Julio A. Afonso Rodríguez, M.B.A., Institutional Economics, Economic Statistics and Econometrics, UNIVERSITY OF LA LAGUNA and UNIVERSITARY INSTITUTE OF REGIONAL DEVELOPMENT, SCHOOL OF BUSINESS AND ECONOMICS. CAMPUS DE GUAJARA. CAMINO LA HORNERA, S/N, LA Laguna, 38071, Spain
Many of the recent work on analysis of high-frequency financial time series mainly focus on three related topics: (a) the heavy-tailed nature of the distribution of returns, (b) the modeling of conditional second or higher moments and, (c) the identification and treatment of outliers and the development of robust inferential procedures. The recent developments in Extreme Value Theory (EVT) for iid and dependent stationary processes allow a convenient treatment of very large sample data from different parent distributions. EVT gives results on approximate distributions and order of convergence of the sample maximum, and limiting distributions of the largest observations of the sequence. Heavy-tailed distributions are prone to generate possibly very extreme order statistics which are very large relative to the central body of the data. With these distributions it seems convenient to distinguish between two types of very large sample data: (1) “extremes” as large realizations of random variables generated by the assumed population distribution, and (2) “outliers” as large sample values which do not belong to the population of interest. Thus, it seems more convenient to define an outlier in terms of tail probabilities refered to a particular distribution. In the case the reference distribution is unknown, one possibility is to use results from the theory of extremes.
Recently, Burridge and Taylor (2006) and Schluter and Trede (2008) consider the detection of additive outliers in heavy-tailed distributions via functions based on extreme order statistics incorporating distributional results from EVT. In the first case, the authors consider test statistics based on spacings of adjacent order statistics and in the latter test statistics based on ratio of consecutive order statistics. In both cases they develop procedures for identification of possible multiple outliers.
In this paper we review both procedures, analyze the functional relationship between spacings and ratios of extreme order statistics and their behavior under different number and configurations of additive outliers. Using the inverse probability integral transform of uniform order statistics under different general representations of heavy-tailed distributions, we get approximate results for expectation and covariance structure of both sets of statistics, and using known results for the representation of spacings from continuous distributions we study how to obtain appropriate scaling factors to avoid the divergence of the distribution in the case of the the spacings. Moreover, we analyse the finite sample performance of such procedures under outlier contamination for dependent processes of the GARCH-type through a simulation study under different distributional assumptions and types of contamination. Under no contamination, simulation results indicate that the finite-sample distribution of both test statistics for GARCH processes is very similar to that the innovation process.
On a second stage, we study the possibility of using similar test statistics based on log-spacings and log-ratios of adjacent order statistics, because in this cases there exist a number of results that establish convenient distributional representations for log-spacings from heavy-tailed distributions of the Pareto type. The finite sample distribution and behavior of such proposed modified statistics under additive outlier contamination is illustrate through a simulation study.