Semiparametric tests for seasonal unit roots based on a semiparametric feasible GLSE

Statistics & Probability Letters 50 (2000) 207 – 218

Semiparametric tests for seasonal unit roots based on a semiparametric feasible GLSE  Dong Wan Shin ∗ , Man-Suk Oh Department of Statistics, Ewha University, Seoul, 120-750 South Korea Received November 1999; received in revised form January 2000

Abstract Semiparametric extensions of the seasonal unit root tests for the model of Dickey et al. (1984, J. Amer. Statist. Assoc. 79, 355) are proposed. Development of semiparametric extensions based on the ordinary least-squares estimator (OLSE) is impossible for the regression of Dickey et al. (1984) since the limiting null distribution of the OLSE-based test statistic is entangled with nuisance parameters under the usual normalization. This is in contrast with the successful development of the OLSE-based semiparametric unit root tests of Phillips (Econometrica 55 (1987) 277). To overcome the diculty, we propose tests based on a feasible generalized least-squares estimator (GLSE), instead of the OLSE, and the spectral decomposition of the generalized sum of products of the regressor variables. The key advantage of the proposed method is that one can construct the feasible GLSE and hence tests of the seasonal unit root without specifying a parametric c 2000 Elsevier Science B.V. All rights reserved model for the error process. Keywords: Seasonal unit roots; Semiparametric tests; Fourier transform; GLSE

1. Introduction Since the work of Dickey et al. (1984), DHF from here on, tests for seasonal unit root have attracted much attention from many researchers. Among the tests developed so far, two important tests are those of DHF and Hylleberg et al. (1990), HEGY from here on. DHF considered models and tests for seasonal unit roots for all the seasonal frequencies. On the other hand, HEGY considered quarterly models and developed methods for testing seasonal unit roots for the seasonal frequencies 0; =2; . Various extensions of the DHF and the HEGY tests have been proposed and properties of the test statistics have been studied, for example, by Beaulieu and Miron (1993), Ghysels et al. (1994), Ghysels et al. (1996), and Shin and So (2001). Also, Breitung and Franses (1998) constructed semiparametric “Phillips–Perron-type” extension of the HEGY tests which are valid for a wide class of autocorrelated errors. 

This work was supported by a grant from BK-21 project. author.

∗ Corresponding

c 2000 Elsevier Science B.V. All rights reserved 0167-7152/00/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 0 ) 0 0 1 0 6 - 1

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D.W. Shin, M.-S. Oh / Statistics & Probability Letters 50 (2000) 207 – 218

Error terms of unit root regressions and seasonal unit root regressions are usually serially correlated. There are basically two strategies for adjusting the serial correlations. One strategy is augmenting regressions with lags of dierenced observations in the context of “augmented Dickey–Fuller” regressions of Dickey and Fuller (1979) and DHF. This approach is parametric in the sense that a parametric model structure of the error process should be speciÿed. Another strategy is the “Phillips–Peron”-type semiparametric modiÿcations proposed by Phillips (1987) and Phillips and Perron (1988). The semiparametric modiÿcations are advantageous over the parametric approach in that they do not require model speciÿcation of the error structure; hence, it can be used for a wide class of serially correlated error processes. For the unit root tests of HEGY, Breitung and Franses (1998) developed a semiparametric extension and showed that their semiparametric tests are robust against particular forms of structural breaks (Amsler and Lee, 1995). For the model of DHF, however, there has been no semiparametric extension despite many advantages of the semiparametric approach. While semiparametric tests for the models of Phillips (1987), Phillips and Perron (1988), and Breitung and Franses (1998) are based on the OLSE, the OLSE in the DHF model deÿes semiparametric modiÿcations. The reason is that, in addition to the one-sided long-run covariance parameter, the limiting null distribution of the OLSE under the usual normalization of subtracting the true value and then multiplying the sample size depends on the covariance parameters of the error process. Thus, instead of the OLSE, we look into asymptotics of the GLSE and develop semiparametric modiÿcations of the DHF tests. Our semiparametric tests are based on a feasible GLSE for which no parametric speciÿcation of the error process is required. The resulting test statistics have the same limiting null distributions as those established by DHF for independent identically distributed (i.i.d.) error. In establishing asymptotics of the GLSE, we utilize a spectral representation of the generalized sum of products of the regressor variables. The representation is based on the Fourier transforms of the observations and their lags up to the order of the period of seasonality. In the remaining of the paper, asymptotics of the OLSE and the GLSE are investigated in Section 2, semiparametric unit root tests based on the GLSE are developed in Section 3, and proofs of the theoretical results are provided in the appendix. 2. Asymptotics for the OLSE and the GLSE for DHF model Consider a seasonal time series model yt = yt−d + ut ;

