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$y_t$ is independent from $x_{1,t}$ and $x_{2,t}$ by definition, otherwise $u_t$ should be deterministic. It means that $\frac{dx_{1,t}}{dy_t}=0$ and $\frac{dx_{2,t}}{dy_t}=0$. All betas are constant so $\frac{du_t}{dy_t}=1$.


If you are predicting the return from time "i" to time "i+l" then you cannot use any information beyond time "i" to train your model. As it appears you are getting returns from "i-5" to "i" and assuming that this same relationship will hold into the future from day "i" to "i+5". In theory there is nothing glaringly bad about this approach, but I would ...

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