Hypothesis Testing in ARIMA(p, 1, q) Models

Abstract

Let the time series { Y t : t ∈ (1, 2, …)} satisfy Y t = ρY t -1 + Z t and Z t + Σ p i =1 a i Z t−1 = e t + Σ q j =1 β j e t-j , where {e t } is a sequence of normal, independently distributed (NID(0, σ 2 )) random variables, and y 0 = 0. Associated with the Z t process are the characteristic equations m p + Σ p i =1 a i m p-i = 0 and m q + Σ q j =1 β j m q-j = 0, the roots of which are assumed to be less than one in absolute value. Thus, using the notation of Box and Jenkins (1976), we would say Y t is an ARIMA( p , 1, q ) process if ρ = 1. Under the assumption that ρ = 1, the limiting distributions of nonlinear least squares regression estimators of the parameters appearing in the preceding model are obtained. Regression t -type statistics for testing the hypothesis that ρ = 1 are discussed. Similar results are obtained for models that allow a nonzero mean. An illustrative example is given.