Breitung, J.

2 ~ r 1 ~ r frequency unit roots 1* ~ r 2* ~ r lemma statistic regression phillip unit root distribution vk ϭ p hypothesis section process series error philip hans franses phillips –perron tests result 2 ϭ 0 vk ͻ p frequency vk model breitung fu ~ vk time series value assumption franse wald statistic journal testing hylleberg philip brownian motions power theory decomposition bivariate regressions proof property theorem function 2 dr ϩ table polynomial hypothesis h 0 bivariate estimate 0+9 h 0 k econometrics t ϫ1 t order test statistics parameter lemma 1+ ~4! l sequence university density –perron least-square remark carlo ~1!+ difference monte uncorrelated errors size bias oxford university press unit root tests problem 0 ϫ g truncation autoregressive nonparametric correction vk յ p