Table 2: Nominal Size and Power of the Inclan and Tiao (1994) test for a single change-point in
the volatility based on a GARCH process.
|
Samples, T Inclan & Tiao Statistic Returns Transformations |
( rt )2 |
500 |
1000 IT |
3000 IT | ||||
|
IT |rt| |
( ut )2 |
( rt )2 | ||||||
|
|rt| |
( ut )2 ( rt )2 |
|rt| |
( ut )2 | |||||
|
H0 : Univariate GARCH, |
r0,t = |
u 0, t.Jh 0, t, h 0, t = |
ω0 + α0u0,t-1 + β0h0,t—1, with (ω0,α0,β0): | |||||
|
DGP1: (0.4, 0.1, 0.5) |
0.133 |
0.000 |
0.038 |
0.131 |
0.000 |
0.041 0.165 |
0.000 |
0.042 |
|
DGP2: (0.1, 0.1, 0.8) |
0.286 |
0.000 |
0.038 |
0.337 |
0.005 |
0.041 0.353 |
0.003 |
0.051 |
H1A : Break in the dynamics of volatility, β0 (increase of 0.1) at 0.5T
DGP1: β0 = 0.5, β1 = 0.6 0.404 0.001 0.047 0.633 0.023 0.047 0.966 0.294 0.058
DGP2: β0 = 0.8,β1 = 0.9 0.989 0.621 0.047 1.000 0.969 0.046 1.000 1.000 0.047
H1B : Break in the constant of volatility, ω0 (increase of 0.1) at 0.5T
DGP1: ω0 = 0.4, ω1 = 0.5 0.276 0.002 0.047 0.519 0.014 0.043 0.883 0.131 0.044
DGP2: ω0 = 0.1, ω1 = 0.2 0.786 0.121 0.043 0.959 0.385 0.047 1.000 0.981 0.032
H C : Break in the error, u 0 - N(0,1) (increase σ u 1 = 1.1)at0.5 Γ
DGP1: uι - N(0,1.1) 0.610 0.057 0.242 0.806 0.133 0.461 0.993 0.685 0.910
DGP2: u 1 - N(0,1.1) 0.599 0.046 0.247 0.823 0.128 0.432 0.994 0.668 0.929
Hd : Outliers in the error, u0 - N(0,1) (μu 1 = 5 every 250 observations).
DGP1: u 1 - N(5,1) 0.357 0.000 0.231 0.271 0.000 0.112 0.219 0.000 0.079
DGP2: u 1 - N(5,1) 0.505 0.001 0.257 0.500 0.002 0.136 0.481 0.002 0.079
Notes: The Inclan and Tiao (1994) statistic IT = T//2 max∣Dk| where Dk = [ζ∑k r2/∑τ r2) - -T- J is specified for independent
processes. It also converges to the sup ofa Brownian Bridge with asymptotic critical value 1.36 at the 5% significance level.
25
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