Table 3: Size, Power and Frequency Distribution of the number of change-points obtained
with the Lavielle and Moulines (2000) test when there is a single break at 0.5T of the sample in
the GARCH process.
|
Samples, T |
1000 |
1000 |
|
Lavielle & Moulines |
BIC |
LWZ BIC LWZ |
|
Segments, tk = 5 | ||
|
Returns |
( rt )2 |
|rt| |
|
Number of Breaks |
01≥20 |
1≥20 1≥20 1≥2 |
H0 : Univariate GARCH, rо,t = uо,tjhoj, hо,t = ω0 + αou2,t-1 + βoho,t-1, with (ωо,αo,βq):
DGP1: (0.4, 0.1, 0.5) 0.96 0.03 0.01 1.00 0.00 0.00 0.98 0.02 0.00 1.00 0.00 0.00
DGP2: (о.1, о.1, о.8) 0.88 о.о7 о.о5 1.00 о.оо о.оо 0.93 о.о7 о.оо 1.00 о.оо о.оо
HA : Break in the dynamics of volatility with parameters (β0, β 1 )
|
DGP1: (0.5,0.6) |
0.72 |
0.24 |
0.04 |
1.00 |
0.00 |
0.00 |
0.79 |
0.20 |
0.01 |
1.00 0.00 0.00 | ||
|
DGP1: (0.5,0.8) |
0.00 |
0.95 |
0.05 |
0.00 |
1.00 |
0.00 |
0.00 |
0.93 |
0.07 |
0.00 |
1.00 |
0.00 |
|
DGP2: (0.8,0.7) |
0.21 |
0.75 |
0.03 |
0.85 |
0.15 |
0.00 |
0.20 |
0.75 |
0.05 |
0.84 |
0.16 |
0.00 |
|
DGP2: (0.8,0.4) |
0.00 |
0.72 |
0.28 |
0.00 |
1.00 |
0.00 |
0.00 |
0.86 |
0.14 |
0.00 |
1.00 |
0.00 |
HB : Break in the constant of volatility with parameters (ω 0, ω 1 )
DGP1: (0.4,0.5) 0.85 0.14 0.01 1.00 0.00 0.00 0.82 0.18 0.00 1.00 0.00 0.00
DGP1: (0.4,0.8) 0.00 0.94 0.06 0.38 0.62 0.00 0.00 1.00 0.00 0.36 0.64 0.00
DGP2: (0.1,0.3) 0.00 0.94 0.06 0.18 0.82 0.00 0.00 0.99 0.01 0.13 0.87 0.00
DGP2: (0.1,0.5) 0.00 0.86 0.14 0.00 1.00 0.00 0.00 0.95 0.05 0.00 1.00 0.00
H C : Break in the variance of the error with parameters (σ u 0, σ u 1 )
DGP1: (0,1.1)
DGP1: (0,1.5)
DGP1: (0,3)
0.01 0.49 0.50
0.00 0.63 0.37
0.00 0.60 0.40
0.01 0.94 0.05
0.00 0.97 0.03
0.00 0.98 0.02
0.01 0.57 0.42
0.00 0.58 0.42
0.00 0.53 0.47
0.01 0.95 0.04
0.00 0.97 0.03
0.00 0.93 0.07
Hf : Outliers in the error, u0 - N(0,1) (μu 1 = 5 every 250 observations).
DGP1: u 1 - N(5,1) 0.99 0.01 0.00 1.00 0.00 0.00 0.99 0.01 0.00 1.00 0.00 0.00
DGP2: u 1 - N(5,1) 0.98 0.02 0.00 1.00 0.00 0.00 0.92 0.06 0.02 1.00 0.00 0.00
Notes: The Lavielle and Moulines (2000) test is described in section 1.2. The Bayesian Information Criterion (BIC) and its
modification by Liu et al. (1997) denoted as LWZ are used. The simulations focus on DGP1, DGP2, T = 1000 for 500 trials. For
comparison purposes the alternative hypotheses of change points are similar to the K&L simulations (Table 1) and extended to larger
breaks. Reported is the frequency distributionn of the breaks detected. The highlighted numbers refer to the true number of
change-points in the simulated process.
26