Table 6: Testing for a single change-point in the volatility of daily Stock Market Indices (SMI)
over the period 1989-2001
Change-point Statistics
SMI Returns
Change-point k * |
Kokoszka & Leipus Test |
Inclan & Tiao Tests | |||||
max Uk * |
max Uk* ^ HAC |
max Uk* GARMA |
IT |
Bt ( C ) | |||
FTSE |
|rt| |
05/06/97 |
1.917 |
5.862* |
6.665* |
4.414* |
0.599 |
( rt )2 |
04/08/97 |
2.238 |
5.266* |
3.511* |
9.195* |
1.249 | |
HSI |
|rt| |
14/08/97 |
3.460 |
4.619* |
5.828* |
4.954* |
0.321 |
( rt )2 |
18/08/97 |
7.104 |
2.181* |
1.291 |
8.583* |
0.556 | |
NIKKEI |
|rt| |
31/07/97 |
1.521 |
3.091* |
3.806* |
2.905* |
0.449 |
( rt )2 |
21/10/97 |
1.836 |
1.972* |
1.305 |
4.427* |
0.684 | |
S&P500 |
|rt| |
04/02/97 |
2.395 |
6.882* |
7.181* |
5.837* |
0.356 |
( rt )2 |
26/03/97 |
2.718 |
4.888* |
1.665* |
11.103* |
0.678 |
Notes: (1) The Stock Market Index (SMI) series refer to the Financial Times Stock Exchange index 100 (FTSE100), the Hang-Seng
Index (HSI), the Nikkei 500 (NIKKEI), the Standards and Poors 500 index (S&P500). The daily sample over the period 4/1/1989 to
19/10/2001 yields T = 3338 observations. The series rt := logpt - log p t-1 represents the returns on each index. The change-point tests
are applied to the (rt)2 and ∣rt∣ transformations as well as (ut)2 where ut is the residual from the GARCH. (2) The Kokoszka and Leipus
(1998, 2000) reported max Uk∙ is the maximum of the statistic, Uk =( ɪ ∑k r2 - ɪ ɪ ∑T, r2 ) which is standardized by
' ' ' ‘ ∖, TT l=1 j ' TT l =k+1 lJ j
nonparametric estimators, Heteroskedastic Consistent (σHAC) and ARMA estimators (σARMA) of squared and absolute returns. The
normalized statistic max Uk∙ /σ converges to the sup of a Brownian Bridge. (3) The Inclan and Tiao (1994) statistic
Dk = (∑j=1 l∑j= 1 r2) - "r specified for iid processes normalized as IT = 1∕T/2 max(Dk ∣also converges to the sup of a Brownian
Bridge and is extended in Kim et al. (2000) for GARCH processes to be Bt(C) = CJT max|.Dk∣ where C2and κ are constants that are
estimated by substituting the quasi-MLEs of the GARCH(1,1) ω, Й, β and T-1 ∑j=1 r4 to ω, α, β and E(rj). (4) k* refers to the
location of the break and the * symbol attached to statistics denotes that the null hypothesis of no structural change is rejected using the
asymptotic critical value of 1.36.
30
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