THE INVARIANCE PRINCIPLE FOR LINEAR PROCESSES WITH APPLICATIONS

Abstract

Let X_t be a linear process defined by X_t = [sum ]_k=0^∞ ψ_kε_t−k, where {ψ_k, k ≥ 0} is a sequence of real numbers and {ε_k, k = 0,±1,±2,...} is a sequence of random variables. Two basic results, on the invariance principle of the partial sum process of the X_t converging to a standard Wiener process on [0,1], are presented in this paper. In the first result, we assume that the innovations ε_k are independent and identically distributed random variables but do not restrict [sum ]_k=0^∞ |ψ_k| < ∞. We note that, for the partial sum process of the X_t converging to a standard Wiener process, the condition [sum ]_k=0^∞ |ψ_k| < ∞ or stronger conditions are commonly used in previous research. The second result is for the situation where the innovations ε_k form a martingale difference sequence. For this result, the commonly used assumption of equal variance of the innovations ε_k is weakened. We apply these general results to unit root testing. It turns out that the limit distributions of the Dickey–Fuller test statistic and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) test statistic still hold for the more general models under very weak conditions.