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PARAMETER ESTIMATION OF BILINEAR PROCESSES USING ABC 137
Here, c0 = 1 and {✏t} is a sequence of independent and identically distributed (iid) random variables (rvs) with zero mean and variance 2.
In this class, the conditional mean E(Yt|Ft 1) is a nonlinear function of yt i,i = 1,2,.., whereas the conditional variance Var(Yt|Ft 1) is constant. Here Ft 1 is the -field generated by (Yt 1, Yt 2, . . . ). Further extension of this class can be made by including terms indexed to `2 = 0 in the last summation of (2.1), in which case these models also account for nonlinear variations in the conditional variance.
The class of bilinear models plays an important role in modeling nonlinearity for various reasons.
(1) The class is an obvious generalization of ARMA models resulting in nonlinear conditional mean and conditional variance.
(2) Under fairly general conditions, bilinear processes approximate finite order Volterra series expansions to any desired order of accuracy over finite time intervals (see Brockett [4]). Volterra series expansion is a dense class within the class of nonlinear time series. Therefore, under fairly general conditions, bilinear processes are also a dense class within nonlinear processes, approximating any nonlinear process to a desired level of accuracy.
(3) The class is fairly well-studied. Much is known regarding the existence of unique and stationary solutions. Although some identification, esti- mation and diagnostic techniques are available, much of the work on the class remains to be completed.
(4) Bilinear processes are often used in control theory although in a some- what di↵erent context in which they are applied in time series. In con- trol theory, the output Yt, as well as the input processes ✏t are observ- able, making the probabilistic structure simple. In the context we use these models, the input random process ✏t is not observed. This some- what restricts the use of these models within the time series framework. In estimation and prediction, it is important to know that the input process ✏t is also measurable with respect to the Ys, s  t, i.e., it is invertible. Unfortunately, the lack of verifiable conditions for invertibil- ity (except for very simple bilinear processes) restricts the use of these processes as models.
(5) A very important feature of the bilinear processes is that they are capable of producing sudden bursts of large values. Hence, they are very appropriate for modeling time series showing burst-like phenomena. This behavior is common in many areas, such as telecommunications and internet tra c (see Resnick [31] and Turkman et al. [38]).