The Analysis of Marked and Weighted Empirical Processes of Estimated Residuals

Abstract

An extended and improved theory is presented for marked and weighted empirical processes of residuals of time series regressions. The theory is motivated by 1step Huber-skip estimators, where a set of good observations are selected using an initial estimator and an updated estimator is found by applying least squares to the selected observations. In this case, the weights and marks represent powers of the regressors and the regression errors, respectively. The inclusion of marks is a non-trivial extention to previous theory and requires refined martingale arguments.

Heteroscedasticity testing after outlier removal

“…Now: (i) By Assumption 2.1 (1 + juj 8 )f(u) is integrable, Lipschitz with a weakly unimodal bound. Then, by Lemma A.7 in Berenguer-Rico, Johansen and Nielsen (2019b) for k = 0; 1; 2 we have that (a) and Nielsen (2019b) we have that (e) u k+2 f(u) is Lipschitz and (f ) kjuj k+1 f(u) < 1 uniformly in c; as required.…”

“…> 0 as required in Lemma B.3 when k = 4: (b) By Assumption 2.1(b); we have that (1 + jcj 8 )f(c) is Lipschitz with a weakly unimodal bound. Therefore, Lemma A.4 in Berenguer-Rico, Johansen and Nielsen (2019b) gives sup c2R (1+jcj 8 )f(c) < 1: In turn, this implies that F is additively and multiplicatively Lipschitz -see Berenguer-Rico, Johansen and Nielsen (2019a) remark 3.1.…”

Uniform Consistency of Marked and Weighted Empirical Distributions of Residuals

“…(2.4) and study F w;p n (a; b; c) uniformly over a; b varying in expanding compact sets depending on n and c 2 R. Speci…cally, we make use of Lemma 3.1 in Berenguer-Rico, Johansen and Nielsen (2019). It states that if lim n!1 P(~ 2 ) > 1 for some estimator~ in a compact set and > 0, then for any function F n ( ; c) of 2 and c 2 R, we have that…”

“…The second tool is a chaining argument which allows us to derive the required uniformity results over a; b; c: The argument is as follows, see also Berenguer-Rico, Johansen and Nielsen (2019). Consider the process F n ( ; c) where 2 and c 2 R. Introduce K grid points c k and cover the set by M balls with centres m with a small radius : The chaining argument is…”

“…Koul and Ossiander (1994), see also Koul (2002), considered the weighted empirical processes, in which the scale is known and p = 0; so that there are no marks. Johansen and Nielsen (2016a) and Berenguer-Rico, Johansen and Nielsen (2019) considered the general case with weights and marks. The marked empirical process of Koul and Stute (1999), and Escanciano (2007) arise when the weights are w in = n 1=2 1 (x i d) and the present indicators 1 (" i c) are set to unity.…”