t = 1; : : : ; n;

where {yt ; t = 1; : : : ; n} is a set of observations, d is a positive integer, and ut is a stationary process having zero mean and ÿnite variance. We reparameterize the above model into zt = yt−d + ut ;

zt = yt − yt−d ;

(1)

where  =  − 1. In DHF model (1),  is the parameter of interest. Especially, testing for  = 0 (the unit root) is of our primary concern. In most cases, the error process ut are serially correlated and their autocovariance structure is highly complicated, making inference on  dicult. Thus, it would be valuable to ÿnd a reasonable estimator of  which does not require speciÿcation of the autocovariance of the error process. Unlike in the models of Phillips (1987), Phillips and Perron (1988), and Breitung and Franses (1998) in which semiparametric modiÿcations are successfully based on the OLSE, we will show in this section that for the DHF model asymptotics of the OLSE of  are substantially dierent from those of the GLSE except for the nonseasonal case of d = 1. This implies that the OLSE for seasonal model (1) suer from substantial eciency loss and the GLSE should be preferred to the OLSE. We will also show that an asymptotic analysis of the GLSE enables us to develop a semiparametric feasible GLSE of  without a parametric speciÿcation for ut .

D.W. Shin, M.-S. Oh / Statistics & Probability Letters 50 (2000) 207 – 218

209

The OLSE and the GLSE of  in DHF model (1) are ˆO = (X 0 X )−1 X 0 Z;

ˆG = (X 0

−1

X )−1 X 0

−1

Z;

respectively, where X = (y1 ; : : : ; yn−d )0 ;

Z = (z1+d ; : : : ; zn )0 ;

U = (u1+d ; : : : ; un )0 ;

= var(U ):

Let h = E(u0 uh ) denote the autocovariance function (ACF) of ut and let f() = (2)

∞ X

−1

h exp(−h);

2 = −1;

h=−∞

denote the spectral density of the error process ut . Let j = 2j=d, j = 0; 1; : : : ; d − 1, denote the frequencies corresponding to the roots of (1 − Ld ) = 0. The main object of this section is to show that, if  = 0, i.e., U = Z, then X0

−1

X=

r X

0 {2f(j )}−1 (Xj0 Xj cos2 j + Xj+r Xj+r sin2 j ) + op (n2 )

(2)

j=0

Z0

−1

X = U0

−1

X=

r X

{2f(j )}−1 (Z 0 Xj cos j + Z 0 Xj+r sin j ) − n + op (n);

(3)

j=0

and characterize asymptotics of the terms in (2) and (3), where r = [d=2], the integer part of d=2, Xj and Xj+r are the Fourier coecients of {Y0 ; : : : ; Yd−1 } deÿned below, and Yj =(y1+j ; : : : ; yn−d+j )0 , j=0; 1; : : : ; d. The term  is deÿned in (12) below and depends on autocovariance parameters of ut only through f(j ); j = 0; : : : ; r, which can be estimated semiparametrically without imposing a parametric model on ut . The limiting null distributions of n−2 X 0 −1 X and n−1 U 0 −1 X are shown to be the same as those for i.i.d. case of = I . We therefore can construct a feasible semiparametric GLSE of   −1 r X 0 ˆ j )}−1 (Xj0 Xj cos2 j + Xj+r  {2f( Xj+r sin2 j ) ˆ+ G = j=0

