Consistent cross-validatory model-selection for dependent data: hv-block cross-validation

This paper considers the impact of Shao's (1993) recent results regarding the asymptotic inconsistency of model selection via leave-one-out cross-validation on h-block cross-validation, a cross-validatory method for dependent data proposed by Burman, Chow and Nolan (1994, Journal of Time Series Analysis 13, 189–207). It is shown that h-block cross-validation is inconsistent in the sense of Shao (1993, Journal of American Statistical Association 88(422), 486–495) and therefore is not asymptotically optimal. A modification of the h-block method, dubbed ‘hv-block’ cross-validation, is proposed which is asymptotically optimal. The proposed approach is consistent for general stationary observations in the sense that the probability of selecting the model with the best predictive ability converges to 1 as the total number of observations approaches infinity. This extends existing results and yields a new approach which contains leave-one-out cross-validation, leave-nv-out cross-validation, and h-block cross-validation as special cases. Applications are considered.