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t-statistic and

asymptotic beh

Statistics & Probability Letters 78 (2008) 622–628

www.elsevier.com/locate/stapro

Behaviour of Dickey–Fuller tests when there is a break under theunit root null hypothesis

Amit Sen�

Department of Economics, 3800 Victory Parkway, Xavier University, Cincinnati, OH 45207, USA

Received 12 February 2007; received in revised form 4 July 2007; accepted 7 September 2007

Available online 26 October 2007

Abstract

We show that Dickey and Fuller’s [1979. Distribution of the estimator for autoregressive time series with a unit root. J.

Amer. Statist. Assoc. 74, 427–431; 1981. Likelihood ratio statistics for autoregressive time series with a unit root.

Econometrica 49, 1057–1072.] normalized estimator, and F-statistics will spuriously reject the unit root null when the true

data generating process is a unit root with a break.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Unit root; Break; Normalized estimator; Dickey–Fuller F-statistics

1. Introduction

Owing to the seminal work of Perron (1989), it is well recognized that conventional Dickey–Fuller statisticsfail to reject the unit root null hypothesis when there is a break under the trend-stationary alternative.1

However, as noted by Leybourne et al. (1998), relatively less attention has been given to the asymptoticbehaviour of conventional Dickey–Fuller unit root tests when there is a break under the null hypothesis.Therefore, Leybourne et al. (1998) derive the asymptotic behaviour of the Dickey–Fuller pseudo t-statistic forHO : r ¼ 1, denoted by tt, based on the following regression:

yt ¼ mþ btþ ryt�1 þ ut (1)

when the break under the unit root null is characterized as follows:

ModelðAÞ : yt ¼ m1 þ dDtðt�Þ þ yt�1 þ et, ð2Þ

ModelðBÞ : yt ¼ m1 þ ðm2 � m1ÞDUtðt�Þ þ yt�1 þ et, ð3Þ

e front matter r 2007 Elsevier B.V. All rights reserved.

l.2007.09.024

3 745 2931; fax: +1 513 745 3692.

ess: sen@xavier.edu

˜ es and Reyes (1998, 1999) and Leybourne and Newbold (2000b) for the asymptotic behaviour of the Dickey–Fuller

the normalized estimator when there is a break under the trend stationary alternative. See Sen (2001, 2003) for the

aviour of the Dickey–Fuller F-statistics under the trend-break stationary alternative.

ARTICLE IN PRESSA. Sen / Statistics & Probability Letters 78 (2008) 622–628 623

where t� 2 ð0; 1Þ determines the time at which the break occurs in the sample, Dtðt�Þ ¼ 1ðt¼½t�T �þ1Þ is a dummyvariable that is equal to 1 when t ¼ ½t�T � þ 1 and zero otherwise, d is the level break magnitude, DUtðt�Þ ¼1ðt4½t�T �Þ is a dummy variable that is equal one if tX½t�T � þ 1 and zero otherwise, m2 � m1 is the drift breakmagnitude, and et�i:i:d:ð0;s2Þ.

Model (A) allows for an exogenous change in the level of the series, and Model (B) allows for an exogenouschange in the rate of growth, see Perron (1989) for further details. Leybourne et al. (1998) show that tt willspuriously reject the unit root null hypothesis if the break in the level is proportional to T1=2, and the break inthe rate of growth is proportional to T�1=2, where T is the sample size. That is, the presence of a break underthe unit root null hypothesis will lead to the ‘converse Perron phenomenon,’ that is, spurious rejection of theunit root null hypothesis.

We consider three additional unit root statistics proposed by Dickey and Fuller (1979, 1981): (a) thenormalized estimator Tðr� 1Þ, (b) the F-statistic for the joint null hypothesis HO2 : m ¼ 0;b ¼ 0;r ¼ 1,denoted by F2, that corresponds to the hypothesis that yt is a random walk, and (c) the F-statistic for the jointnull hypothesis HO3 : b ¼ 0;r ¼ 1, denoted by F3, that corresponds to the hypothesis that yt is a random walkwith drift. Specifically,

