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WALD TESTS OF I(1) AGAINST I(d) ALTERNATIVES:

SOME NEW PROPERTIES AND AN EXTENSION TO

PROCESSES WITH TRENDING COMPONENTS

By Juan J. Doladoa, Jesus Gonzaloa, and Laura Mayoralb ?

aDept. of Economics, U. Carlos III de Madrid;bInstitut d ´Análisi Económica (CSIC).

Abstract

This paper analyses the behaviour of a Wald-type test, i.e., the (E¢cient) Fractional

Dickey-Fuller (EFDF) test of I(1) against I(d); d < 1; relative to LM tests. Further, it extends

the implementation of the EFDF test to the presence of deterministic trending components in

the DGP. Tests of these hypotheses are important in many macroeconomic applications where

it is crucial to distinguish between permanent and transitory shocks because shocks die out in

I(d) processes with d < 1. We show how simple is the implementation of the EFDF in these

situations and argue that, under …xed alternatives, it is preferred to the LM test in Bahadur´s

sense. Finally, an empirical application is provided where the EFDF approach allowing for

deterministic components is used to test for long-memory in the GDP p.c. of several OECD

countries, an issue that has important consequences to discriminate between alternative growth

theories.

JEL Clasi…cation: C12 C22 O40

Keywords: Deterministic components, Fractional processes, Power, Unit roots.

?Corresponding E-mail:jesus.gonzalo@uc3m.es. We are grateful to an anonymous referee for very helpful

comments and to Claudio Michelacci and Bart Verspagen for making their data available to us. We also

thank Javier Hidalgo, Javier Hualde, Francesc Mármol, Peter Robinson, Carlos Velasco. Financial support

from the Spanish Ministry of Education through grants SEJ2006-00369, SEJ2004-04101 and SEJ2007-04325

and also from the Barcelona Economics Program of CREA is acknowledged. The usual disclaimer applies.

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1. INTRODUCTION

Typically, tests of I(1) vs. I(0) processes have problems in rejecting the null that a time

series fytg is I(1) when the true DGP is a fractionally integrated, I(d); process, particularly

if 0:5 < d < 1. This issue can have serious consequences for the analysis of the medium

and long- run properties of macroeconomic and …nancial variables. For instance, (i) shocks

could be identi…ed as permanent when in fact they die out eventually, and (ii) two series

could be considered as spuriously cointegrated when they are independent at all leads and

lags (see Gonzalo and Lee, 1998). Further, these mistakes are more likely to occur in the

presence of deterministic components as, e.g. in the case of trending economic variables.

In view of this problem, the goal of this paper is threefold. First, we discuss the power

behavior of a recently proposed Wald test of I(1) vs. I(d), d 2 [0;1) relative to the one

achieved by well-known LM tests. In particular, we use the concept of Bahadur´s asymptotic

relative e¢ciency (henceforth, ARE; see Gourieroux and Monfort, 1995) to derive new

analytical results regarding the non-centrality parameters of both types of tests under …xed

alternatives. Secondly, we extend the Wald-type testing procedure, originally derived for

driftless processes, to the more realistic case where deterministic components are present.

Finally, we present a feasible linear single-step regression estimation approach to deal with

serially correlated errors.

Speci…cally, we focus on a modi…cation of the Fractional Dickey-Fuller (FDF) test by

Dolado, Gonzalo and Mayoral (2002; DGM hereafter) recently proposed by Lobato and

Velasco (2007; LV hereafter) which achieves a slight improvement in e¢ciency over the

former. This test, henceforth denoted as the EFDF (e¢cient FDF) test, generalizes the

traditional DF test of I(1) against I(0) processes without deterministic components to the

broader framework of testing I(1) against I(d) with 0 ? d < 1. The EFDF (and the FDF)

test belongs to the family of Wald tests and relies upon the principle underlying the popular

Dickey-Fuller (DF) approach. The idea is to test for the statistical signi…cance of the slope

coe¢cient, '; by means of its t-ratio, t', in a regression where the dependent variable and

the regressor are …ltered so as to become I(0) under the null and the alternative hypothesis,

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respectively.1Both DGM and LV set ?ytas the dependent variable, where ? = (1 ? L):2

As regards the regressor, whereas DGM choose ?dyt?1, LV show that zt?1(d) = (1 ?

