Detecting Multiple Breaks in Financial Market Volatility Dynamics



Table 1: Nominal Size and Power of the Kokoszka and Leipus (2000) test for a single
change-point in the volatility based on a GARCH process.

Samples, T

KokoszkaLeipus Statistic

Returns Transformations

500

Umaχlσ ηac
(rt)2 |rt|

1000

Umaxlσ HAC
( rt ) 2 | rt |

3000

Umaxlσ HAC

( rt )2

|rt|

H0 : Univariate GARCH, r0,t =

ξ u 0, tjh 0, t, h 0, t = ω 0

+ α0u2,t-1 + β0h0,t-1, with (ω0,α0,β0):

DGP1: (0.4, 0.1, 0.5)

0.059 0.072

0.061 0.078

0.061

0.067

DGP2: (0.1, 0.1, 0.8)

0.171 0.165

0.187 0.185

0.212

0.205

H1A : Break in the dynamics of volatility, β0 (increase of 0.1) at 0.5T.

DGP1: β0 = 0.5 to β1 = 0.6

0.273 0.280

0.492 0.473

0.945

0.926

DGP2: β0 = 0.8 to β1 = 0.9

0.978 0.978

0.999 0.999

1.000

1.000

H1A : Break in the dynamics of volatility, β0 (increase of 0.1) at 0.3T.

DGP1: β0 = 0.5 to β1 = 0.6

0.190 0.204

0.382 0.390

0.838

0.825

DGP2: β0 = 0.8 to β1 = 0.9

0.934 0.942

0.996 0.999

1.000

1.000

H1B : Break in the constant of volatility, ω0 (increase of 0.1) at 0.5T.

DGP1: ω0 = 0.4 to ω 1 = 0.5

0.210 0.204

0.353 0.353

0.809

0.787

DGP2: ω0 = 0.1 to ω 1 = 0.2

0.718 0.702

0.913 0.915

1.000

1.000

H1B : Break in the constant of volatility, ω0 (increase of 0.1) at 0.3T.

DGP1: ω0 = 0.4 to ω 1 = 0.5

0.148 0.153

0.254 0.262

0.674

0.634

DGP2: ω0 = 0.1 to ω 1 = 0.2

0.552 0.573

0.851 0.844

0.999

0.999

H1C : Break in the

error, u0 - N(0,1) (increase σu 1 =

1.1) at 0.5T

DGP1: u ι - N(0,1.1)

0.287 0.277

0.548 0.520

0.921

0.917

DGP2: u 1 - N(0,1.1)

0.449 0.437

0.710 0.700

0.982

0.975

H1C : Break in the

error, u0 - N(0,1) (increase σu 1 =

1.1) at 0.3T

DGP1: u 1 - N(0,1.1)

0.195 0.199

0.329 0.333

0.833

0.804

DGP2: u 1 - N(0,1.1)

0.376 0.386

0.548 0.548

0.932

0.923

HD : Outliers in the error, u0 - N(0,1) (μu 1

= 5 every 250 observations).

DGP1: u 1 - N(5,1)

0.019 0.046

0.015 0.039

0.005

0.044

DGP2: u 1 - N(5,1)

0.039 0.115

0.046 0.134

0.062

0.145

Notes: The Kokoszka and Leipus (2000) test statistic is defined as Uk =jk=1 rj - T- ɪ 'τTk+1 rj2J. The max UT(k) is
standardized by the VARHAC estimator,
σHAC, which is applied to Xt that represents either squared or absolute returns of the GARCH
model. The normalized statistic
UmaxlσHAC converges to the sup of a Brownian Bridge with asymptotic critical value 1.36 at the 5%
significance level. The Normal GARCH (1,1) model is simulated (1,000 replications) where the superscirpts 1 and 0 in the variables
and coefficients in the Table denote the cases with and without change-points, respectively.

24



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