F2 ¼ ð3s2Þ�1ðS

2� S

2Þ, ð4Þ

F3 ¼ ð2s2Þ�1ð ~S

2� S

2Þ, ð5Þ

where r is the OLS estimate of the coefficient on yt�1 in (1), S2¼PT

t¼1ðyt � m� bt� ryt�1Þ2, s2 ¼

ðT � 3Þ�1S2, S

2¼PT

t¼1ðyt � yt�1Þ2, and ~S

2¼ S

2� T�1ð

PTt¼1ðyt � yt�1ÞÞ

2. The critical values forthese statistics are tabulated in Dickey and Fuller (1979, 1981). We derive the limiting behaviourof Tðr� 1Þ, F2, and F3 when there is a break under the unit root null hypothesis. In addition toModel (A) and Model (B) described above, we consider a simultaneous break in the level and the rate ofgrowth given by

ModelðCÞ : yt ¼ m1 þ dDtðt�Þ þ ðm2 � m1ÞDUtðt�Þ þ yt�1 þ et (6)

2. Asymptotic results and simulation evidence

In this section, we describe the limiting distribution of the normalized estimator ðTðr� 1ÞÞ, and theDickey–Fuller F-statistics (F2 and F3) under the data generating process described in Eqs. (2), (3), and (6).Following Leybourne and Newbold (2000a), we allow the break magnitudes d and m2 � m1 to be proportionalto the sample size, that is, d ¼ k1T

1=2, and m2 � m1 ¼ k2T�1=2. The limiting behaviour are expressed as

functions of the Wiener Process, denoted by W ðrÞ for r 2 ½0; 1�, defined on the unit interval. All proofs arerelegated to an appendix.

Theorem 1. Suppose the true data generating process for the time series fytgTt¼0 is given by Eq. (2). Then,

(a)

(c)

Tðr� 1Þ ) 1jjVAjjfMA

31A11 þMA32A12 þMA

33A13g,

(b)

F2 )1

3ðk2þs2ÞfA0V�1A AÞg,

1

1 0 �1 2

F3 ) 2ðk21þs2ÞfA V A A� fk1 þ sW ð1Þg g,

where A is 3� 1 matrix with A½1; 1� ¼ A11 ¼ sW ð1Þ þ k1, A½2; 1� ¼ A21 ¼ sW ð1Þ � sR 10

W ðrÞdrþ k1t�,

A½3; 1� ¼ A31 ¼s22½W ð1Þ2 � 1� þ k1sW ð1Þ, MA

31 ¼112

k1ð�1þ 4t� � 3t�2Þ þ 12sR 10 rW ðrÞdr� 1

3

R 10 W ðrÞdr, MA

32

¼ 12k1ð�t� þ t�2Þ þ 1

2sR 10

W ðrÞdr� sR 10

rW ðrÞdr, MA33 ¼

112, VA is a 3� 3 symmetric matrix with VA½1; 1� ¼ 1,

V A½1; 2� ¼ 12, V A½1; 3� ¼ k1ð1� t�Þ þ s

R 10 W ðrÞdr, V A½2; 2� ¼ 1

3, V A½2; 3� ¼ 1

2k1ð1� t�2Þ þ s

R 10 rW ðrÞdr, and

V A½3; 3� ¼ k21ð1� t�Þ þ 2k1s

R 1t� W ðrÞdrþ s2

R 10

W ðrÞ2 dr, and jjVAjj is the determinant of VA.

ARTICLE IN PRESSA. Sen / Statistics & Probability Letters 78 (2008) 622–628624

Theorem 2. Suppose the true data generating process for the time series fytgTt¼0 is given by Eq. (3), then

(a)

(c)

(c)

Tðr� 1Þ ) 1jjVBjjfMB

31B11 þMB32B12 þMB

33B13g,

(b)

F2 )12 fB0V�1BÞg,

3s B

1 0 �1 � 2

F3 ) 2 fB VB B� fk2ð1� t Þ þ sW ð1Þg g, 2s

where B is 3� 1 matrix with B½1; 1� ¼ B11 ¼ sW ð1Þ þ k2ð1� t�Þ, B½2; 1� ¼ B21 ¼ sW ð1Þ � sR 10