 ×

r X

 ˆ j )}−1 (Z 0 Xj cos j + Z 0 Xj+r sin j ) − nˆ {2f(

j=0

ˆ j ); j = 0; : : : ; r and . ˆ It also enables us to construct seasonal unit by using semiparametrically estimated f( root tests Detailed expressions for the Fourier coecients are −1

X0 = d

d X

Yi−1 cos(i0 );

i=1

Xj = 2d−1

d X

Yi−1 cos(ij );

j = 1; : : : ; rc ;

i=1

Xj+r = 2d−1

d X i=1

Yi−1 sin(ij );

j = 1; : : : ; rc ;

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D.W. Shin, M.-S. Oh / Statistics & Probability Letters 50 (2000) 207 – 218

and, if d is even, additionally, Xd=2 = d−1

d X

Yi−1 cos(id=2 )

i=1

where rc = [(d − 1)=2]. Denote the tth element of Xj by xjt+d so that Xj = (xj1+d ; : : : ; xjn )0 , j = 0; : : : ; d − 1. We now state conditions required for our results. P∞ C1. ut is a strong mixing sequence satisfying supt E|ut |+ ¡ ∞ and m=1 cm1−2= ¡ ∞ for some ¿2 and ¿0, where cm ’s are the mixing coecients.P n C2. ut is a stationary sequence satisfying h = 0 h| h | = o(n). C3. f()¿0 for all  ∈ [0, ]. Condition C1 is for the invariance principle of the partial sums of ut adopted by Phillips (1987) for the study of the semiparametric tests for unit roots. This condition can be replaced by other conditions for the invariance principle such as those given in Phillips and Solo (1992). Condition C2 is for existence of the matrix given for somePtechnical aspects of the GLSE. Noting that abP∞in Theorem 1 belowPand n ∞ solute summability h = 0 | h | ¡ ∞ implies h = 0 h| h |6n h = 0 | h | = O(n), we can say that C2 is slightly more restrictive than the usual absolute summability. The class of errors speciÿed by C2 are much more general than the class of stationary invertible ARMA errors because C2 states algebraic decline of ACF while ACF of ARMA processes decay exponentially. In order to analyze asymptotics of the GLSE, we need another condition C3 on the spectral density. Condition C3 states that [f()]−1 is well-deÿned and ut is invertible. Condition C3 also guarantees invertibility of the covariance matrix . Note that stationary and invertible autoregressive moving average (ARMA) models satisfy conditions C1–C3. Limiting null distribution of nˆO under C1 and C2 is ÿrst established in Theorem 1 below. Theorem 1. Assume the DHF model (1). Under conditions C1 and C2;   −1  d−1 Z 1 d−1 Z 1 X X L Wj2   Wj dWj + d
 ; nˆO → d  0

j=0

j=0

0

(4)

L

0 with covariance where → denotes convergence in distribution; P∞ W = (W0 ; : : : ; Wd−1 ) is a BrownianPmotion ∞ matrix with (i; j) element !|i−j| ; !k = h=−∞ dh+k ; k = 0; : : : ; d − 1; and
= h = 1 dh .

Implications of Theorem 1 in the context of eciency and unit root test are discussed at the end of this section and at the beginning of Section 3, respectively. We next investigate asymptotics of the GLSE. By deÿnition of the Fourier coecients, we have X = Y0 =

r X

(Xj cos j + Xj+r sin j ):

(5)

j=0

Simple algebra gives (1 − L)

d−1 X

!

d−i

= (1 − Ld )L;

L

(6)

i=0

2

(1 − 2L cos j + L )

( d−1 X i=0

) L

d−i

cos (ij )

= (1 − L cos j )(1 − Ld )L;

D.W. Shin, M.-S. Oh / Statistics & Probability Letters 50 (2000) 207 – 218 2

(1 − 2L cos j + L )

( d−1 X

211

) d−i

L

sin (ij )

= −sin j (1 − Ld )L2 ;

