# Block Bootstrapping Relative Returns

I want to run a block bootstrap on the relative returns, but I'm not sure if subtracting the mean is important.

A bootstrap sequence is a synthetic sequence generated using the original sequence. If you're backtesting or estimating certain measures, such as volatility or the mean, then you can use the bootstrap to get a confidence interval on these values. The issue is in the way the bootstrap sequence is generated. You have to cut the original sequence into blocks, for the block bootstrap, and select blocks uniformly at random and place them in the sequence they were selected until you have a new bootstrap sequence of length n. You can't use the original prices, so you have to use the relative prices. These relative prices are the same as 1 day returns. I am wondering if mean centering is important in practice. I currently take the difference of their logs,

$\log\left(\frac{r_t}{r_{t-1}}\right)$

but it seems that some posts have suggested subtracting the mean return from the difference of logs

$\log\left(\frac{r_t}{r_{t-1}}\right) - \mu_r$

Most of the papers have an assumption of zero mean sequences. It's easy to zero-mean, but I'm afraid that this introduces some lookahead bias as the mean of the entire sequence is only known with knowledge of the full sequence. Why is subtracting the mean useful/important in practice?

• What is the purpose of the bootstrap? – John Jun 5 '13 at 20:54
• " You can't use the original prices, so you have to use the relative prices." Do you mean returns? – Quartz Jun 7 '13 at 12:01
• Anyway for risk a 0 mean return is a common assumption. Not only using a drift is an arbitrary decision (just like 0 drift, but slightly less so) and the process might not be stationary, but there's $\mu_r$'s estimation error too. As a compromise you could blend in $\mu_r$ weighting it by (an estimate of) the estimate precision, not just strictly statistical precision but also sample relevance taking into account how far in the past you're measuring (200 days or weeks are different in two ways). – Quartz Jun 7 '13 at 12:11
• I'm using one period returns, so yes. sorry for the confusion and thank you for your comment. – Amir Sani Jun 10 '13 at 12:49
• You could simply do density kernel smoothing estimation of the underlying probability distribution of each separate block, generate the data from that and then pick subsets of the resulting data. This should not depend on the mean of the sequence. – SMeznaric Jun 15 '13 at 15:16

It obviously depends on what you're trying to do but since we're speaking about returns zero centering is what's usually done because of the null hypothesis claiming that expected excess returns are zero. You zero center the distribution because you want to obtain a distribution satisfying the null hypothesis. In this distribution you then plug your sample mean and get a p-value.

This comes in handy in performance evaluation. It's what Aronson does in Evidence Based Technical Analysis when measuring the significance of the observed profits. It's also what White does in A Reality Check for Data Snooping when calculating the p-value for each model. White calculates for example those two V-values. For a single model you have

$$\bar V_1 = n^{1/2} \bar f_1$$

which is basically the sample mean (see the paper if you don't get the $n^{1/2}$ value) and you also have the bootstrapped distribution

$$\bar V_{1,i}^* = n^{1/2} (\bar f_{1,i}^* - \bar f_1)$$

which as you can see is zero centered through mean subtraction. The p-value is then obtained by plugging in $\bar V_1$ in the $\bar V_{1,i}^*$ distribution.

I also wouldn't say this adds any forward looking bias as you're not using the mean information to make any kind of decision. You're simply trying to determine whether the observed returns over a certain period beat the expected returns of a random strategy.

When you do bootstrapping, the actual problem is that you have too few events to really get an actual distribution. There are X methods to get the original set enriched, reinserting the events based not only on the uniform distribution when selecting the nth event out of the N total ones.

As the mean is one of the first characteristics of a distribution (the 1st moment), substracting a mean (whatever reasons you have), you implicitly throw information away. You might well calculate whatever mean you want at the very end and do with it the intended stuff.