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

Abstract Let the time series Yt: t ∈ (1, 2, …) satisfy Yt = ρY t-1 + Z t and Zt + Σ p i=1 a i Zt−1 = et + Σ q j=1 β j et-j, where e t is a sequence of normal, independently distributed (NID (0, σ2) ) random variables, and y 0 = 0. Associated with the Zt process are the characteristic equations mp + Σ p i=1 aimp-i = 0 and mq + Σ q j=1 βjmq-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 Yt 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. Key Words: Time seriesNonstationarityDifferencingUnit roots