(7)

i=0

j = 1; : : : ; rc and, if d is even, additionally, ) ( d−1 X Ld−i (−1)i−1 = (1 − Ld )L: (1 + L) i=0

Therefore, xjt satisfy (1 − L)x0t = ut−1 =d;

(8)

(1 − 2L cos j + L2 )xjt = 2(ut−1 − cos j ut−2 )=d;

(9)

(1 − 2L cos j + L2 )xj+rt = −2 sin j ut−2 =d;

(10)

j = 1; : : : ; rc , and, if d is even, additionally, (1 + L)x(d=2)t = ut−1 =d;

(11)

where L is the lag operator such that Lzt = zt−1 . If d is even, elements of the transformed vectors X0 , {(Xj , Xj+r ); j = 1; 2; : : : ; rc }, Xd=2 , are seasonally integrated with dierent frequencies 0, {j ; j = 1; 2; : : : ; rc }, . If d is odd, the previous statement holds with Xd=2 deleted. Hence, Xj are nearly orthogonal in the sense that Xj0 Xi is of smaller order than Xi0 Xi if i and j correspond to dierent frequencies. (Chan and Wei, 1988, Theorem 3:4:1). Therefore, (5) is an orthogonal decomposition and we have the following lemma for decompositions of X 0 −1 X and U 0 −1 X , which are useful in investigating asymptotics of the GLSE. Lemma 1. Assume C1–C3; Pr the DHF model (1). Under conditions 0 0 Xj cos j sin j + Xj+r Xj+r sin2 j ) + op (n2 ), (i) X 0 −1 X = j = 0 {2f(j )}−1 (Xj0 Xj cos2 j + 2Xj+r P r (ii) U 0 −1 X = j = 0 {2f(j )}−1 (U 0 Xj cos j + U 0 Xj+r sin j ) − n + op (n); where ∞ X h h =d; =

(12)

h=1

 h = 1=2f(0) − d (−1)h−1 =2f() + 2

rc X

 (cos j

jh−1

−

;

jh−2 )=2f(j )

j=1 jh

= sin(h + 1)j =sin j ;

(13)

for h = −1; 0; 1; : : : ; j = 1; : : : ; rc ; and d = 1 if d is even and d = 0 if d is odd. In the following lemma, asymptotics for the elements given in Lemma 1 are established. Lemma 2. Assume DHF model (1). Let C1–C3 hold. Then; jointly Z 1 L W0∗2 ; (i) n−2 X00 X0 → 2f(0)d−2 0

n

−2

L Xj0 Xj → 4d−2 Aj L

2

sin j ;

0 Xj+r → 4d−2 Aj sin2 j ; n−2 Xj+r p

0 Xj → 0; n−2 Xj+r

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D.W. Shin, M.-S. Oh / Statistics & Probability Letters 50 (2000) 207 – 218

and; if d is even; additionally; L

0 Xd=2 → 2f()d−2 n−2 Xd=2

Z 0

( L

n−1 U 0 X0 → d−1

(ii)

1

Z

2f(0) (

L

n−1 U 0 Xj → 2d−1

Bj +

∗2 Wd=2 ;

1

0

∞ X

W0∗ dW0∗ +

)

h

;

h=1

∞ X

jh−1 h

− cos j (Bj cos j − Cj sin j +

h=1

n

−1

L

0

−1

U Xj+r → −2d

sin j

Bj cos j − Cj sin j +

jh−2 h

) jh−2 h )

;

h=1

!

∞ X

∞ X

;

h=1

and; if d is even; additionally; ( n

−1

0

L

−1

U Xd=2 → d

Z

2f()

1

0

∗ ∗ Wd=2 dWd=2

+

∞ X

) h−1

(−1)

h

:

h=1

p

where → denotes convergence in probability; and ! Z 1 Z 1 2 ∗2 ∗2 Wj + Wj+r ; Aj = {2f(j )=4 sin j } 0

Z Bj = {2f(j )=2 sin j }

1

0

Z Cj = {2f(j )=2 sin j }

1

0

(14)