W ðrÞdrþ

12k2ð1� t�2Þ, B½3; 1� ¼ B31 ¼

s22½W ð1Þ2 � 1� þ k2s

R 1t� W ðrÞdrþ k2sðW ð1Þ �

R 10 W ðrÞdr� t�fW ð1Þ �W ðt�ÞgÞ

þ12k22ð1� 2t� þ t�2Þ, MB

31 ¼112

k2t�ð1� 2t� þ t�2Þ þ 12sR 10 rW ðrÞdr� 1

3sR 10 W ðrÞdr, MB

32 ¼112

k2ð�1þ 3t�2

�2t�3Þ þ 12sR 10 W ðrÞdr� s

R 10 rW ðrÞdr, MB

33 ¼112, V B is a 3� 3 symmetric matrix with VB½1; 1� ¼ 1, VB½1; 2�

¼ 12, VB½1; 3� ¼ 1

2k2ð1� 2t� þ t�2Þ þ s

R 10 W ðrÞdr, V B½2; 2� ¼ 1

3, V B½2; 3� ¼ 1

6k2ð2� 3t� þ t�3Þ þs

R 10 rW ðrÞdr,

and VB½3; 3� ¼ 13k22ð1� 3t� þ 3t�2 � t�3Þ � 2k2t�s

R 1t� W ðrÞdrþ 2k2s

R 1t� rW ðrÞdrþ s2

R 10 W ðrÞ2 dr, and jjV Bjj

is the determinant of V B.

Theorem 3. Suppose the true data generating process for the time series fytgTt¼0 is given by Eq. (6), then

(a)

Tðr� 1Þ ) 1jjVC jjfMC

31C11 þMC32C12 þMC

33C13g,

(b)

F2 )1

3ðs2þk2ÞfC0V�1C CÞg,

1

1 0 �1 � 2

F3 ) 2ðs2þk21ÞfC V C C � fk1 þ k2ð1� t Þ þ sW ð1Þg g,

where C is 3� 1 matrix with C½1; 1� ¼ C11 ¼ sW ð1Þ þ k1 þ k2ð1� t�Þ, C½2; 1� ¼ C21 ¼ sW ð1Þ � sR 10

W ðrÞdr

þk1t�þ 12k2ð1� t�2Þ, C½3; 1� ¼C31¼ k1sW ð1Þþk2s

R 1t� W ðrÞdrþk1k2ð1�t�Þþ 1

2k22ð1� 2t� þ t�2Þþk2sðW ð1Þ

�R 10 W ðrÞdr� t�fW ð1Þ �W ðt�ÞgÞ þ s2

2½W ð1Þ2 � 1�, MC

31 ¼12V C ½2; 3� � 1

3V C ½1; 3�, MC

32 ¼12V C ½1; 3�� V C ½2; 3�,

MC33 ¼

112, V C is a 3� 3 symmetric matrix with VC ½1; 1� ¼ 1, V C ½1; 2� ¼ 1

2, VC ½1; 3� ¼ 1

2k2ð1� 2t� þ t�2Þ

þk1ð1� t�Þ þ sR 10 W ðrÞdr, V C ½2; 2� ¼ 1

3, V C ½2; 3� ¼ 1

6k2ð2� 3t� þ t�3Þ þ 1

2k1ð1� t�2Þ þ s

R 10 rW ðrÞdr, and

VC ½3; 3� ¼ s2R 10 W ðrÞ2 drþ k2

1ð1� t�Þ þ 13k22ð1� 3t� þ 3t�2 � t�3Þ þ k1k2ð1þ t�2Þ þ 2k1s

R 1t� W ðrÞdr� 2k2t�

sR 1t� W ðrÞdrþ 2k2s

R 1t� rW ðrÞdr, and jjV Cjj is the determinant of V C .

These results indicate that the asymptotic behaviour of Tðr� 1Þ, F2, and F3 depends on the breakmagnitudes (k1 and k2) as well as the location of the break-date ðt�Þ. In order to evaluate whetherthese statistics exhibit the ‘converse Perron phenomenon,’ we calculated their size in finite samples viasimulations using the following parameter values: m1 ¼ 0, y0 ¼ 0, s2 ¼ 1, T ¼ 100, d ¼ f2:5; 5; 10g,m2 ¼ f0:5; 1; 3g, t� ¼ f0:01; 0:02; 0:03; . . . ; 0:99g, and et�i:i:d:Nð0; 1Þ. For Model (C), we used all combi-nations of d and m2. The statistics F2 and F3 are calculated using (4) and (5), and in all cases 10,000replications were used. The size of Tðr� 1Þ, F2 and F3 are calculated using finite sample critical valuesat the 5% significance level from Dickey and Fuller (1979, 1981). In order to save space, we only report theresults corresponding to d ¼ 2:5 and m2 ¼ 1. The complete set of simulation results are availablefrom the author upon request. First, consider the size of all statistics when the data generating processis given by Model (A) with d ¼ 2:5 is shown in Fig. 1. We have included the size of the Dickey–Fuller t-statistic ðttÞ for comparison. While all unit root statistics will spuriously reject the unit rootnull in the presence of an early break, the size distortions associated with the normalized estimator islowest in all cases. Also, as expected, we found that the size distortions increase with the breakmagnitude d.