d)?1(?d?1? 1)?ytimproves the e¢ciency of the test. Non-rejection of H0: ' = 0 against

H1: ' < 0, implies that the process is I(1) and, conversely, rejection of the null implies

that the process is I(d):

In order to compute either ?dyt?1or zt?1(d), an input value for d is required. One could

either consider a (known) simple alternative, HA : d = dA < 1 or, more realistically, a

composite one, H1: d < 1. We focus here on the last case where DGM and LV show that

it su¢ces to use a T?-consistent estimate (with ? > 0) of the true integration order to get

a N(0;1) limiting distribution of the resulting test-statistic:

Under a sequence of local alternatives approaching H0: d = 1 from below at a rate of

T?1=2, LV (2007, Theorem 1) prove that, with Gaussianity, the EFDF test is asymptotically

equivalent to the uniformly most powerful invariant (UMPI) test, i.e., the LM test intro-

duced by Robinson (1991, 1994) and later adapted by Tanaka (1999) for the time domain.

We …rst show that, when the alternative is …xed, the former has a larger non-centrality

parameter than the latter, in line with the standard result about the better power properties

of Wald tests relative to LM tests (see Engle, 1984). Moreover, when compared to other

tests of I(1) vs. I(d) which rely on direct inference about semiparametric estimators of

d, the EFDF test exhibits better power properties in general, under a correct speci…cation

of the stationary short-run dynamics of the error term in the auxiliary regression. This is

due to the fact that the semiparametric estimation procedures often imply larger con…dence

intervals of the memory parameter, in exchange for less restrictive assumptions on the error

term and robustness in case of misspeci…cation.3By contrast, the combination of a wide

1In the DF setup, these …lters are ? = (1 ? L) and ?0L = L; so that the regressand and regressor are

?yt and yt?1, respectively.

2As shown in DGM (2002), both regressors can be constructed by applying the truncated version of the

binomial expansion of the …lter (1 ? L)din the lag operator L to yt (t = 0;1;:::), so that ?d

?i(d) yt?i; where ?i(d) is the i-th coe¢cient in that expansion, de…ned at the end of this Introduction. In

the sequel, we will refer to this truncated …lter simply as ?d:

3See, e.g., Velasco (1999), Robinson (2003), Abadir et al. (2005), Shimotsu and Phillips (2005) and

+yt =Pt?1

i= 0

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range of semiparametric estimators for the input value of d with an auxiliary parametric

regression, as the one discussed above, yields a parametric rate for the Wald tests. Thus,

in a sense, the Wald tests combine the favorable features of both approaches to improve

power, while reducing the danger of misspecifying short-run dynamics.

Next, we investigate how to implement the EFDF test when some deterministic compo-

nents are considered in the DGP, a case which was neither considered by LV nor by DGM.

Although we will analyze other types of trends, we will mainly focus on the role of a linear

trend since many (macro) economic time series exhibit this type of trending behavior in

their levels. Our main result is that, in contrast with what happens with most tests for

I(1) against I(0), the EFDF test remains being e¢cient in the presence of deterministic

components and it maintains the same asymptotic distribution, insofar as they are cor-

rectly …ltered. In this respect, this result mimics the one found for LM tests in this case;

cf. Robinson (1994), Tanaka (1999) and Gil-Alaña and Robinson (1997).

Lastly, we extend the results obtained for a DGP with i:i:d: error terms to the case where

they are autocorrelated, as in the (augmented) DF case (ADF henceforth). In this respect,

DGM (2002, Theorems 6 and 7) have proved that, in order to remove the autocorrelation, it

is su¢cient to augment the set of regressors in the auxiliary regression of the FDF test with

k lags of the dependent variable such that k " 1 as T " 1; and k3=T " 0, as in Said and

Dickey (1984). This leads to the augmented FDF (AFDF) test. As regards the EFDF test,

we conjecture that a similar result holds, although we will con…ne our discussion below, as

in LV (2007), to the case of …nite-lag AR processes. The procedure based on the EFDF test

turns out to be much simpler than accounting for serial correlation in the LM framework.

