Asymptotic Distribution of Least Squares Estimators for Purely Unstable Arma (m,∞)

Residual empirical processes for long and short memory time series

“…This paper first establishes a uniform expansion of the residual empirical process of {ε t } under a general framework. The result is used to study the stochastic regression model of Robinson and Hidalgo (1997) and the unstable AR model of Chan and Terrin (1995), Truong- Van and Larramendy (1996) and Wu (2006). It is shown that the test statistic (1.5) of Ho and Hsing (1996) is no longer valid when the stochastic regression model includes an unknown intercept or when the characteristic polynomial of the unstable AR model has a unit root.…”

“…When φ(z) = (1 − z), δ n = n −1 and X t = y t−1 = t−1 i=1 ε i . By Theorem 6.1 of Chan and Terrin (1995) and Theorem 3.1 of Truong- Van and Larramendy (1996) or Theorems 3 and 4 of Wu (2006), Assumption 2.2(a) holds. By Lemma 4.1(c) and (d), we see that Assumption 2.2(b) and (c) holds.…”

Asymptotic laws of successive least squares estimates for seasonal arima models and application

“…In this case, it may be assumed without loss of generality that U ( B )= C 0 ( B ) with L 2. Consider the following processes ( Y n , h , n =1,2,…) for h =0,1,…, L , that are associated with the prime factors of C 0 ( B ): Then, from Chan and Wei (1988) for q =0 and from Truong‐van and Larramendy (1996) for q 1, we quote the following lemma.…”

“…Truong‐van and Larramendy (1996) extended the work by Chan and Wei (1988) to stable–unstable ARMA( p , q ) processes when using a martingale approximation technique. It can be mentioned that Ling and Li (1998) applied the approach in Chan and Wei (1988) to obtain asymptotic distribution of maximum likelihood estimators for unstable GARCH models.…”

“…The process x j , t closest to the must verify the condition h j −1 < i h j . More precisely, following Tiao and Tsay (1983), we decompose U ( B ) as a product of factors A k ( B ) whose zeroes are all simple, that is and to consider, unlike Chan and Wei (1988) and Truong‐van and Larramendy (1996), the particular processes x j , t for j =1,2,…, m , whose characteristic polynomials are . Since for j m −1, the x j , t ‐processes are not classical unstable ARMA because their error terms are nonstationary, we extend the technique in Chan and Wei (1988) to establish that, for j =1,2,…, m −1, the OLS estimators of the AR parameters of the x j , t ‐processes are distributed as linear combination of ratios of inner products of iterated integrals of a Brownian motion.…”