C .-C. Hsu Economics Letters 70 2001 147 –155
149
k T
2 2
ˆ ˆ
9 9
RSSk 5
O
y 2 x b k 1
O
y 2 x b k ,
t t
1 t
t 2
t 51 t 5k 11
ˆ ˆ
and b k and b k are the pre- and post-change least-squares estimators of b and b . The estimated
1 2
1 2
ˆ break fraction, t, is defined as
]
ˆ t 5 inf t :t 5 argmin
RSS[Tt ] , 3
h j
t [[t,t ] ]
ˆ ˆ
and we may write t 5 k T. It is easy to verify that
T 2
˜
9
RSS[Tt ] 5
O
´ 1 x l1 2 H [Tt ],
t t
ht.[Tt ]j T
t 51
where
21 21
˜ ˜
˜ ˜
˜
9 9
H [Tt ] 5 E 0,tM 0,tE 0,t 1 E t,1M
t,1E t,1, 4
T T
T T
T T
T
with
[Tt ] [Tt ]
2 2
˜
9 9
E t ,t 5
O
x ´ 1 x l1 ,
M t ,t 5
O
x x .
T 1
2 t
s
t t
ht.[Tt ]j
d
T 1
2 t
t t 5[Tt ]11
t 5[Tt ]11
1 1
Hence, the least-squares estimator of 3 is the same as that obtained from maximizing 4. Bai 1997 ˆ
establishes the consistency result for t in regression models with weakly dependent data; Kuan and Hsu 1998 also prove consistency for the location model with ARFIMA data. In this section, we
generalize these consistency results to regression models with ARFIMA regressors and disturbances.
P
ˆ
Theorem 2.1. Given DGP 1, suppose that Conditions [A1]–[A3] hold. Then t
→ t as T
→ `,
ˆ where t is given by 3.
We report some Monte Carlo results in Fig. 1. The data y are generated according to 1 with
t
b 5 [1,0.5]9, b 5 [2,1]9, and t 5 0.5. Four combinations of d and d are considered: d ,d 5
1 2
x ´
x ´
0.4,0.4, 20.4,0.4, 0.4, 2 0.4, 20.4, 2 0.4. In each experiment, T 5 100, and the number of ˆ
replications is 50000. Fig. 1 shows that the empirical distributions of k are more concentrated around the true change point, although their precisions vary with d and d . These simulations confirm the
x ´
result of Theorem 2.1. However, it seems that the value of d is more crucial in determining the
´
ˆ performance of k. In view of Fig. 1 a and c or b and d, we can see that for d 5 0.4
x
ˆ d 5 2 0.4, k performs better when d is negative. On the other hand, comparing Fig. 1 a and b,
x ´
ˆ we observe that given d 5 0.4, k performs similarly for d 5 0.4 and d 5 2 0.4; in fact, positive d
´ x
x x
yields a slightly better result. The same conclusion holds for Fig. 1 c and d.
3. Spurious change
We have shown that the least-squares estimator is capable of locating the change point when it exists. In this section, however, we demonstrate that it is difficult to know if there is indeed a
150 C
.-C. Hsu Economics Letters 70 2001 147 –155
ˆ Fig. 1. Empirical distributions of k for regression models when k 5 50. a d 5 0.4, d 5 0.4 b d 5 2 0.4, d 5 0.4 c d 5 0.4,
x ´
x ´
x
d 5 2 0.4 d d 5 2 0.4, d 5 2 0.4.