Further, we point out that the two-step procedure proposed by LV (2007) can be simpli…ed

to a feasible linear single-step estimation approach. An empirical application dealing with

testing the possibility that long GNP per capita series for several OECD countries may follow

nonstationary I(d) processes, yet with shocks that die out (supporting the hypothesis of

beta-convergence) instead of I(1) (no convergence), illustrates our proposed methodology.

The rest of the paper is structured as follows. Sections 2 brie‡y overviews the properties

Shimotsu (2006).

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of the EFDF test when the process is a driftless random walk under the null and derives

new results about the power of this test relative to the LM test under …xed alternatives

using Bahadur´s ARE. Section 3 extends the previous results to the case where the process

contains trending deterministic components (e.g., a linear trend), considering both the case

of i:i:d. and autocorrelated errors. Section 4 discusses an empirical application of the

previous test. Finally, Section 5 draws some concluding remarks.

Proofs of the theorems are collected in the Appendix.

In the sequel, the de…nition of a I (d) process that we will adopt is the one used by

Akonom and Gourieroux (1987) where a fractional process is initialized at the origin. This

corresponds to Type-II fractional Brownian motion (see the previous discussion in footnote

3) and is similar to the de…nitions of an I(d) process underlying the LM test proposed Robin-

son (1994) and Tanaka (1999). Moreover, the following conventional notation is adopted

throughout the paper: ?(:) denotes the Gamma function, and f?i(d)g represents the se-

quence of coe¢cients associated to the expansion of (1 ? L)din powers of L ,

?i(d) =

?(i ? d)

?(?d)?(i + 1):

The indicator function is denoted by 1(:): Finally,

w

! denotes weak convergence in D[0;1]

! means convergence in probability.endowed with the Skorohod J1topology, and

p

2. THE EFDF TEST

2.1 De…nitions

Like Robinson (1994) we consider a process fytg that is generated by an additive model,

namely as the sum of a deterministic component, ?(t); and an I(d) component, ut; so that

yt= ?(t) + ut; (1)

where ut= ??d"t1t>0is a purely stochastic I (d) process and "tis a zero-mean i.i.d. random

variable.

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When ?(t) ? 0;4DGM introduced a Wald-type (FDF) test for testing the null hypothesis

of H0 : d = 1 against the composite alternative H1 : 0 ? d < 1; based on the t-ratio

associated to the hypothesis ? = 0 in the OLS regression

?yt= ??d?yt?1+ ?t: (2)

where d?? 0 is an input value needed to perform the test. If d?is chosen such that

d?=bdT; wherebdTis a T?-consistent estimator of d, with ? > 0; DGM (2002) and LV (2006)

Recently, LV (2007) have proposed the EFDF test based on a modi…cation of regression

have shown that the asymptotic distribution of the resulting t-statistic, t?is N (0;1).

(2) that permits to achieve higher e¢ciency s under the assumption of ?(t) ? 0 (or known).

More speci…cally, their proposal is to compute the t-statistic, t'; associated to the null

hypothesis ' = 0 in the regression

?yt= 'zt?1(d?) + "t; (3)

where zt?1(d?) is de…ned as5

zt?1(d?) =

??d??1? 1?

(1 ? d?)

?yt;

such that ' = (d?? 1): Note that, when ' = 0; the model becomes a random walk, i.e.,

?yt= "t; while, when ' = (d?? 1) < 0 , it becomes a pure fractional process, ?dyt= "t:

The insight for the higher e¢ciency of the EFDF test is that, under H1, the regression

model considered in (2) can be written as ?yt = ?1?d"t = "t+ (d ? 1)"t?1+ 0:5d(d ?

1)"t?2+ ::: = ??dyt?1+ "t+ 0:5d(d ? 1)"t?2+ ::: with ? = d ? 1. Thus, the error term ?t

= "t+0:5d(d?1)"t?2+::: in (2) is serially correlated. Although OLS provides a consistent

estimator of ?, since ?t is orthogonal to the regressor ?yt?1 = "t?1; it is not the most

e¢cient one. By contrast, the regression model used in the EFDF test does not su¤er

from this problem since, by construction, yields an i:i:d. error term. Finally, note that

4Alternatively, ?(t) could be considered to be known. In this case, the same arguments go through after

substracting it from yt to obtain a purely stochastic process.

5A similar model was …rst proposed by Granger (1986) in the more general context of testing for cointe-

gration with multivariate series, a modi…cation of which has been recently considered by Johansen (2005).