The size of all statistics when the data generating process is given by Model (B) with m2 ¼ 1 isshown in Fig. 2. It should be noted that F2 is not an appropriate test owing to the presence of adrift. F3 exhibits the most severe size distortions that occur over a considerably larger set of break-dates.While the normalized estimator reveals some size distortions, they are much smaller compared to the

ARTICLE IN PRESS

0.15

0.10

0.05

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00

Siz

e

Break Fraction, ∗

t

T(p − 1)

φ2

φ3

Fig. 1. Empirical size of tt, Tðr� 1Þ, f2, and f3 at the 5% significance level with d ¼ 2:5.

0.90

0.75

0.60

0.45

0.30

0.15

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.00

Siz

e

Break Fraction, ∗

φ2

φ3

T(p − 1)

t

Fig. 2. Empirical size of tt, Tðr� 1Þ, f2, and f3 at the 5% significance level with m2 ¼ 1.

0.90

0.75

0.60

0.45

0.30

0.15

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.00

Siz

e

φ2

φ3

Break Fraction, ∗

T(p − 1)

t

Fig. 3. Empirical size of tt, Tðr� 1Þ, f2, and f3 at the 5% significance level with d ¼ 2:5 and m2 ¼ 1.

A. Sen / Statistics & Probability Letters 78 (2008) 622–628 625

Dickey–Fuller t-statistic, and occur over a much smaller range of break-dates (usually t�o0:06). Also,the size distortions with the normalized estimator increase more slowly compared to F3 and tt, with anincrease in m2.

ARTICLE IN PRESSA. Sen / Statistics & Probability Letters 78 (2008) 622–628626

Finally, the size of all unit root tests when the data generating process evolves according to Model (C) withd ¼ 2:5 and m2 ¼ 1 is shown in Fig. 3. Here again, the size distortions with the normalized estimator are thesmallest, and with F3 are the largest. The size distortions in F3 occurs over a much larger range of break-dates,and increase fairly rapidly with the break-magnitudes. However, the size distortions with the normalizedestimator are the smallest, they occur over a small range of break-dates very early in the sample, and do notincrease by much with the break magnitudes d and m2.

3. Conclusion

We have derived the asymptotic behaviour of the Dickey–Fuller (1979, 1981) unit root statistics, namely,the normalized estimator, and the F-statistics (denoted by F2 and F3) for three different characterizations ofbreak under the unit root null hypothesis. Our results complement the analysis of Leybourne and Newbold(2000a) and Leybourne et al. (1998) who find that the Dickey–Fuller t-statistic will spuriously reject the unitroot null hypothesis in the presence of a break. We find that the normalized estimator and the F-statistics alsoexhibit this ‘converse Perron phenomenon.’ However, we found that the severity of size distortions is smallestfor the normalized estimator and largest for F3. Our results reinforce the findings of Leybourne and Newbold(2000a) and Leybourne et al. (1998), and caution the practitioner regarding the use of conventionalDickey–Fuller unit root tests without modeling a break under the null hypothesis.

This research was partially supported by a Faculty Development Grant at Xavier University. We would like tothank an anonymous referee for helpful comments that improved the exposition of our paper.

Appendix A

Let y ¼ ðm; b;rÞ0, then the OLS estimator of y based on regression (1) is y ¼ ðX 0X Þ�1X 0Y where X is thematrix of explanatory variables and Y is the vector of the dependent variable implied by Eq. (1), and the error

sums of squares is given by S2¼PT

t¼1ðyt � m� bt� ryt�1Þ2. S

2¼PT

t¼1ðyt � yt�1Þ2 and ~S

2¼ S

2�

T�1ðPT

t¼1ðyt � yt�1ÞÞ2 are the error sums of squares from the reduced models implied by the null hypotheses

HO2 and HO3, respectively. The true location of the break-date is denoted by t�, and the vector of error termsfetg is denoted by e . All summations are taken over the sample, that is, from t ¼ 1 to T unless otherwise

specified. The results are based on the functional weak convergence result s�1T�1=2P½rT �

t¼1et )W ðrÞ8r 2 ½0; 1�where is W ðrÞ is the Wiener Process defined on the unit interval, and ‘‘)’’ denotes weak convergence. DefineDT ¼ diagðT1=2;T3=2;TÞ, Dðt�Þ and DUðt�Þ are vectors based on Dtðt�Þ and DUtðt�Þ, respectively. Withoutloss of generality, we set m1 ¼ 0, and y0 ¼ 0.

