# Overlap in time series training sets

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?

• I want to comment on your statement that you feel that you do not make best use of the available data. – Skullduggery Oct 24 '13 at 18:09
• You want to forecast a variable which itself is an average over $T$ variables. You surely have reason for this; E.g., maybe $T$ covers the length of a seasonal cycle and hence the average covers seasonal information and period-to-period information. So why do you want to use the $y_{t-s}$, $s=0,1,2, \ldots$ as a predictor and not the lagged averages $\overline{y}_{t-s}$, $s=0,1,2, \ldots$? Besides the question on how to exploit your data most effectively I do not understand the "overlapping" issue. – Skullduggery Oct 24 '13 at 18:17
• To be more precise: Your first question seems to address the issue of selecting suitable predictors, the second on how to set up a suitable rolling-window validation scheme. – Skullduggery Oct 24 '13 at 18:22
• @Skullduggery Feel free to ignore the part about selecting predictors. Just assume that at time $t$ I have a bunch of predictors $x_t$ that I've already selected. I'm mainly interested in how to set up a rolling-window scheme (or indeed, if I should use rolling windows at all). – Chris Taylor Oct 24 '13 at 18:35
• I can't think of a reason to use rolling windows. I assume you'd also use rolling windows on your inputs? You're not making new information, just interpolating. – Jessica Collins Oct 25 '13 at 3:54