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I've been poring over this paper written by Ledoit and Wolf regarding their approach to constructing hypothesis tests for Sharpe Ratios.

In short, they see that running circular block bootstrap resamples of the sample returns provided are a good approach as the distribution of the returns are unknown, but the autocorrelation and volatility clustering of the returns need to be accounted for. But before doing that, they must decide which block length must be used beforehand. For this, they recommend a calibration process where you offer a handful of block sizes ($b_1$, $b_2$, $b_3$, $...$, $b_n$)and the function outputs which block size is most ideal.

Within this calibration function, two different bootstraps are happening:

  1. On the original sampled data, a semi-parametric model $\hat{P}$ is fitted via a vector autoregression (VAR) with a stationary bootstrap being ran on the residuals. $K$ such models are created.

  2. For each $\hat{P}$, $M$ circular block bootstraps are being ran on the returns for each block size $b_i$ provided.

For each $\hat{P}$ (of which there are $K$), the test statistics under these two return streams (The $\hat{\theta}$ and $\hat{\theta}^*$, respectively) are then compared over all $M$ simulations, and a p-value ($p$) is calculated in the following way:

$$p = \frac{\text{# of times } \hat{\theta}^*\geq\hat{\theta}}{M}$$

As a reminder, there are going to be $K$ values of $p$. An "empirical rejection probability" is then calculated as follows:

$$\text{emp_reject_prob} = \frac{\text{# of times } p < \alpha}{K}$$

The ideal block size is the one which has an "empirical rejection probability" closest to the $\alpha$ (typically 0.05) specified.

My questions are as follows:

  • Why fit a semi-parametric model in such a way (VAR w/ stationary bootstrap)?

  • Why am I comparing $\hat{\theta}$ and $\hat{\theta}^*$? I've seen some answers such as this, but still can't get my head around it.

  • Why choose a block size resulting in an emp_reject_prob close to $\alpha$?

I know this is a lengthy question - I appreciate any help available.

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  • $\begingroup$ I would recommend first testing with less 'fancy' apparatus, like the standard tests for the Sharpe based on the $t$ statistic. It is likely you will fail to reject in that case, and there is no need to bother with Ledoit & Wolf. $\endgroup$ Jan 14 '20 at 22:56

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