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

ThusandZ _g n (s) > C n (s) ,1 Z 1Roger Koenker and Zhijie Xiao 33 Z _g n (r)dv n (r) ds , _g(s) > C(s) ,1 Z 1_g(r)dv n (r) > ds0s0sZ = _g n (s) > C n (s) ,1 Z 1[_g n (r) , _g(r)] dv n (r) > ds0sZ + _g n (s) > C n (s) ,1 , C(s) ,1 Z 1_g(r)dv n (r) > ds0sZ + _gn (s) > , _g(s) > C(s) ,1 Z 1_g(r)dv n (r) > ds0s= o p (1);v n () > ,and the result follows.Z 0= v n () > ,_g n (s) > C n (s) ,1 Z 1_g n (r)dv n (r) > dsZ 0sg(s) > C(s) ,1 Z 1g(r)dv n (r) > ds + o p (1);sAppendix B. Asymptotic Critical ValuesLike many other Kolmogorov-Smirnovtype tests (see, e.g. Andrews (1993), the limiting distributionsup 2T kw0()k is dependent on the norm jjjj, the pre-specied T and the dimension parameterq. Notice that the transformation is generally unstable in the extreme right tails, and the uniformconvergence of existing estimators of the density and score (f(F ,1 (s)) and f 0 =f(F ,1 (s))) usuallyrequires that T be bounded away from zero and one, we consider a subset of [0; 1] whose closure liesin (0; 1):We calculated the 1%, 5%, and 10% critical values for the test statistic sup 2T kev n ()k basedon simulations where the Brownian motion was approximated by a Gaussian random walk, usinga sample size n = 2000 and 20; 000 replications. For the norm kk, we use the `1 norm for a q-dimensional vector x; kxk = P qj=1 jx jj. Table 1 covers T =["; 1,"] for " =0:05; 0:1; 0:15, 0:2, 0:25,0:3; and q =1; 2;::::::; 20. Although conventionally we consider symmetric intervals T =["; 1 , "]for some small numbers ", amuch wider range of intervals T may be considered for the proposedtests. Critical values based other choices of the interval T and the dimension parameter q can besimilarly calculated. Gauss programs are available from the authors upon request.