´ x
´
structural change in Id data. In practice, researchers usually first test the existence of a change and then decide if they should proceed to estimate the change point. Unfortunately, as noted by Kuan and
Hsu 1998, many well-known tests for structural change have significant size distortions when data have long memory. In contrast with Tang and MacNeill 1993, these size distortions do not result
from ignoring nuisance parameters of weak dependence. In fact, the null of parameter constancy will be rejected with probability approaching one when data have strong positive correlations. In
regressions with Id variables, consider the Wald statistic for the null hypothesis b 5 b against the
1 2
alternative 1. For each possible break k, the Wald statistic is
21
ˆ ˆ
ˆ ˆ
ˆ ˆ
W k 5 b k 2 b k9V1,k 1 Vk,T b k 2 b k,
T 1
2 1
2 21
ˆ where Vt ,t 5 M
t ,t RSS T. Under the null hypothesis, l 5 0, and hence
1 2
T 1
2 [Tt ]
2
˜ ˜
E t ,t 5 E t ,t 5
O
x ´ , H [Tt ] 5 H [Tt ]
T 1
2 T
1 2
t t
T T
t 5[Tt ]11
1
21 21
9 9
5 E 0,tM 0,tE 0,t 1 E t,1M
t,1E t,1.
T T
T T
T T
It follows that
C .-C. Hsu Economics Letters 70 2001 147 –155
151
H [Tt ] 2 H
T T
]]]]]]] W [Tt ] 5
,
T T
2
O
´ 1 H [Tt] T
S D
t T
t 51 21
9
where H 5 E 0,1M 0,1E 0,1. The next result shows that W [Tt ] diverges when d . 0,
T T
T T
T ´
regardless of the value of d .
x
Theorem 3.1. Given DGP 1 with l 5 0, suppose that Conditions [A2] and [A3] hold with
] 0 , d , 0.5. Then for each t [ [t, t],
´
]
2 2
k [B t 2 tB 1]
´ d
d D
´ ´
22d
´
]]]]]]] T
W [Tt ] →
5
T
g 0t1 2 t
´
as T →
`, where g 0 is the variance of ´ .
´ t
Theorem 3.1 shows that for a positive d , W [Tt ] diverges in probability. By the continuous
´ T
mapping theorem, the supremum Wald test of Andrews 1993 and mean Wald and exponential Wald tests of Andrews and Ploberger 1994 also diverge to infinity. Thus, when there is no change, the
Wald-type tests would reject the null of parameter constancy with probability approaching one as long as the disturbances have long memory i.e., 0 , d , 0.5, and hence may incorrectly suggest that a
´
change has taken place. As the result of Theorem 3.1 does not depend on d , this conclusion holds
x
even when the regressor is antipersistent i.e., 2 0.5 , d , 0. The Lagrange multiplier and
x
ˆ likelihood ratio tests also suffer from the same problem. Moreover, when V is replaced with the
Newey-West estimator to account for weak dependence, we can show that W still diverges as d . 0.
T ´
Our simulations also confirm the asymptotic analysis of Theorem 3.1. The DGP is generated according to 1 with b 5 b 5 [1,2]9 for different combinations of d and d : d 5 2 0.45, . . . ,0.45
1 2
x ´
x
and d 5 0.1, . . . ,0.45 with the increment 0.05. Thus, there are 162 experiments; for each experiment,
´
T 5 200, the number of replications is 5,000, and the nominal size is 5. We compute the supremum ˆ
Wald test of Andrews 1993. In computing the Wald statistic, we estimate V by the Newey-West estimator. The bandwidth is determined by the data-dependent formula of Andrews 1991 with
AR1 specification. The rejection frequencies of the supremum Wald test are summarized in Fig. 2. It can be seen that the type I errors increase from approximately 25 for d 5 0.1 to about 95 for
´
d 5 0.45, regardless of the value of d . Therefore, as long as d is positive, the existing tests suffer
´ x
´
from serious size distortions. The previous results suggest that, whether or not there is a change, it is likely that one would be led
to estimate the change point after conducting standard structural-change tests. From the proof of Theorem 3.1, we know that
22d
´
]
ˆ t
5 argmax T
H [Tt ]
t [[t,t ] T
] 2
2 2
2
k B t k [B 1 2 B t]
´ d
´ d
d D
´ ´
´
]
]]] ]]]]]]
→ argmax
1
t [[t,t ]
t 1 2 t
] ]
; argmax G t,
t [[t,t ] d
´
]
152 C
.-C. Hsu Economics Letters 70 2001 147 –155
Fig. 2. Empirical sizes of sup-Wald-type tests for regression models.
