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 Kokoszka & Leipus Statistic Returns Transformations |
500 Umaχlσ ηac |
1000 Umaxlσ HAC |
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 = (ɪ ∑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.
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