I have a time series prediction problem where the aim is to forecast the average value of $y_t$ over the next $T$ periods, given all the information available up to point $t$. For example, I want to forecast
$$\bar{y}_t = \frac{1}{T}\sum_{k=1}^T y_{t+k}$$
as a function of a bunch of other variables $x_t$ which are available at time $t$.
When building a training set from the data, I could ensure that I have no overlapping responses by considering
$$t = 0, T, 2T, 3T, \dots$$
However, I feel that this may not be making best use of the available data, and result in models with a lot of variance. An alternative is to use overlapping responses, for example
$$t=0, \tfrac{1}{2}T, T, \tfrac{3}{2}T, \dots$$
but I worry that this may create a lot of bias in the trained model.
Are there known results about how using overlapping data affects the bias/variance tradeoff? Is there a "best" level of overlap to use?