How can you calculate actual values when all you have is rolling averages?

Let's say you have a set of data that is rolling 6 month averages of the actual monthly data. Good data collection would mean you saved the actual values and then calculated the rolling averages, but unfortunately for you, there wasn't good data collection and all you have are rolling averages.

If the first "rolling average" in your data was actually just an average of the first month (i.e. your dataset is monthly widget sales and the first month is when the store opened), it would be easy to calculate the original data. The first month's actual data would be the same as the average. Then the second month's actual data would be the second month's average times two, minus the first month's data. And so on.

But unfortunately for you the first month's rolling average is actually an average of that month and the five months prior.

Is there any way to calculate the actual values? It seems that there are infinite possibilities that could lead to the same rolling averages, so an exact calculation is out of the question.

Barring an exact calculation, is there anything that you could do to estimate the actual values effectively? Some sort of statistical method?

• As you have discovered, there are infinitely many possibilities: to be precise, the first five months could be anything, and then the other months can all be determined in terms of the first five. I suppose that among all the possibilities there would be one that would have the least variance, and maybe there's a formula to find that one. But whether having minimal variance makes it better than any other estimate, I couldn't say. – Gerry Myerson May 29 '12 at 2:51