As we noted in the last post here , the site is under transfer to a new (and improved) location! We shall be rebranding soon! This is the cause of the small delay in new posts. Be that as it may, I am back with another idea on how to get your returns rolling when your run dry of how to beat the market!

In this post I consider the notion of standardization of a random variable and how this can effect the results of a trading strategy. Consider the returns of an asset [math] y_{t}[/math] and think of another random variable, say [math] s_{t}[/math] everywhere positive and drawn from some distribution independently of [math] y_{t}[/math]. This new random variable acts like a standard deviation in regular standardization but not quite in our case! Suppose, as I do in the implementation, that we have [math]s_{t} \sim U[o, b][/math] and so [math] s_{t}[/math] is uniformly distributed in the interval [0, b] for some upper bound [math] b[/math].

Then, consider the new random variable [math] z_{t} \doteq y_{t}/s_{t}[/math] and compute the first-order autocorrelation of it, i.e., [math] \rho_{z}(1) = \left\{\mathsf{E}(z_{t}-\mu_{z})(z_{t-1}-\mu_{z})\right\}/\sigma_{z}^{2}[/math]. Now, if the scaling by [math] s_{t}[/math] was such that the standardized variable would have small(er) autocorrelation than the original variable we could easily (and safely) predict [math] z_{t}[/math] by its sample mean - and then reverse the standardization and predict [math] y_{t}[/math]! But we still have no idea what the "right" or "appropriate" standardizing variable should be, even with b fixed at some value; but we can do a good job via simulation and resampling, as follows:

Step 1. Generate [math] j=1, 2, \dots, B[/math] samples [math] s_{tj}[/math] of random draws of independent random variables from the [math] U[0, b][/math] distribution of size [math] n[/math], the sample available for [math] y_{t}[/math].

Step 2. Form the standardized variables [math] z_{tj} \doteq y_{t}/s_{tj}[/math] and compute their first-order autocorrelation [math]\widehat{\rho}_{j}(1)[/math].

Step 3. Rank the standardized variables in increasing order of their absolute first-order autocorrelation and select the minimum, i.e., select [math] z_{t(1)} = argmin_{j\in\left\{1, 2, \dots B\right\}}\widehat{\rho}_{j}(1)[/math] and compute its sample mean over a rolling window of [math] R[/math] observations [math]\bar{z}_{t(1)} \doteq R^{-1}\sum_{\tau=t-R+1}^{t}z_{\tau(1)}[/math]

Step 4. Compute the prediction of the original variable as [math] \widehat{y}_{t+1|t} \doteq 0.5\cdot b\cdot \bar{z}_{t(1)}[/math] and trade it to obtain the return [math] r_{t+1|t} \doteq sgn(\widehat{y}_{t+1|t})y_{t+1}[/math]. That's it!

And, you ask, does this approach generate excess returns??! You bet it does, and the results below will certainly illustrate that! In Table 1 below I have collected the backtesting results from a trading exercise with weekly returns over the two years 2022-2023, across various values of the pair [math](b, R)[/math] and [math] B=100[/math] replications (yes, results will change in terms of combinations if you use different values of [math] B[/math], you should try it!). You will immediately note that the method works across different combinations and, therefore, one can devise some form of cross-validation to "optimally" select these parameters - but that's an exercise for the reader. The ETFs that I am using should be standard fare for the readers, and the Python code is always to be found at my github repository.