Block Bootstrap for Parameter Estimation Error when Parameters are recursively estimated

In real time forecasting, the sample is usually split into an estimation period of R observations and a prediction period of P observations, where T=R+P. Parameters are often estimated in a recursive manner, initially using R observations, then R+1 observations and so on until T-1 observations are used and sequence of P estimators are constructed. This paper provides a new block bootstrap procedure that mimics the limiting distribution of the scaled sum of the difference between the P estimators and their probability limit. We consider the case of m-estimators. In the recursive case, earlier observations are used more frequently than temporally subsequent observations. This introduces a bias to the usual block bootstrap, as any block from the original sample has the same probability of being selected. We circumvent this problem by first forming blocks as follows. Resample R observations from the first R observations, and then concatenate onto this vector an additional P resampled observations from the remaining sample. Construct a sequence of P bootstrap estimators, using the resampled series. Thereafter, construct the sum (scaled by sqrt P) of the difference between the P bootstrap estimators and the P actual estimators, and add an adjustment term in order to ensure that the sum of the two has the same limiting distribution as the sum (scaled by sqrtP) of the difference between the P (actual) estimators and their probability limits. This recursive block bootstrap 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) for the case of non-vanishing parameter estimation error. The second is an out of sample version of the integrated conditional moment 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 for assessing the relative out of sample accuracy of multiple conditional distribution models. This procedure can be viewed as an extension of the Andrews (1997) conditional Kolmogorov test.