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

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 (r_t)² and ∣r_t∣ transformations as well as (u_t)² where u_t 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 l}J                                   ^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(D_k ∣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.

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