0

∗ Wj∗ dWj+r

Wj∗

dWj∗

Z −

0

Z +

1

0

1

! ∗ Wj+r

dWj∗

;

(15)

! ∗ Wj+r

∗ dWj+r

;

(16)

∗ are independent standard Brownian motions. the index j runs through j = 1; 2; : : : ; rc and W0∗ ; : : : ; Wd−1

As a consequence of Lemma 1 and Lemma 2, we get the limiting null distribution of nˆG , which is the same as that of nˆO under i.i.d. ut established by DHF. Thus, one can use the table given by DHF for probabilities associated with nˆG . Theorem 2. Assume DHF model (1). Let C1–C3 hold. Then −1  d−1 Z 1 d−1 Z 1 X X L ∗2   Wj Wj∗ dWj∗ ; nˆG → d j=0

0

j=0

0

(17)

∗ )0 is a d-dimensional standard Brownian motion. where W ∗ = (W0∗ ; : : : ; Wd−1

The fact that the limiting null distribution of nˆO in Theorem 1 is dierent from that of nˆG in Theorem 2 implies that the OLSE, even if adjusted for the shift parameter
, is less ecient than the GLSE which is optimal. This is in contrast with the non-seasonal case of d = 1 in which nˆO , if adjusted for
, has the same limiting distribution as nˆG , implying asymptotic eciency of nˆO upon some simple adjustment for
.

D.W. Shin, M.-S. Oh / Statistics & Probability Letters 50 (2000) 207 – 218

213

3. Semiparametric tests for DHF model In this section, we develop semiparametric tests for the DHF model (1). Our interest is the unit root hypothesis H0 :  = 0. According to Theorem 1, one may consider a “Phillips–Perron-type” modiÿcation of 2 0 −1 ˆ ˆ the DHF test statistic, based on the OLSE, as given by nˆ+ O = nˆO − n (X X )
; where
is a consistent estimator of shift parameter
. However, the limiting null distribution (4) of nˆO , even if adjusted for
, is complicatedly related with the ACF when d ¿ 1 because the Brownian motions (W0 ; W1 ; : : : ; Wd−1 ) are correlated, having covariance matrix . Therefore, the modiÿcation fails to give a null distribution which is free from nuisance parameters. On the other hand, the limiting null distribution of the GLSE given in Theorem 2 does not involve nuisance parameters. According to Lemma 1 and Lemma 2, under H0 , we have (2) and (3). Therefore, the following statistic  −1 r X 0 ˆ j )}−1 (Xj0 Xj cos2 j + Xj+r  {2f( Xj+r sin2 j ) nˆ+ G =n j=0

 ×

r X

 ˆ j )}−1 (Z 0 Xj cos j + Z 0 Xj+r sin j ) − nˆ {2f(

(18)

j=0

is asymptotically equivalent to nˆG and has the limiting distribution given in (17) under H0 and C1–C3, where ˆ j ) and ˆ are consistent estimators of f(j ) and . In addition, the -statistic f(  −1=2 r X 0 ˆ j )}−1 (Xj0 Xj cos2 j + Xj+r  {2f( Xj+r sin2 j ) ˆ+ G = j=0

 ×

r X

 ˆ j )}−1 (Z 0 Xj cos j + Z 0 Xj+r sin j ) − nˆ {2f(

(19)

j=0

has the limiting null distribution given by −1=2  d−1 Z 1 d−1 Z 1 X X ∗2   Wj Wj∗ dWj∗ : j=0

0

j=0

0

These distributions are the same as those of DHF for i.i.d. ut . Thus, we can use the percentage points tabulated by DHF. Appendix Proof of Theorem 1. We can easily show that, as m → ∞, with ‘ = [mr]; L

m−1=2 (yd‘ ; yd‘+d−1 ; : : : ; yd‘−d+1 )0 → W (r): Now, the result follows from (A.1) and that, if n = md; (n−2 Pm−1 2 Pd−1 −1 Pm−1 −1 ‘ = 1 yd‘−d+j , d j=0 m ‘ = 1 yd‘−d+j zd‘+j ).

