Graphical Methods for Investigating the Finite-sample Properties of Confidence Regions: A Gap in the Literature? A New Proposal

In the literature, there are not satisfactory methods for measuring and presenting the performance of confidence regions. In this paper, techniques for measuring effectiveness of confidence regions and for the graphical display of simulation evidence as regards the coverage and effectiveness of confidence regions are developed and illustrated. Three types of figures are discussed: called coverage plots, coverage discrepancy plots, and coverage effectiveness curves, that permits to show the “true” effectiveness, rather than a spurious nominal effectiveness. We prove that when simulations are run to compute the coverage for only one confidence level, which is usually done in the literrature for classical presentations in tables, all the information useful for computing the coverage for all the levels is present. Thus, there is absolutely no loss of computing time by using this method, whereas it provides more information than the corresponding tabular presentations. These figures are used to illustrate the finite sample properties of autoregressive parameter confidence regions in the context of AR(1) processes. Particularly, asymptotic, percentile, and percentile-t confidence intervals, as well as confidence intervals based on inverting bootstrap tests are presented and commented. Monte Carlo results assessing the performance of these confidence intervals for various situations are also presented. We show that classical confidence intervals have very poor performances, even the percentile-t interval, whereas confidence intervals based on inverting bootstrap tests have quite satisfactory performance. An application is made on stock market indices.