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Abstract
Financial advisors commonly recommend that the investment horizon should be rather long in order to benefit from the ‘time diversification’. In this case, in order to choose the optimal portfolio, it is necessary to estimate the risk and reward of several alternative portfolios over a long-run given a sample of observations over a short-run. Two interrelated obstacles in these estimations are lack of sufficient data and the uncertainty in the nature of the return generating process. To overcome these obstacles researchers rely heavily on block bootstrap methods. In this paper we demonstrate that the estimates provided by a block bootstrap method are generally biased and we propose two methods of bias reduction. We show that an improper use of a block bootstrap method usually causes underestimation of the risk of a portfolio whose returns are independent over time and overestimation of the risk of a portfolio whose returns are mean-reverting.
Keywords:
Acknowledgements
The authors are grateful to two anonymous referees, Peter Hall, Cedric Heuchenne, Jochen Jungeilges, Steen Koekebakker, Jean-Pierre Urbain and seminar participants at the University of Liege HEC Management School for their comments and suggestions on the earlier drafts of this paper. The usual disclaimers apply.
Notes
†The usual justification for some particular block length is to say that ‘it is probably long enough to pick up most of the possible time dependencies’ (Hansson and Persson Citation2000, p. 57).
†To the best of the authors's knowledge, we are the first to present a statistically significant evidence of mean reversion in the prices of portfolios of value stocks.
†For the sake of motivation, consider what happens to the estimate for Var(x t,t+m ) when m → n. Obviously in the limit, when the period length converges to the sample length, there is only one available block of data to estimate Var(x t,t+m ). Therefore, regardless of the nature of the data generating process, Var(x t,t+m ) → 0 as m → n.
‡We do not correct the estimates for the bias, because in order properly to correct for the bias we need to make some specific assumptions about the nature of the data generating process. This will be demonstrated later in the paper.
†The reasonable value of φ should be close to but less than 1.0, see the discussion in Fama and French (Citation1988).
†See also the results of the simulation analysis presented by Politis and Romano (Citation1994). These authors also demonstrate that the estimates provided by the stationary bootstrap are much less variable than those provided by the moving block bootstrap.
‡The first sample length in the table equals the length of our empirical data set, see subsection 3.2.2. The second sample length is roughly twice the length of our data set. This choice is made to demonstrate that a longer sample length decreases the estimation bias.
§See Orcutt and Irwin (Citation1948) and Marriott and Pope (Citation1954), who describe in detail why the use of overlapping blocks of data induces negative serial dependence.
†Note that the estimate for the mean return in our model is not biased for any block length.
‡The values in the table are based on the results of simulations using the moving block bootstrap method. The use of the stationary bootstrap method for the case with the ‘optimal’ block length (for the returns on small and value stocks) produces virtually identical results. Similarly, instead of using the ‘optimal’ block length in a block bootstrap method one can employ the standard bias correction method described in subsection 4.3. The magnitudes of the bias in the estimation of the variance ratio for the case where l = m can be found in table (when n = 63). This bias correction method also produces results close to the results reported in Panel D of table .
