Detecting Multiple Breaks in Financial Market Volatility Dynamics



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
Ukis 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 lj= 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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