(A.1) Pn−1

t=d

2 yt−d , n−1

Pn−1

t=d

yt−d zt ); =(d−2

Pd−1

j=0

m−2

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D.W. Shin, M.-S. Oh / Statistics & Probability Letters 50 (2000) 207 – 218

Lemma A.1 (Shin and Oh, 1999). Let vt be a sequence of real numbers. Let V = (v1 ; : : : ; vn )0 . Assume P ∞ h = 0 | h | ¡ ∞. Then the following identities hold (i) V = 2f(0)V + H (0) (0)G − H (0)’(0)vn ; (ii) V = 2f()V + H () ()G − (−1)n H ()’()vn ; (iii) for any  6=  and  ∈ (0, 2); V = 2f()V + Im[ − (sin )−1 e− H () ()G − exp(n)H ()’()vn ]; where Im(z) is the imaginary part of a complex number z; G = (g1 ; : : : ; gn )0 ; gt = (vt − vt−1 ); in (i); gt = (−1)t (vt + vt−1 ); in (ii); gt = et (1 − 2vt cos  + vt−2 ); in (iii); H () = diag[exp(−); : : : ; exp(−n)];   R2 () : : : Rn−1 () −R1 () R1 ()    −R2 () −R1 () R1 () : : : Rn−2 ()       =  −R3 () −R2 () −R1 () : : : Rn−3 ()  :    ::: ::: ::: ::: :::    −Rn () −Rn−1 () −Rn−2 () : : : −R1 () Ri () =

∞ X

h exp(h);

i = 1; 2; : : : ;

h=i

vt = 0 for t60; and Ri () is the complex conjugate of Ri (). Lemma A.2. Let C1–C3 hold. Let vt be a seasonally integrated process satisfying (1−L)vt =ut ; (1+L)vt =ut ; or (1 − 2 cos L + L2 )vt = ut . Let G; H; ; and ’ be deÿned in Lemma A:1. Then |V | = Op (n);

|H () ()G| = op (n);

0

1=2

where |V | = (V V )

|H ()’()vn | = op (n);

is the Euclidean norm of the vector V = (v1 ; : : : ; vn )0 .

Proof. We ÿrst give a proof for the case (1 − L)vt = ut . Now gt = ut and H (0) = I . The order for |V | is a direct consequence of the fact that n−1=2 v[nr] converges in distribution to a Brownian motion. We note that, due to C2, ’(0)0 ’(0) =

n X

|Ri (0)|2 =

i=1

62

‘ ∞ X X

∞ X ∞ n X X

‘ h =

i=1 ‘=i h=i

min(h; n) ‘ h 62

‘=1h=1

∞ min(‘;h; ∞ X X X n) ‘=1h=1

n X

‘| ‘ | + n

‘=1

‘ h

i=1 ∞ X

! | ‘ |

‘ = n+1

∞ X

! | h |

= o(n):

(A.2)

h=1

∗ ∗ Let R∗‘ = −R‘+1 (0) for ‘¿0, RP ‘ = R−‘ (0) for ‘ ¡ 0. Then the (‘; h)th element of = (0) is R‘−h . Now, n ∗ the ith row of G is ( G)i = ‘ = 1 R‘−i u‘ . Observe that, by (A.2),

E(| G|2 ) =

n n X n X X i=1 ‘=1h=1

R∗‘−i R∗h−i ‘−h 64n( )

n X

R2‘ (0) = o(n2 );

‘=1

where (B) is the maximum eigenvalue of a symmetric matrix P∞B. We have used the fact that, if B is a positive deÿnite matrix, |a0 Bb|6(B)|a||b| and the fact supn ( )6 h=−∞ | h | ¡ ∞ due to the Gershgorin’s Theorem

D.W. Shin, M.-S. Oh / Statistics & Probability Letters 50 (2000) 207 – 218

215

(Horn and Johnson, 1985). Thus, | G| = op (n). Also, by (A.2), |’(0)| = o(n1=2 ) and hence |’(0)vn | = op (n). This completes the proof for the case (1 − L)vt = ut . Proofs for the other cases are similar and are omitted. Proof of (i) of Lemma 1. According to Lemmas A.1 and A.2 and noting (8) – (10), −1

