshortrun dynamics. Working Paper Number 281, The Rochester Center for Economic Research.

Pesaran, M.H., Potter, S.M., 1997. A #oor and ceiling model of U.S. output. Journal of EconomicDynamics and Control 21, 661}695.

Shin, Y., 1994. A residual based test of the null of cointegration against the alternative of nocointegration. Econometric Theory 10, 91}115.

Stinchcombe, M., White, H., 1998. Consistent speci"cation testing with nuisance parameters presentonly under the alternative. Econometric Theory 14, 295}324.

Swanson, N.R., 1999. LM tests and nonlinear error correction in economic time series. StatisticaNeerlandica, 53, 76}95.

TeraK svirta, T., Anderson, H.M., 1992. Characterizing nonlinearities in business cycles using smoothtransition autoregressive models. Journal of Applied Econometrics 7, S: S119}S136.

Tong, H., 1990. Non-linear Time Series: A Dynamical System Approach. Oxford University Press,New York.

White, H., 1984. Asymptotic Theory for Econometricians. Academic Press, New York.


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