Proof of Theorem 1. The data generating process described by Model (A) implies that

DT ðy� yÞ ¼ ½D�1T ðX0X ÞD�1T �

�1½D�1T X 0dDðt�Þ þD�1T X 0e�, ðA:1Þ

S2� S

2¼ e0X ðX 0X Þ�1X 0eþ d2Dðt�Þ0X ðX 0X Þ�1X 0Dðt�Þ þ 2dDðt�Þ0X ðX 0X Þ�1X 0e, ðA:2Þ

~S2� S

2¼ e0X ðX 0X Þ�1X 0eþ d2Dðt�Þ0X ðX 0X Þ�1X 0Dðt�Þ

þ 2dDðt�Þ0X ðX 0X Þ�1X 0e� T�1d2� T�1

Xet

� �2� 2T�1d

Xet

� �. ðA:3Þ

The data generating process given by Model (A) and d ¼ k1T1=2 imply that: T�1=2d ! k1, T�3=2dð½t�T �þ1Þ!k1t�, T�1dy½t�T �)k1sW ðt�Þ, T�3=2

Pyt�1)s

R 10 W ðrÞdrþk1ð1� t�Þ, T�5=2

Ptyt�1)s

R 10 rW ðrÞdrþ

k1ð1� t�2Þ, T�2P

y2t�1 ) s2

R 10 W ðrÞ2 drþ k2

1ð1� t�Þ þ 2k1sR 1t� W ðrÞdr, T�1

Pyt�1et ) s2=2½W ð1Þ2 � 1�þ

k1sfW ð1Þ �W ðt�Þg.

Using the limiting behaviour of these moments in (A.1)–(A.3) and using T�1S2¼asy

T�1ðd2þ e0eÞ yield the

results in Theorem 1 by noting that: DT ðy� yÞ ¼ V�1A A, ðT � 3Þ�1S2) k2

1 þ s2, S2� S

2) A0V�1A A, and

~S2� S

2) A0V�1A A� fk1 þ sW ð1Þg2. &

ARTICLE IN PRESSA. Sen / Statistics & Probability Letters 78 (2008) 622–628 627

Proof of Theorem 2. The data generating process described by Model (B) implies that

DT ðy� yÞ ¼ ½D�1T ðX0X ÞD�1T �

�1½D�1T X 0m2DUðt�Þ þD�1T X 0e�, ðA:4Þ

S2� S

2¼ e0X ðX 0X Þ�1X 0eþ m2DUðt�Þ0X ðX 0X Þ�1X 0DUðt�Þm2 þ 2m2DUðt�Þ0X ðX 0X Þ�1X 0e, ðA:5Þ

~S2� S

2¼ e0X ðX 0X Þ�1X 0eþ m2DUðt�Þ0X ðX 0X Þ�1X 0DUðt�Þm2

þ 2m2DUðt�Þ0X ðX 0X Þ�1X 0e� T�1m22ðT � ½t�T �Þ2 � T�1

Xet

� �2

� 2T�1m2ðT � ½t�T �Þ

Xet

� �. ðA:6Þ

The data generating process given by Model (B) and m2 ¼ k2T�1=2 imply that: T�1=2m2ðT � ½t�T �Þ