] for any t [ [t, t]. By invoking the functional law of iterated logarithm Taqqu, 1977, we have
]
2 2
B t [B 1 2 B t]
d d
d
´ ´
´
]] ]]]]]]
lim sup 5lim sup
5 0 a.s.
t 1 2 t
t →
t →
1 2
2
Thus, the asymptotic behavior of G t is well-defined on [0,1] with G 0 5 G 1 5 k B 1 . It
d d
d ´
d
´ ´
´ ´
D
ˆ ˆ
follows that t has a limiting distribution with support equal to [0,1], and that t →
argmax G t.
t [[0,1] d
´
However, the maximum of G t is not attained at 0 or 1 with probability one, since
d
´
2 2
k [tB 1 2 B t]
d d
d
´ ´
´
]]]]]]] G 0 2 G t 5 G 1 2 G t 5 2
, 0,
d d
d d
´ ´
´ ´
t1 2 t for any t [ 0,1. This proves the following theorem.
Theorem 3.2. Given DGP 1 with l 5 0, suppose that Conditions [A2] and [A3] hold with
D
] ˆ
0 , d , 0.5. If t 5 0 and t 5 1, then t →
argmax G t and G 0 5 G 1 , G t for any
´ t [[0,1]
d d
d d
´ ´
´ ´
] t [ 0,1 with probability one.
Ideally, when there is no change, we expect that the probability mass of the change-point estimator
P
ˆ concentrates around the two end points of the sample; i.e., t
→ h0, 1j. Theorem 3.2 shows that,when
ˆ ˆ
d . 0, t is unlikely to be close to 0 and 1; it is more likely that t incorrectly identifies a change point
´
in the middle of the sample even when there is none. This is the problem of spurious change, in the sense of Nunes et al. 1995. Nunes et al. 1995 and Bai 1998 showed that the I1 regressors and
errors are responsible for the occurrence of spurious break and spurious regression. Tsay and Chung 2000 also pointed out that the spurious regression may occur when d 1 d . 0.5. From Theorem
x ´
3.2, we can see that the spurious-change problem may still exist even if the regressors and disturbances are stationary and d 1 d , 0.5. That is, a spurious change may arise when there is no
x ´
spurious regression. We examine some simulations which are conducted as that in Section 2 but b 5 b 5 [1,0.5]9. Four
1 2
cases are considered: d ,d 5 0.35,0.35, 20.35,0.35, 0.35, 2 0.35, 20.35, 2 0.35. Fig. 3
x ´
C .-C. Hsu Economics Letters 70 2001 147 –155
153
ˆ Fig. 3. Empirical distributions of k for regression models when there is no change. a d 5 0.35, d 5 0.35 b d 5 2 0.35, d 5 0.35 c
x ´
x ´
d 5 0.35, d 5 2 0.35 d d 5 2 0.35, d 5 2 0.35.
x ´
x ´
ˆ shows that, for cases a and b, k has the lowest rejection frequencies at the two end points 1 and
T . Hence, the least-square estimator will most likely find a change point in the middle of the sample rather than two end points; this confirms the result of Theorem 3.2. Note that when d 1 d , 0.5 Fig.
x ´
3 b, the spurious-change problem still occurs. Moreover, we also consider the cases that d , 0. In
´
ˆ contrast with Fig. 3 a and b, c and d show that the empirical distributions of k now have more
probability mass at the two end points. Although we do not give a formal proof, it seems that the spurious-change problem occurs only for a positive d . As the results of Section 2, the fractionally
´
differencing parameter of disturbances, d , is also crucial for a spurious change.
´
4. Conclusions
