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Most of the prior posts and papers I've read on this subject deal with autoregressive models but mine is currently of the form:

$$y(t+h) = X(t) + e$$

where $X$ represents the set of predictor variables that does not include $y(t)$.

I've also tried machine learning techniques with the same outcome and predictor variables.

I'd like to use k-fold cross-validation or something similar rather than just having a single training sample and holdout test sample; however, both $y$ and $X$ exhibit some degree of autocorrelation (for instance, for both variables $t$ and $t+1$ have a significant correlation). Thus, I'm wondering if it's problematic to have months randomly assigned to the folds, and if so, what the possible remedies are?

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So long as the errors are uncorrelated, this is fine. See https://robjhyndman.com/publications/cv-time-series/. Although that article is specifically about autoregression, the same argument works if you can set up your model as a regression $y \sim X\beta + e$.

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    $\begingroup$ Thanks for the feedback. To clarify, what if there is correlation between the errors at t and t+1, but the residual autocorrelation dies off before lag h. In this instance, I can't incorporate any of the residual information from time t into my model to improve the forecast/prediction of y(t+h); however, the observation at t used to forecast t+h and observation at t+1 used to forecast t+h+1 will not be independent. $\endgroup$
    – Kman
    Commented Jun 4, 2018 at 19:02

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