The Block Bootstrap for Parameter Estimation Error In Recursive Estimation Schemes, With Applications to Predictive Evaluation

This paper introduces a new block bootstrap which is valid for recursive m-estimators, in the sense that its use suFFIces to mimic the limiting distribution of (1/P^.5)(SUM(t=R to T-1)(THETA-t-hat - THETA-plus)); where R denotes the length of the estimation period, P the number of recursively estimated parameters, bµt is a recursive m¡estimator constructed using the first t observations, and THETA-t-plus is its probability limit. In the recursive case, earlier observations are used more frequently than temporally subsequent observations. This introduces a bias to the usual block bootstrap. We circumvent this problem by first resampling R observations from the initial R sample observations, and then concatenating onto this vector an additional P resampled observations from the remaining sample. Thereafter, THETA-hat-t-star is constructed using the resampled series, and an adjustment term is added to 1/P^.5)(SUM(t=R to T-1)(THETA-hat-t-star - THETA-t-hat)); in order to ensure that the distribution of this sum is the same as the distribution of (1/P^.5)(SUM(t=R to T-1)(THETA-t-hat - THETA-plus)). This parameter estimation error bootstrap for recursive estimation schemes can be used to provide valid critical values in a variety of interesting testing contexts, and three such leading applications are developed. The first is a generalization of the reality check test of White (2000) that allows for non vanishing parameter estimation error. The second is an out-of-sample version of the integrated conditional moment (ICM) test of Bierens (1982,1990) and Bierens and Ploberger (1997) which provides out of sample tests consistent against generic (nonlinear) alternatives. Finally, the third is a procedure assessing the relative out-of-sample predictive accuracy of multiple conditional distribution models. This procedure is based on an extension of the Andrews (1997) conditional Kolmogorov test. The main findings from a small Monte Carlo experiment indicate that: (i) the adjustment term used in the suggested bootstrap substantially improve coverage rates relative to a bootstrap without adjustment, and (ii) the suggested bootstrap is as reliable as the standard block bootstrap within the context of full sample estimation.