Comput Econ (2009) 34:365–382 DOI 10.1007/s10614-009-9180-8 Tests of Random Walk: A Comparison of Bootstrap Approaches Eduardo J. A. Lima · Benjamin M. Tabak Accepted: 5 May 2009 / Published online: 28 May 2009 © Springer Science+Business Media, LLC. 2009 Abstract This paper compares different versions of the multiple variance ratio test based on bootstrap techniques for the construction of empirical distributions. It also analyzes the crucial issue of selecting optimal block sizes when block boot- strap procedures are used. The comparison of the different approaches using Monte Carlo simulations leads to the conclusion that methodologies using block bootstrap methods present better performance for the construction of empirical distributions of the variance ratio test. Moreover, the results are highly sensitive to methods employed to test the null hypothesis of random walk. Keywords Resample · Bootstrap · Variance ratio · Random walk 1 Introduction Among the different methods developed to test the presence of serial correlations in time series, the variance ratio test (VR) became quite popular after the studies of Lo and Mackinlay (1988, 1989), 1 Poterba and Summers (1988) and Cochrane (1988). It has been frequently utilized to test the random walk hypothesis (RWH) not only in financial time series, but also in macroeconomic data. The Lo and Mackinlay (1988) VR methodology, for testing the RWH against stationary alternatives exploits the fact that the variance of random walk increments 1 It is worth mentioning that several studies, using variance ratios in different contexts, preceded the research of Lo and Mackinlay (1988). However, none of these previous studies formalized the sample theory for the test statistics. For this reason, most researchers attribute the variance ratio test to Lo and Mackinlay (1988). E. J. A. Lima · B. M. Tabak (B ) Banco Central do Brasil, Brasilia, Brazil e-mail: benjamin.tabak@bcb.gov.br B. M. Tabak Universidade Catolica de Brasilia, Brasilia, DF, Brazil 123