UNIT ROOT TESTING ON BUFFERED AUTOREGRESSIVE MODEL.

Note that A T () 1 p T max t T jy t 1 R t ()j 1 p T T X t=1 w 0 t (0 0)[R t (0) R t ()] In Li et al. (2015), the convergence of the estimated thresholds was shown to be T (^ r L r L0) = O p (1) and T (^ r U r U0) = O p (1). Under the unit root hypothesis, for any we have ~ t 1 () = ~ N T () + ~ A T () q ~ D T ()^ 2 where ~ N T () = 1 T T X t=1 ~ y t 1 ~ e t ~ R t () 1 T T X t=1 ~ y t 1 ~ w t 1 ~ R t () T X t=1 ~ w 2 t 1 ~ R t () ! Note that X 0 X is a special case of X 0 ()X and denote M =X 0 X=T. From Lemma 1.2, it can be obtained that M() is asymptotically block diagonal, then LR T () is LR T () =S 0 1 ()[M 11 () M 11 ()M 1 11 M 11 ()] 1 S 1 () +S 0 2 ()[M 22 () M 22 ()M 1 22 M 22 ()] 1 S 2 () +o p (1) = S 0 1 () p T M 11 () T M 11 () T M 11 T 1 M 11 () T 1 S 0 p T +S 0 2 ()[M 22 () M 22 ()M 1 22 M 22 ()] 1 S 2 () +o p (1) where 1 p T S 1 () = 1 p T T X t=1 y t 1 p T e t R t () 1 p T M 11 ()M 1 11 T X t=1 y t 1 p T e t) Z 1 0 W (s)dW (s;R()) R() Z 1 0 W (s)dW (s); S 2 () = 1 p T T X t=1 w t e t R t () 1 p T M 22 ()M 1 22 T X t=1 w t e t)G() 1 G; M 11 () T M 11 () T M 1 11 T M 11 () T)R()(1 R()) Z 1 0 W 2 (s)ds; and M 22 () M 22 ()M 1 22 M 22 ()!. [Extracted from the article]