Evaluation of extremal properties of GARCH(p,q) processes

Generalized autoregressive conditionally heteroskedastic (GARCH) processes are widely used for modelling features commonly found in observed financial returns. The extremal properties of these processes are of considerable interest for market risk management. For the simplest GARCH(p,q) process, with max(p,q) = 1, all extremal features have been fully characterised. Although the marginal features of extreme values of the process have been theoretically characterised when max(p, q) >= 2, much remains to be found about both marginal and dependence structure during extreme excursions. Specifically, a reliable method is required for evaluating the tail index, which regulates the marginal tail behaviour and there is a need for methods and algorithms for determining clustering. In particular, for the latter, the mean number of extreme values in a short-term cluster, i.e., the reciprocal of the extremal index, has only been characterised in special cases which exclude all GARCH(p,q) processes that are used in practice. Although recent research has identified the multivariate regular variation property of stationary GARCH(p,q) processes, currently there are no reliable methods for numerically evaluating key components of these characterisations. We overcome these issues and are able to generate the forward tail chain of the process to derive the extremal index and a range of other cluster functionals for all GARCH(p, q) processes including integrated GARCH processes and processes with unbounded and asymmetric innovations. The new theory and methods we present extend to assessing the strict stationarity and extremal properties for a much broader class of stochastic recurrence equations

The clustering of extreme values for some asymmetric GARCH-type models

Several models with conditional heterosckedasticity have been studied in financial econometrics, with the simple GARCH(1,1) with Gaussian innovation representing the standard benchmark. There is evidence of asymmetry in some daily data and more flexible models, which take such an asymmetry into account, have become recently popular. Understanding the extremal behaviour of asymmetric processes becomes very important to build proper inference about extremal events. For processes satisfying mild mixing conditions the clustering of extreme values is characterzied by a single key-parameter, known as the extremal index, which represents the average clusters size of values which exceed a high-level threshold. An approach extending results for the GARCH(1,1) is presented, with skew-t innovation

Store flyers: managing spatial distribution under budget constraints

PurposeThe study develops a decision support system for the spatial distribution of store flyers, identifying a number of factors related to the demand and the competition influencing the complexities of their allocation to the target population.Design/methodology/approachThe model was developed incorporating the insights found in existing marketing literature and bypassing the limitations of the managerial practices. To this end, an in-depth discussion with a panel of retailers was held. The model was tested in collaboration with a retail chain.FindingsThe proposed system is flexible and provides an almost endless array of solutions in accordance with the retailer's strategic approach to the market. It captures the key trade-offs that need to be made during the decision-making process of a retailer with limited marketing resources.Practical implicationsThe traditional managerial approach, based on a set of operational steps, is overtaken by a model that systematically considers the interrelationships between the decision-making factors involved.Originality/valueThis is the first attempt to analyse spatial distribution of store flyers, a topic that has yet to be explored in retail marketing research. The paper conceptualises the key variables which affect the optimisation problem and reviews the different streams of extant research to obtain the appropriate insights

Clusters of Extreme Observations and Extremal Index Estimate in GARCH Processes

Several methods have been proposed for identifying clusters of extreme values leading to estimators of the extremal index; the latter represents,in the limit, the mean-size of each cluster of thresholds exceedances. The detection of clusters of extremes is relevant for the class of processes commonly used in financial econometrics, such as GARCH processes. The paper illustrates a novel approach to the above identification that exploits additional knowledge of the trajectory of the process around extreme events, and compares it to traditional approaches, using simulation from a GARCH process. We assess the relative performance of estimators in terms of bias, mean square error and distributional properties.

Robustness for multilevel models: Fraud detection with the forward search

Several methods using multiple regression or classification tools are commonly adopted to identify outliers which are, perhaps, the most important statistical units for anti-fraud detection. For data in the European Union, which are here analysed, the presence of clusters of several firms and several countries, may hide structures and information, making standard and classical tools often unreliable. Moreover, even the parameters estimation of classical models can be severely biased by influential observations or outliers. A methodological solution is to exploit the natural hierarchical structure of multilevel models to take into account th time-varying evolution of quantities traded, and their price, for each country. Multilevel models, however, are not robust as they simply generalize linear models and ANOVA. A forward search algorithm is presented to make parameter estimation robust in the presence of outliers and avoiding masking and swamping, leading to a more accurate identification of suspicious firms. The influence of outliers, if any is inside the dataset, will be monitored at each step of the sequential procedure, which is the key element of the forward search. Preliminary results on simulated data have highlighted the benefit of adopting the forward search algorithm, which can reveal masked outliers, influential observations and show hidden structures. An application to real data is also illustrated