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application of L´ Hôpital rule to zt?1(d?) in the limit case as d?! 1 leads to a regressor

equal to ?ln(1?L)?yt= ?1

section 2.3).

j=1j?1?yt?j, which is the one used in Robinson’s LM test (see

Theorem 1 in LV (2007), which we reproduce below for completeness, establishes the

asymptotic properties of t':

Theorem 1 Under the assumption that the DGP is given by yt= ??d?t1(t>0), "tis i:i:d.

with …nite fourth moment, the asymptotic properties of the t-statistic t'for testing ' = 0

in (3), where the input of zt?1(c

a) Under the null hypothesis (d = 1),

dT) is a T??consistent estimator of d?; for some d?> 0:5

with ? > 0; are given by

t'(bdT)

w

! N (0; 1):

b) Under local alternatives, (d = 1 ? ?=pT);

t'(bdT)

w

! N (??h(d?); 1);

q

where h(d?) = ?1

c) Under …xed alternatives (d 2 [0;1) < 1), the test based on t'(bdT) is consistent.

LV (2007) show that the function h(:) achieves a global maximum at 1 where h(1) =

p?2=6, and that h(1) equals the noncentrality parameter of the locally optimal Robinson’s

LM test (see subsection 2.2 below). Thus, insofar as a T?-consistent estimator of d is used

j=1j?1?j(d?? 1)=

?1

j=1?j(d?? 1)2; d?> 0:5; d?6= 1:

as input of zt?1(d?) with ? > 0, the EFDF test is locally asymptotically equivalent to

Robinson´s LM test. In practice, the estimate of d could be smaller than 0:5. If such is the

case, the input value can be chosen according to the following rule:edT= max{bdT;0:5+#g,

easily obtained by applying some semiparametric estimators. Among them, the estimators

where # is a small number, e.g., # = 0:001: A power-rate consistent estimate of d can be

proposed by Abadir et al. (2005), Shimotsu (2006) and Velasco (1999) provide convenient

choices since they also cover the case where deterministic components are present, as we do

in section 3.

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2.2 Asymptotic relative e¢ciency of Wald and LM tests.

As discussed earlier, the closer competitor to the Wald (FDF and EFDF) tests is the LM

test proposed by Robinson (1991, 1994) in the frequency domain, subsequently extended

by Tanaka (1999) to the time domain. In this section we discuss the power properties of

the three competing tests under …xed alternatives.6The comparison is done in Bahadur’s

ARE sense.

We start with the LM test, denoted as LMT; which considers the null hypothesis of ? = 0

against the alternative ? 6= 0 for the DGP ?d0+?yt = "t. In line with the hypotheses

considered in this paper, we will focus on the particular case where d0= 1 and ?1 ? ? < 0:

Assuming that "t? N(0;?2), the score-LM test is computed as

r

j=1

where b ?j=PT

score-LM test is as the t-ratio (t?) ofb?olsin the regression

?yt= ?x?

LMT=

6

?2T1=2

T?1

X

j?1b ?j

w

! N (0;1); (4)

t=j+1?yt ?yt?j=PT

t=1(?yt?j)2(see Robinson, 1991 and Tanaka, 1999).

Breitung and Hassler (2002) have shown that an alternative simpler way to compute the

t?1+ et; (5)

where x?

under H0 : ? = 0; b ?e tends to ? and plim T?1P(x?

Under a sequence of local alternatives of the type ? = ?T?1=2with ? > 0 for H0 :

t?1=Pt?1

j=1j?1?yt?j: Intuitively, since t?=P(?ytx?

t?1)=b ?e(P(x?

t?1)2)1=2and,

t?1)2= ?2=6; then t?has the same

limiting distribution as LMT:

d0= 1, the LMT (or t?) is the UMPI test. However, as discussed above, the EFDF test is

asymptotically equivalent to the UMPI under the appropriate choice ofbdTearlier discussed.

the limiting distribution of the the EFDF test is identical to that of the LM test, i.e.,

N(??h(d); 1) where h(:) is ?=p6 for d = 1. DGM (2002, Theorem 3) in turn obtained that

6The available results in the literature only establish the consistency of the Wald and LM test under …xed

Hence, as stated in Theorem 1 above, when ?(t) ? 0 (or known) and d = 1 ? ?T?1=2;

alternatives. Yet, they do not derive the non-centrality parameters as we do below.

