The invariance principle for linear processes with applications

Let Xt be a linear process defined by [refer paper], where [refer paper] is greater than or equal to 0 is a sequence of real numbers and (ek, k = 0, plus or minus 1, plus or minus 2, ...) is a sequence of random variables. Two basic results, on the invariance principle of the partial sum process of the Xt converging to a standard Wiener process on [0,1], are presented in this paper. In the first result, we assume that the innovations ek are independent and identically distributed random variables but do not restrict [refer paper]. We note that, for the partial sum process of the Xt converging to a standard Wiener process, the condition [refer paper] or stronger conditions are commonly used in previous research. The second result is for the situation where the innovations ek form a martingale difference sequence+ For this result, the commonly used assumption of equal variance of the innovations ek 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. Disciplines Physical Sciences and Mathematics Publication Details This article was originally published as Wang, Q, Lin, YX and Gulati, CM, The invariance principle for linear processes with applications, Econometric Theory, 18, 2002, 119-139. Copyright Cambridge University Press. Original article available here. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/423 THE INVARIANCE PRINCIPLE FOR LINEAR PROCESSES WITH APPLICATIONS QIYING WANG, YAN-XIA LIN, AND CHANDRA M. GULATI University of Wollongong Let Xt be a linear process defined by Xt 5 (k50 ` ck et2k , where $ck, k $ 0% is a sequence of real numbers and $ek, k 5 0,61,62, + + + % is a sequence of random variables+ Two basic results, on the invariance principle of the partial sum process of the Xt converging to a standard Wiener process on @0,1# , are presented in this paper+ In the first result, we assume that the innovations ek are independent and identically distributed random variables but do not restrict (k50 ` 6ck 6 , `+ We note that, for the partial sum process of the Xt converging to a standard Wiener process, the condition (k50 ` 6ck 6 , ` or stronger conditions are commonly used in previous research+ The second result is for the situation where the innovations ek form a martingale difference sequence+ For this result, the commonly used assumption of equal variance of the innovations ek 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+