Xj = {2f(j )}−1 (Xj −

−1

Qj );

j = 0; : : : ; d − 1;

with Qj such that |Qj | = op (n). From (5), we thus have X0

−1

X=

r X

{2f(j )}−1 (X 0 Xj cos j + X 0 Xj+r sin j ) + op (n2 )

(A.3)

j=0

because, due to C3, {( )}−1 6{inf  2f()}−1 ¡∞ (Brockwell and Davis, 1990, Proposition 4:5:3) and we have |X 0

−1

Qj |6{( )}−1 |X ||Qj | = op (n2 ):

We now observe that 0 X 0 Xj = Xj0 Xj cos j + Xj+r Xj sin j + op (n2 )

(A.4)

X 0 Xj+r = Xj0 Xj+r cos j + Xj+r Xj+r sin j + op (n2 )

(A.5)

and because products of Xi and Xj corresponding to dierent frequencies are all op (n2 ) by Theorem 3:4:1 of Chan and Wei (1988). Combining (A.3) – (A.5), we get the result. Proof of (ii) of Lemma 1. According to Lemma A.1, U0

−1

X=

r X

{2f(j )}−1 (U 0 Xj cos j + U 0 Xj+r sin j

j=0

−U 0

−1

aj cos j − U 0

−1

aj+r sin j );

where a0 = d−1 (0)U1 − ’(0)x0n ;

(A.6)

aj = Im[ − 2d−1 (sin j )−1 e−j H (j ) (j )H (−j )(U1 − cos j U2 ) −exp(nj )H (j ) (j )xjn ];

(A.7)

aj+r = Im[2d−1 e−j H (j ) (j )H (−j )U2 − exp(nj )H (j )’(j )xj+rn ]; and, if d is even, additionally, ad=2 = d−1 H () ()U1 − (−1)n H ()’()xd=2n ; j = 1; : : : ; r; U1 = (ud ; : : : ; un−1 )0 ; U2 = (ud−1 ; : : : ; un−2 )0 . Now, n−1

r X

{2f(j )}−1 (U 0

j=0

−1

aj cos j + U 0

−1

p

aj+r sin j ) → 

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D.W. Shin, M.-S. Oh / Statistics & Probability Letters 50 (2000) 207 – 218

and the result follows if we show ∞ X p

h ; n−1 U 0 −1 a0 → d−1

(A.8)

h=1

n−1 U 0

−1

∞ X

p

aj → 2d−1

(

jh−1

− cos j

jh−2 ) h ;

(A.9)

h=1

n−1 U 0

−1

p

aj+r → −2d−1

∞ X

sin j

jh−2 h ;

(A.10)

h=1

and, if d is even, additionally, n−1 U 0

−1

p

ad=2 → d−1

∞ X

(−1)h−1 h :

(A.11)

h=1

We now show (A.8). According to (A.6), U0

−1

a0 = d−1 U 0

−1

(0)U1 − U 0

−1

’(0)x0n :

Since ut is a zero mean stationary process with absolutely summable ACF, n−1 U 0

−1

p

(0)U1 → lim n−1 E[U 0

−1

n

  = lim n−1 tr (0) n

0

0 0

In−d−1

(0)U1 ]



∞ X

= R1 (0) =

h ;

where In−d−1 is the (n − d − 1) × (n − d − 1) identity matrix. We have E[U 0 (A.2), var[U 0

−1

’(0)] = ’(0)0

We thus have U 0 n−1 U 0

−1

−1

−1

(A.12)

h=1 −1

’(0)] = 0 and, according to

’(0)6’(0)0 ’(0){( )}−1 = o(n):

’(0) = op (n1=2 ). Also, x0n = Op (n1=2 ). Therefore,

’(0)x0n = op (1)