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the corresponding distribution of the FDF under local alternatives test is N(??; 1): Hence,

in the case of local alternatives, the asymptotic e¢ciency of the FDF test relative to the

LM and EFDF tests is 0:78 ('p6=?).

In the rest of this section, we analyze the case with …xed alternatives where, to our

knowledge, results are new. In particular, we derive the non-centrality parameters of the

FDF, EFDF and LM tests under an I(d) alternative where the DGP is assumed to be

?dyt= "twith d 2 (0; 1): Hence, ?yt= ??b"twhere b = d ? 1 < 0. These non-centrality

parameters correspond (in square terms) to the approximate slopes of the tests in Bahadur’s

sense. The following result holds.

Theorem 2 If ?dyt= "twith d 2 [0;1); the t-statistics associated to the EFDF and FDF

tests, denoted as t'and t?, respectively, verify,

T?1=2t'

p

! ?

??(3 ? 2d)

(1 ? d)?(2 ? d)

[?(3 ? 2d) ? (d ? 1)2?2(2 ? d)]1=2? cFDF(d);

?2(2 ? d)? 1

?1=2

? cEFDF(d);

T?1=2t?

p

! ?

while, under the same DGP; the LM test de…ned in (4) satis…es that,

r

(1 ? d)?(d ? 2)

T?1=2LMT

p

! ?

6

?2

?(2 ? d)

1

X

j=1

?(j + d ? 1)

j?(j + 2 ? d)? cLM(d);

where cEFDF(d); cFDF(d) and cLM(d) denote the non-centrality parameter under the …xed

alternative d 2 (0;1) of the EFDF, FDF and LM tests, respectively:

Figure 1 displays the three non-centrality parameters for d 2 (0;1): In Bahadur’s sense,

the ratio of the approximate slopes of the tests (e.g., ARE(EFDF;LM;d) = [cEFDF(d)=(cLM(d)]2)

de…nes the asymptotic relative e¢ciency (ARE) of one test versus the other. When ARE

is greater than 1, it is said that the …rst test is asymptotically preferred (or asymptot-

ically more powerful) in Bahadur’s sense to (than) the second one (see section 23.2.3 in

Gourieroux and Monfort, 1995). As expected, the noncentrality parameters of the EFDF

and the LM tests behave similarly for values of d very close to H0, whereas the one of the

FDF test is slightly smaller for these local alternatives. Nonetheless, the LM test performs

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signi…cantly worse than both Wald-type tests when the alternative is not local. The EFDF

tests performs slightly better than the FDF test in line with LV’s (2007) arguments about

e¢ciency. The intuition for the worse power performance of the LM test is that there does

not exist any value for ? in (5 ) that makes etboth i:i:d. and independent of the regressor

for …xed alternatives, implying that x?

t?1does not maximize the correlation with ?yt. Fig-

ure 2 depicts the Bahadur´s ARE of the EFDF and FDF tests with respect to the LM, plus

the one between the two Wald tests, in the range d 2 (0:5;1): The message to be drawn

from this Figure is similar to that in Figure 1.

In sum, for …xed alternatives with (approximately) d < 0:9, using the above ARE criteria,

these tests can be ranked in decreasing asymptotic power order as EFDF>FDF>LM.

0.00.10.2 0.30.4 0.5

d

0.60.70.8 0.91

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

LM

EFD F

FD F

Fig 1. Non-centrality parameters of LM and Wald tests

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:

0.510.56 0.61 0.660.71 0.76

d

0.81 0.860.91 0.96

0.5

1

1.5

2

2.5

3

Bahardur's ARE

EFDF-LM

EFDF-FDF

FDF-LM

FIG 2. Bahadur’s Asymptotic Relative E¢ciency

As regards semiparametric estimators, both the Fully Extended Local Whittle (FELW,

see Abadir et al., 2005) and the Exact Local Whittle estimators (ELW, see Shimotsu, 2006)

verify the asymptotic propertypm(bdT?d)

H1: d < 1 is the nonparametric rate Op(pm) which is smaller than the Op(pT) parametric

w

! N?0;1

! N (0;1). Therefore, their rate of divergence under

4

?for m = o(T

4

5): Test statistics for unit

roots are based on ?d= 2pm(bdT?1)

rate achieved by the Wald test. Of course, this loss of power is just the counterpart of their

w

higher robustness against misspeci…cation.

