In this section, we discuss the properties of the estimator. In order to obtain these properties, we need the following assumptions:
(A1) is a sequence of independent and identically distributed random vectors.
(A2) , .
The following two lemmas are given by Diananda .
Lemma 4.1 Let , be a sequence of random variables such that the distribution of is independent of t for every and n and such that this collection is independent of for every and p if . Assume , , where p and m are positive integers. Then has a limiting normal distribution with mean 0 and variance .
Lemma 4.2 Suppose that , , are sequences of random variables. Let , , such that , , , as , at every continuity point. Then at every continuity point of .
Now, we state our main results in the following theorems.
Theorem 4.1 Under the conditions of (A1)-(A3), let . Then , as , where , the notation stands for convergence in distribution.
Theorem 4.2 Under the conditions of (A1)-(A3), let . Then , as , where , where the notation stands for convergence in probability.
Theorem 4.3 Under the conditions of (A1)-(A3), we have as .
Suppose that we have observations from model (1.1). Based on the estimator of , we can obtain the following forecasting procedure:
One-step forecasting: ;
Two-step forecasting: ;
n-step forecasting: .