(A.13)

and we get (A.8) from (A.12) – (A.13). Similarly we get (A.11). Let  = j 6= 0, . We observe that !# " 0 0 p −1 0 −1 −1 H () ()H (−)U1 → lim n tr H () ()H (−) n U n In−d−1 0 = e R1 () = e

∞ X

h e−h :

(A.14)

h=1

and, similarly, n−1 U 0

−1

p

H () ()H (−)U2 → e2 R2 () = e2

∞ X h=2

By the same argument for (A.13), n−1 U 0

−1

H () ()’()xin = op (1);

i = j; j + r:

h e−h :

(A.15)

D.W. Shin, M.-S. Oh / Statistics & Probability Letters 50 (2000) 207 – 218

217

Therefore, from (A.7), (A.14) and (A.15), we get (A.10) as " !# ∞ ∞ X X p −1 0 −1 −1 −1 −  −h 2 −h aj → Im −2d (sin ) e e

h − cos e e

h e n U h=1

= 2d−1 (sin )−1

∞ X

h=2

{sin(h) − cos  sin(h − 1)} h = 2d−1

h=1

∞ X

(

− cos 

jh−1

jh−2 ) h :

h=1

Similarly, we get (A.11) that n−1 U 0

−1

∞ X

p

aj+r → −2d−1

sin 

jh−2 h :

h=1

Proof of (i) of Lemma 2. The ÿrst and the last results are obvious from (8) and (11). Lemma 1 and proof of Theorem 1 of Breitung and Franses (1998) P Pshow that, if x˜t = 2 cos x˜t−1 − x˜t−2 + x˜t for a stationary process x˜t , then the weak limits of n−2 x˜2t−1 , n−2 x˜t−1 x˜t are A and A cos , respectively, where A is the same as Aj in (14) with f replaced by the spectral density of x˜t and j replaced by . Noting that xjt satisfy (9) we get the third result of L

n−2 Xj0 Xj → 4d−2 {Aj − 2 cos j (Aj cos j ) + cos2 j Aj } = 4d−2 sin2 j Aj : ∗ } are Similarly, we get the remaining two results. By Theorem 2:2 of Chan and Wei (1988), {W0∗ ; : : : ; Wd−1 independent standard Brownian motions.

Proof of (ii) of Lemma 2. The ÿrst and the last results are obvious from (8) and (11). Let =j 6= 0; .PLemma 1 and proof of Breitung and Franses (1998) also P show that the weak limits of n−1 x˜t−1 x˜t P of Theorem 1P ∞ ∞ −1 x˜t−2 x˜t are B + h = 1 h−1 x˜h and B cos  − C sin  + h = 1 h−2 x˜h , respectively, where B, C, h and n are the same as Bj , Cj , jh in (15), (16) and (13) with the obvious modiÿcations and x˜h = E(x˜0 x˜h ). Now, in view of (9) and (10), we get the remaining two results. Proof of Theorem 2. According to Lemma 1(i) and Lemma 2(i),  ! Z 1 Z 1 Z 1 rc d−1 Z X X L ∗2  = d−2 W0∗2 + Wj∗2 + Wj+r n−2 X 0 −1 X → d−2  0

j=1

0

0

j=0

0

1

Wj∗2 :

R1 L By Lemma 2(ii) and (A.8), n−1 (U 0 X0 − U 0 −1 a0 ) → d−1 0 W0∗ dW0∗ . Similarly, n−1 (U 0 Xd=2 − U 0 R1 ∗ L ∗ → d−1 0 Wd=2 dWd=2 , if d is even. Also, by Lemma 2(ii), and (A.10) and (A.11), n−1 (U 0 Xj cos j + U 0 Xj+r sin j − U 0

−1

aj cos j − U 0

−1

(A.16) −1

ad=2 )

L

aj+r sin j ) → 2d−1 sin j Cj :

Hence, we get n−1 U 0

−1

L

X → d−1

d−1 Z X j=0

0

1

Wj∗ dWj∗ :

Now, the result follows from (A.16) and (A.17).

(A.17)

218

D.W. Shin, M.-S. Oh / Statistics & Probability Letters 50 (2000) 207 – 218

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