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I have a problem where there are two time series $\{x_t\}_{t\geq 1}$ and $\{z_t\}_{t\geq 1}$ . These two time series are correlated for fixed time instant but uncorrelated with each other across time. We assume that time series $\{z_t\}_{t\geq 1}$ is completely known to us. Is there a way to incorporate this knowledge into an RNN model for multistep prediction of $\{x_t\}_{t\geq 1}$ . My main idea is that if e.g., I want to predict for 10 time steps ahead (predict $x_{t+1},…,x_{t+10}$), I can use a look-back window of size 10 on $x_{t−10},…,x_{t−1}$ and on $z_{t+1},…,z_{t+10}$, so a feature dimension $D=2$. My main concern is what would happen if I wanted to use a longer look-back window on {xt}t≥1 while still wanting to predict the next 10 values of $\{x_t\}_{t\geq 1}$ . In this case the features of my input would require different look-back windows? Is this possible or is there a trick to bypass this?

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    $\begingroup$ "These two time series are correlated for fixed time instant but uncorrelated with each other across time" - wth is this supposed to mean? $\endgroup$ – Aksakal Apr 26 at 18:56

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