148 C
.-C. Hsu Economics Letters 70 2001 147 –155
approaching one and hence may incorrectly suggest that a change has occurred. Moreover, the least-squares estimator may also identify a spurious change point in the middle of the sample.
Interestingly, these phenomena are irrelevant to the dependence structure of regressors. In contrast with the spurious regression results of Tsay and Chung 2000, our finding indicates that a spurious
change may still occur even when there is no spurious regression. This paper proceeds as follows. Least-squares consistency is established in Section 2. The spurious-change results of test and
estimation are discussed in Section 3. Section 4 gives some concluding remarks. All proofs are given in Appendix A.
2. Consistency
We consider the data generating process DGP of y which has a one-time change at unknown
t
point k :
9 9
y 5 x b 1 x l1 1
´ , t 5 1, . . . ,T,
1
t t
1 t
ht.k j t
where x 5 [1,x ]9, b 5 [a ,b ]9, i 5 1,2, l 5 b 2 b 5 [l ,l ]9 denotes the magnitude of parameter
t t
i i
i 2
1 1
2
changes, and 1 is the indicator function. Also let k 5 [Tt ], the integer part of Tt , so that t
represents the relative position of k in the sample. Given 1, we impose the following conditions. ]
] ]
[A1]. t [ [t, t], where t ,t and [t,t] is a proper subset of [0,1].
] ]
]
[A2]. Both x and ´ are stationary and invertible ARFIMA autoregressive, fractionally integrated,
t t
moving average processes:
d d
x ´
F L1 2 L x 5 Q Lv , F L1 2 L ´ 5 Q Lu ,
x t
x t
´ t
´ t
where FL and QL are polynomials in the lag operator L such that their roots lie outside the unit circle, and 2 0.5 , d ,d , 0.5.
x ´
[A3]. The processes
hv j and hu j are two mutually independent sequences of i.i.d. random variables
t t
2 2
d
1
with zero mean, finite variances s and s ,
Euv u , ` with d maxh4, 2 8d 1 1 2d j, and
v u
t 1
x x
d
2
Euu u , ` with d maxh4, 2 8d 1 1 2d j.
t 2
´ ´
Conditions [A2] and [A3] ensure that x and ´ obey the functional central limit theorem FCLT:
t t
[Tt ] [Tt ]
1 1
]]] ]]]
O
x ⇒
k B t,
O
´ , ⇒
k B t, 2
0.51d t
x d
0.51d t
´ d
x x
´ ´
T T
t 51 t 51
where ⇒
denotes weak convergence, k is a positive constant, and B is the fractional Brownian
d P
D
motion. In what follows we also write →
as convergence in probability and →
as convergence in distribution.
ˆ For each hypothetical change point k, the least-squares change-point estimator solves k 5
argmin RSSk, where
1k T
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