3. THE EFDF TEST FOR TRENDING I(D) PROCESSES

3.1 i.i.d. case

In this section, we extend the EFDF testing approach to the more realistic case where

?(t) 6= 0 and unknown. Our goal is to examine how this (unknown) deterministic term

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should be taken into account when implementing the test.

Following Elliott et al. (1996), we consider two di¤erent types of ?(t):

Slowly Evolving Deterministic component

Condition A. (Slowly evolving trend). The deterministic component ?(t) veri…es

?(t) = O(t?); ? < 0:5:

Condition A is immediately satis…ed if ?(t) is a constant term but also holds for a variety

of time functions, such as slowly increasing trends, (e.g., t?; ? < 0:5 or logt):

In this case, it is straightforward to show that the stochastic component in ytdominates

the deterministic term when T is large. Hence, ?(t) has no e¤ect either on the asymptotic

distribution of the t-ratio statistic or on the e¢ciency properties of the test in the absence

of ?(t). Therefore, one can proceed to run regression (3) ignoring the presence of these

slowly evolving trends.

The following theorem presents the properties of the EFDF test when the DGP is given

by (1) and ?(t) veri…es Condition A.

Theorem 3 (Slowly evolving trends) Under the assumption that the DGP is given by yt=

?(t) + ??d?t1(t>0), where d ? 1, ?tis i:i:d. with …nite fourth moment, and ?(t) veri…es

Condition A, the asymptotic properties of the t-statistic t'for testing ' = 0 in (3) (denoted

by EFDF?test), where the input of zt?1(bdT) is a T??consistent estimator of d?; for some

d?> 0:5 with ? > 0; are identical to those stated in Theorem 1.

Evolving Deterministic Components

Condition B. (Evolving trend). The deterministic component ?(t) veri…es.

?(t) = O(t?); ? ? 0:5;

with ? known.

Under Condition B, the DGP is allowed to contain trending regressors in the form of,

say polynomials (of known order) of t: Hence, when the coe¢cients of ?(t) are unknown,

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the test described above are unfeasible. Nevertheless, it is still possible to obtain a feasible

test with the same asymptotic properties as in Theorem 1 if a consistent estimate of ?(t) is

removed from the original process. Indeed, under H0; the relevant coe¢cients of ?(t) can

be consistently estimated by OLS in a regression of ?yton ??(t): For instance, consider

the case where the DGP contains a linear time trend, that is,

yt= ? + ?t + ??d?t; (6)

which, under H0: d = 1; corresponds to the popular random walk with drift case. Taking

…rst di¤erences, it follows that ?yt = ? + ?1?d"t: The OLS estimate of ?;^?; (i.e., the

sample mean of ?yt) is consistent under both H0and H1: In e¤ect, under H0;^? is a T1=2

-consistent estimator of ? whereas, under H1; it is T3=2?d-consistent with 3=2?d > 0:5 (see

Hosking 1996; Theorem 8). Hence, the following theory holds.

Theorem 4 (Evolving trends) Under the assumption that the DGP is given by yt= ?(t)+

??d?t1(t>0), where d ? 1, ?tis i:i:d. with …nite fourth moment, and ?(t) satis…es Condition

B, the asymptotic properties of the t- statistic t'for testing ' = 0 in the regression

g

?yt= 'g

zt?1

?^dT

?

+ et

(7)

(denoted by EFDF? test), where the input^dT of g

?yt= ?yt? ?^ ?(t), g

asymptotic properties of the t-statistic t' for testing ' = 0 in (7) are identical to those

zt?1

?^dT

?^dT

?

?

=

is a T?? consistent estimator

?

(1?^dT)

of d?> 0:5 with ? > 0;g

zt?1

?^ dT?1?1

?

(?yt? ?^ ?(t)); and

the coe¢cients of ?^ ?(t) are estimated by an OLS regression of ?yt on ??(t); then the

stated in Theorem 1.

