Is there an issue with estimating future returns from autocorrelated returns?

Question

I have a time series $X_t$ generated from a standard GBM

$$dS_t = \mu S_t dt + \sigma S_t dW_t$$

If I take the log returns over a rolling window of length $l$

$$r^{(l)}_i = \log \left( \frac{S_i}{S_{i-l}} \right)$$

then the $r^{(l)}_i$'s will be highly autocorrelated.

For example, in python, we can calculate a 5 day rolling return ($l=5$) by

df  # pandas dataframe
>>> date    price
2006-03-01  65.72
2006-03-02  62.91
...


df["rolling_5_day_returns"] = np.log(df["price"].shift(-5)) - np.log(df["price"])

Given the autocorrelation, is there any technical issue in training a model $f$ to estimate the return $\hat r_{i + l}$ at time $i$? That is, at time $i$ we will have an estimate of price difference between times $i$ and $l$.

Explicitly, we have

$$ f(\vec r^{(l)}_i) = \hat r_{i+l} =\log \left( \frac{S_{i+l}}{S_i} \right) $$ $$ f(\vec r^{(l)}_{i+1}) = \hat r_{i+l+1} =\log \left( \frac{S_{i+l+1}}{S_{i+1}} \right) $$ $$ ... $$

where $\vec r^{(l)}_i$ is a vector of historic rolling window returns up to and including time $i$, and $ \hat r_{i+l}$ is the estimate of the return between the current time $i$ and future time $i+l$.

EDIT: When I say technical issue I mean, is it incorrect to train a model on such data, or will the model suffer in performance if the data is autocorrelated?

Also, I plan to train an LSTM model, but should it matter what model we train whether it's a neural net, regression or ARIMA? The data is always the same.

Answer 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 probably advise you, to be conservative, to get the return from "i-6" to "i-1", and then use this to predict the return from "i" to "i+5" to avoid problem of knowing the return at the close of day "i", and then instantly transacting in the same stock right away. I would also suggest reading up on basic momentum strategies, where you use past returns to predict future returns. Good luck!