Page 156 - Textos de Matemática Vol. 47

146 P. DE ZEA BERMUDEZ, M. A. AMARAL TURKMAN, AND K. F. TURKMAN
are presented in Table 2. The results given by ABC for model I are not very good, specially for a. This situation is possibly due to the relatively low sample size.
ABC
Minimum Q0.25 Median Mean Q0.75 Maximum s
------- 0.1089 0.4542 0.5729 0.5625 0.6749 0.8968 0.1523 0.3067 0.6143 0.7443 0.7802 0.9161 1.8460 0.2302
Model I True values a=0.2 b=0.4 2 =1.0
Table 2. Statistics for the sample of size kN = 1000 and the CML estimates of b and 2 obtained before.
CML
aˆ=0.1796
ˆb=0.3224 ˆ2 =1.0597
Let us consider now that the innovations ✏t are iid rvs with N(μ, 2), μ = 0 and
Model II: Xt = 0.6Xt 1 + 0.4Xt 1✏t 1 + ✏t, 2 = 2.0 (a2 + b2 2 = 0.68 < 1),
Model III: Xt = 0.6Xt 1 + 0.4Xt 1✏t 1 + ✏t, 2 = 3.0 (a2 + b2 2 = 0.84 < 1),
Model IV: Xt = 0.6Xt 1 + 0.7Xt 1✏t 1 + ✏t, 2 = 1.0 (a2 + b2 2 = 0.85 < 1).
All these models satisfy the conditions a2 + b2 2 < 1 and |a| < 1, which guarantee stationarity. However, neither of the models satisfy the invertibility condition given in (2.3). Due to this constraint, the CML method implemented before will not be used here. The simulated samples of size n = 5000 are pre- sented in Figure 3 and the statistics calculated with the simulated parameters are presented in Table 3 (N = 106 and kN = 1000). The kernel density esti- mates of a, b and 2 are presented in Figures 4, 5 and 6. For this large sample size, the good agreement between the true values of a, b and 2 and the es- timates obtained clearly show the benefits of using an ABC algorithm in the framework (of the complex) bilinear models. The results obtained for models II
and III indicate that the estimation of a and b is mostly a↵ected by the value of 2.