1. Introduction - Econometrics at Illinois - University of Illinois at ...

Roger Koenker and Zhijie Xiao 31thusv n () = p n' n ()[R n R > ] ,1=2 [R ^() , r , ()]= p n'0()[RR > ] ,1=2 [R ^() , r , ()] + o p (1)) v0()+():Proof of Theorem 2 We may write,^v n () = p n'0()[R n R > n ] ,1=2 [R n ^() , rn , ()]= '0()[R n R > n ] ,1=2p n[R ^() , r , ()]+'0()[R n R > n ] ,1=2p n[r n , r]+'0()[R n R > n ] ,1=2p n[R n , R] ^()= '0()[RR > ] ,1=2p n[R ^() , r , ()]+'0()[RR > ] ,1=2p n[r n , r]+'0()[RR > ] ,1=2p n[R n , R]()+o p (1)Since () = + F ,1 ();^v n () = '0()[RR > ] ,1=2p n[R ^() , r , ()]n+'0() [RR > ] ,1=2p n[r n , r]+[RR > ] ,1=2p on[R n , R]+'0()F ,1 ()[RR > ] ,1=2p n[R n , R]+o p (1)= v n ()+Z > n ()+o p(1)where () =('0(); '0()F ,1 ()) > andZ n = [RR > ] ,1=2p n[r n , r]+[RR > ] ,1=2p n[R n , R]; [RR > ] ,1=2p n[R n , R] > = O p (1):And thus by Theorem 1,^v n () , Z > n () ) v0()+():Proof of Corollary 2 Similar to that of Corollary 1.Proof of Theorem 3 By Theorem 2,Denote the transformation based on _g as^v n () =v0()+Z > n ()+()+o p (1):Q g (h()) = h() ,Since Q g is a linear operator, we haveZ 0_g(s) > C(s) ,1 Z 1s_g(r)dh(r) ds;ev n () > = Q g^v n () > = Q g v0() > + Q g () > Z n + Q g () > + o p (1):By construction, Q g (() > ) = 0, and by Khmaladze (1981), Q g v0() > ) w0() > ; where w0 is a q-variate standard Brownian motion. ThusUnder the null hypothesis,sup2Tev n () ) w0()+e():kev n ()k )sup kw0()k :2T