In a recent paper, Cavaliere et al., [Cavaliere G, 2012] develop bootstrap implementations of the popular likelihood-based co-integration rank tests and associated sequential rank determination procedures of Johansen [Johansen S, 1996]. By using estimates of the parameters of the underlying co-integrated VAR model obtained under the restriction of the null hypothesis, they show that consistent bootstrap inference can be obtained for processes whose deterministic component is either zero, a restricted constant or a restricted trend. In this article, we extend their bootstrap approach to allow the deterministic component to follow the practically relevant cases of either an unrestricted constant or an unrestricted trend from Johansen [Johansen S, 1996]. A full asymptotic theory is provided for these two cases, establishing the asymptotic validity of the resulting bootstrap tests. Our results, taken together with those in Cavaliere et al., [Cavaliere G, 2012], therefore show that the bootstrap approach based on imposing the reduced rank null hypothesis is valid for all five of these deterministic settings. Monte Carlo evidence demonstrates the improvements that the proposed bootstrap methods can deliver over the corresponding asymptotic procedures.

In this article, we investigate the behaviour of a number of methods for estimating the co-integration rank in VAR systems characterized by heteroskedastic innovation processes. In particular, we compare the efficacy of the most widely used information criteria, such as Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) , with the commonly used sequential approach of Johansen [Likelihood-based Inference in Cointegrated Vector Autoregressive Models (1996)] based around the use of either asymptotic or wild bootstrap-based likelihood ratio type tests. Complementing recent work done for the latter in Cavaliere, Rahbek and Taylor [Econometric Reviews (2014) forthcoming], we establish the asymptotic properties of the procedures based on information criteria in the presence of heteroskedasticity (conditional or unconditional) of a quite general and unknown form. The relative finite-sample properties of the different methods are investigated by means of a Monte Carlo simulation study. For the simulation DGPs considered in the analysis, we find that the BIC-based procedure and the bootstrap sequential test procedure deliver the best overall performance in terms of their frequency of selecting the correct co-integration rank across different values of the co-integration rank, sample size, stationary dynamics and models of heteroskedasticity. Of these, the wild bootstrap procedure is perhaps the more reliable overall as it avoids a significant tendency seen in the BIC-based method to over-estimate the co-integration rank in relatively small sample sizes.

This book deals with many of the most relevant topics in price index numbers theory and practice. The problem of the harmonization of CPIs and the time-space integration of baskets is analyzed at the Eu-zone level, with methodological and actual proposals on how to proceed for an overall treatment of the matte. Likewise, the construction of sub-indexes for households economic and social groups is investigated, in order to obtain specific inflation measurement instruments. Evidence from most updated databases is given. The questions of the spatial comparisons of price levels through PPPs and th.