! k2ð1� t�Þ, T�3=2m2P

t! 12k2ð1� t�2Þ, T�3=2

Pyt�1 ) s

R 10 W ðrÞdrþ 1

2k2ð1� 2t� þ t�2Þ, T�5=2

Ptyt�1

) sR 10 rW ðrÞdrþ 1

6k2ð2� 3t� þ t�3Þ, T�1m2

PT½t�T �þ1yt�1 ) k2s

R 1t� W ðrÞdrþ 1

2k22ð1� 2t� þ t�2Þ,

T�1X

yt�1et )s2

2½W ð1Þ2 � 1� þ k2s W ð1Þ �

Z 1

0

W ðrÞdr

� �� t�fW ð1Þ �W ðt�Þg

� �

T�2X

y2t�1 ) s2

Z 1

0

W ðrÞ2 drþ1

3k22ð1� t�Þ3 þ 2k2s

Z 1

t�rW ðrÞdr� t�

Z 1

t�W ðrÞdr

� �

Using the limiting behaviour of these moments in (A.4)–(A.6) and using T�1S2¼asy

T�1e0e yield the results of

Theorem 2 by noting that: DT ðy� yÞ ¼ V�1B B, ðT � 3Þ�1S2) s2, S

2� S

2) B0V�1B B, ~S

2� S

2) B0V�1B B�

fk2ð1� t�Þ þ sW ð1Þg2. &

Proof of Theorem 3. The data generating process described by Model (C) implies that

DT ðy� yÞ ¼ ½D�1T ðX0X ÞD�1T �

�1½D�1T X 0dDðt�Þ þD�1T X 0m2DUðt�Þ þD�1T X 0e� ðA:7Þ

S2� S

2¼ e0X ðX 0X Þ�1X 0eþ dDðt�Þ0X ðX 0X Þ�1X 0Dðt�Þd

þ 2dDðt�Þ0X ðX 0X Þ�1X 0DUðt�Þm2 þ 2dDðt�Þ0X ðX 0X Þ�1X 0e

þ m2DUðt�Þ0X ðX 0X Þ�1X 0DUðt�Þm2 þ 2m2DUðt�Þ0X ðX 0X Þ�1X 0e ðA:8Þ

~S2� S

2¼ e0X ðX 0X Þ�1X 0eþ dDðt�Þ0X ðX 0X Þ�1X 0Dðt�Þd

þ m2DUðt�Þ0X ðX 0X Þ�1X 0DUðt�Þm2 þ 2m2DUðt�Þ0X ðX 0X Þ�1X 0e

þ 2dDðt�Þ0X ðX 0X Þ�1X 0DUðt�Þm2 þ 2dDðt�Þ0X ðX 0X Þ�1X 0e

� T�1d2� T�1m22ðT � ½t

�T �Þ2 � 2T�1dm2ðT � ½t�T �Þ

� 2T�1dX

et

� �� 2T�1m2ðT � ½t

�T �ÞX

et

� �� T�1

Xet

� �2ðA:9Þ

The data generating process given by Model (C), d ¼ k1T1=2, and m2 ¼ k2T

�1=2 imply that T�1=2d ! k1,

T�3=2dð½t�T �þ1Þ!k1t�, T�1dy½t�T �)k1sW ðt�Þ, T�1=2m2ðT�½t�T �Þ!k2ð1�t�Þ, T�3=2m2

Pt! 1

2k2 ð1� t�2Þ,

T�1m2PT½t�T �þ1yt�1)k2s

R 1t� W ðrÞdrþk1k2ð1�t�Þþ 1

2k22ð1�2t

�þt�2Þ, T�3=2P

yt�1) sR 10 W ðrÞdrþk1ð1�t�Þ

þ12k2ð1� 2t� þ t�2Þ, T�5=2

Ptyt�1 ) s

R 10 rW ðrÞdrþ 1

2k1ð1� t�2Þþ 1

6k2ð2� 3t� þ t�3Þ,

ARTICLE IN PRESSA. Sen / Statistics & Probability Letters 78 (2008) 622–628628

T�2X

y2t�1 ) s2

Z 1

0

W ðrÞ2 drþ k21ð1� t�Þ þ

1

3k22ð1� t�Þ3 þ k1k2ð1þ t�2Þ

þ 2k1sZ 1

t�W ðrÞdrþ 2k2s

Z 1

t�rW ðrÞdr� t�

Z 1

t�W ðrÞdr

� �

T�1X

yt�1et )s2

2½W ð1Þ2 � 1� þ k1sfW ð1Þ �W ðt�Þg

þ k2s W ð1Þ �

Z 1

0

W ðrÞdr

� �� k2t�s W ð1Þ �W ðt�Þ

� .

Using the limiting behaviour of these moments in (A.7)–(A.9) and using T�1S2

C ¼asy

T�1ðd2þ e0eÞ yield the

results of Theorem 3 by noting that: DT ðy� yÞ ¼ V�1C C, ðT � 3Þ�1S2) ðk2

1 þ s2Þ, S2� S

2) C0V�1C C,

~S2� S

2) C0V�1C C � fk1 þ k2ð1� t�Þ þ sW ð1Þg2. &

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