THE INVARIANCE PRINCIPLE FOR LINEAR PROCESSES WITH APPLICATIONS
研究了线性过程部分和收敛到标准维纳过程的不变性原理，在弱化创新序列独立同分布或鞅差序列条件下推广了经典结果，并应用于单位根检验。
Let X t be a linear process defined by X t = [sum ] k=0 ∞ ψ k ε 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.