### Linear Regression learning for VXX, OLS estimator, 2D

As the continuation of the previous post, let’s study the Linear Regression in which we have not only 1, but 2 variables.

Let’s assume we want to forecast the next day %change of VXX as an output variable, based on the today %change of the VXX and the yesterday %change

The linear equation would look like this.

**Y = beta0 + beta1*X1 + beta2*X2**

where

X1 = yesterday %change,

X2 = today %change,

Y = next day %change.

The unknown is the beta0, beta1, beta2. We want to determine (learn) them.

Let’s suppose that learn them by looking back in the history by D days, where D can be 20, 50, 100, 200 days.

The corresponding equity curve charts:

Conclusion:

– not much. **They are all similar. That is good, because they are consistent**

– the max. DD is: from 3.5 to 1.5: -57% (scary)

– in the 1D case, the lookback50 was the best, in the 2D case the lookback20 is the best. (probably it is just randomness)

– ** they are similar to the 1D charts. So it seems that introducing another variable (yesterday %change) doesn’t give more useful information for the prediction. It gave more information, but that information was not useful for extra the profit. This can be typical for machine learning. If we introduce a completely random extra variable (as a new dimension) (a non-dependent variable), it can even destroy the prediction power of the simpler case.**

– ** Based on these charts, we would stick to a simpler 1D Linear Regression than the 2D version. That may have a little better profit potential.**

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