As mentioned above, Shimotsu’s (2006) semiparametric estimator provides power rate

consistent estimators of d ? 1 for the case where the DGP contains a linear or a quadratic

trend whereas Velasco’s (1999) estimator is invariant to a linear (and possibly higher order)

time trend.

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3.2 Serial correlation case: The invariant AEFDF test

Next, we generalize the DGP considered in (1) by assuming that utfollows an stationary

linear AR(p) process, namely, ?p(L)ut= ?t1t>0where ?p(L) = 1 ? ?1L ? ::: ? ?pLpwith

?p(z) 6= 0 for jzj ? 1: This motivates the following nonlinear regression model

?yt= '[?p(L)zt?1(d)] +

p

X

j=1

?j?yt?j+ ?t; (8)

which is similar to (3), except for the inclusion of the lags of ?ytand for the …lter ?p(L) in

the regressor whose signi…cance is tested. Estimation of this model is complicated because

of the nonlinearity in the parameters ' and ?p= (?1;:::; ?p): Compared with the i:i:d case,

the practical problem arises because the vector ? is unknown and therefore the regressor

[?p(L)zt?1(d)] is unfeasible. For this reason LV (2007) recommended to apply a two-step

procedure that allows one to obtain e¢cient tests also with autocorrelated errors.

3.2.1 Two-step procedure.—

For the case where ?(t) ? 0 (or known), LV (2007) implement the two step procedure as

follows. In the …rst step, the coe¢cients of ?p(L) are estimated (under H1) by OLS in the

equation

?bdTyt=

p

X

t=1

?j?bdTyt?j+ at;(9)

wherebdTsatis…es the conditions stated in Theorem 1. The estimator of ?p(L) is consistent

with a convergence rate which depends on the rate ?: Second, estimate by OLS the equation

?yt= '[b?p(L)zt?1(^dT)] +

p

X

j=1

?j?yt?j+ vt;(10)

whereb?p(L) is the estimator from the …rst step, andbdT denotes the same estimated input

augmented regression is still both normally distributed and locally optimal. The test will

used in that step as well. As LV (2007, Theorem 2) have shown, the t' statistic in this

be denoted by AEFDF (augmented EFDF) test in the sequel.

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For the case where the coe¢cients of ?(t) are considered to be unknown, a similar pro-

cedure as that described in section 2.1 can be implemented and e¢cient tests will still be

obtained. If ?(t) is a slowly moving trend satisfying Condition A, the test based on regres-

sion (10) can be implemented and the asymptotic properties stated in LV (2007, Theorem

2) still hold through. For the case where ?(t) satis…es Condition B; as discussed earlier,

one needs to remove these terms from the original variables prior to computing regressions

(9) and (10); where the coe¢cients of ?(t) can be estimated by OLS under the null. For

instance, if the DGP is de…ned as in (6), a consistent estimator of ? is obtained from the

OLS estimator of a regression of ?yton a constant term. Clearly, this estimator has the

same properties in this case as those described in Section 3.1. Then, regression (9) simply

becomes

?bdT(yt?^?t) = [1 ? ?p(L)]?bdT(yt?^?t) + at;

whereas regression (10) would be

g

?yt= '[b?p(L)g

zt?1

zt?1

?^dT

?

] +

p

X

t=1

?j^

?yt?j+ vt;(11)

andg

trend and so forth for higher-order time trends.

?yt= ?yt?^? and g

?^dT

?

=

?

?^ dT?1?1

(1?^dT)

?

(?yt?^?): In the case where ?(t) in the

DGP contains a quadratic term, ?ytshould be regressed on a constant and a linear time

LV (2007) have shown that the asymptotic properties of the two-step AEFDF test is

identical to those in Theorems 1 and 2, except that, under local alternatives (d = 1??=pT;

with ? > 0); we have that t'(d)! N (??!;1) and t (d)

!2=?2

6

such that { = ({1;:::; {p)0with {k=P1

ww

! N (?!;1) where

? {0??1{;

j=kj?1cj?k; k = 1;:::; p, cj’s are the coe¢cients

t=0ctct+jk?jj; k; j = 1; :::; p; of Ljin the expansion of 1=?(L); and ? = [?k;j]; ?k;j=P1

denotes the Fisher information matrix for ?(L) under Gaussianity. Note that !2is identical

to the drift of the limiting distribution of the LM test under local alternatives (see Tanaka,

1999).

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