Bilinear (p, 0, 1, 1) Model and its Consistent Identification

This paper is concerned with the BL(p, 0, 1, 1) model $$\sum_{j=0}^{p}a_jx(t-j)=e(t)+bx(t-1)e(t-1), \left\{e(t)\right\}$$ are iid, $$Ee(t)=0,Ee(t)^2=\sigma ^2$$ A concept of weak irreducibility is introduced. It is shown that the bilinear model can be expressed as an ARMA(p, 1) model in the sense of weak stationarity provided that weak irreducibility is satisfied. Using this expression we prove that the moment estimates of a j are unique and consistent. If, in addition, e(t) is JV(0, σ 2) or its first few order moment structure are known a prior, using up to the third order moments, b and σ 2 can also be estimated consistently. Furthermore, these estimates are shown to be asymptotically normal and have a convergence rate of O(ln ln N/N)1/2, where N is the number of observations. In contrast, the usual MLE estimates of the parameters have not been justified to have such asymptotic properties without assuming the finiteness of any order moments of the bilinear series and their infinite linear combinations. Finally, based on the ARMA expression, a consistent indentification procedure for the order p is proposed. Though the model discussed here is somewhat specialized, we hope that the ideas demonstrated here could highlight the possibility of making consistent inference without first solving the higher order moment problem.