The multivariate linear process bootstrap

The linear process bootstrap (LPB) for univariate time seri es has been introduced by McMurry and Politis (2010) and it is called LPB because it generates l inear processes in the bootstrap domain. However, it does not assume that the data are themselves a sam ple from a linear process. They use tapered and banded estimates for the autocovariance matrix of the whole data stretch [cf. Wu and Pourahmadi (2009)] and i.i.d. resampling of appropriately standardized residuals. Under a physical dependence assumption [cf. Wu (2005)], they show validity o f the LPB for the sample mean. In this paper, we generalize their approach to the case of mul tivariate time series and show its validity for the sample mean under different assumptions (mi xing, weak dependence, linearity). To complement the theory of McMurry and Politis (2010), we show that the multivariate LPB works also for spectral density estimation, but that it fails gene rally for sample autocovariances. Even in the univariate case and under assumed linearity, this is sti ll the case. However, we show consistency of the LPB for univariate, causal and invertible linear processes for higher order statistics as e.g. autocovariances and for autocorrelations in the univariate case under linearity only. Moreover, the validity of the LPB for the multivariate sample mean can be used in order to correct the invalidity of the univariate LPB for higher order statistics as e.g. autocovariances in the general case. For this purpose, an LPB-of-blocks bootstrap scheme is proposed