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P. DE ZEA BERMUDEZ, M. A. AMARAL TURKMAN, AND K. F. TURKMAN
(6) It is known that the heavy-tailed behavior of bilinear models is a con- sequence of the tail weight of the innovation process, as well as the multiplicative, nonlinear, features of the di↵erence equation given in (2.1). For instance, when the innovations ✏t are regularly varying, the tail of the bilinear model BL(0, 0, 1, 1),
Xt = cXt 1✏t 1 + ✏t,
where c is a constant, satisfies the following relation
P(|Xt |>x)⇡P(✏21 >x). For a general BL(p, q, m, k), we have
P(|X |>x)⇡P(✏k+1 >x). t1
Hence, while for linear models, Xt and ✏t are tail equivalent, for BL(p, q, m, k) models X is tail equivalent to ✏k+1 (see Resnick [31]).
tt
When innovations have lighter tails, for example Gaussian, then the relationship between the tails of input-output series is more compli- cated. However, typically the tail of the output series Xt is regularly varying when the innovations are Gaussian (Turkman and Amaral Turkman [40]). These results clearly indicate why and how bilinear processes produce extreme observations and show how useful they can be in modelling heavy-tailed dependent data.
These features are evident in the simulated data sets that are plotted in Figure 1. When the innovation process is Gaussian (light-tailed), the simulated sample does not exhibit very large values (left panel). On the other hand, when
Xt
−5 0 5
Xt
0 50 100 150 200 250 300 350
Xt
0 10000 20000 30000 40000 50000
0 100 200 300 400 500 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 ttt
Figure 1. Simulated bilinear data sets - N(0,1) innovations (left), Pareto(2.5) innovations (center) and Pareto(1.5) innovations (right).