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Many economic series are released monthly, but only as YoY (year-over-year) rates, that is, the change in an underlying index over the previous year, expressed in percentages. I wish to convert these releases into MoM (month-over-month) values, so that I can accurately recreate at least part of the underlying index. In the chart below, for example, I'm looking for the red line slope, given the two green line slopes, which would allow me to recreate the underlying index.

enter image description here

In R:

library("xts")
set.seed(212)
mm = (runif(24) - 0.3) / 100
idx = c(1, cumprod(mm + 1))
yy = rollapply(idx, 13, function(x) (x[13]/x[1]) - 1)

So given only yy, I would like to find the mm's, or at least the last 12 of them (or idx directly which is my ultimate goal). How do I go about this? Is it a big set of 12 simultaneous equations that I need to solve?

Note that though I have only drawn 2 YoY lines on the chart, I actually have all of the ones for the past 13 months (going back 12 months each time), and using them I'd like to calculate as many of the MoM values as possible.

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I also believe it cannot be done. If you use your technique of 12 simultaneous equations, you will have more MoM unknowns than YoY knowns, always. Each YoY uses a different set of 12 MoMs.

Think of each YoY as dropping the previous first MoM and adding the new MoM. Each known YoY produces two unknown MoMs. Even after 12 months, you are still outnumbered on MoMs.

I disagree with Brent's argument. You do not need to know the absolute level to solve these for these values on a percent basis.

On a side note, most BEA releases give you the absolute values anyhow, so you can look them up and calculate MoM percentages yourself. http://research.stlouisfed.org/fred2/ also contains different ratios and values for most economic indicators.

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  • $\begingroup$ I'm not arguing that one needs to know the absolute level; I simply pointed out that for every month there exist two sequences of indices that give rise to the same YoY values but distinct MoM values for that month, which proves that none of the MoM values can be derived from the YoY values. The argument from the number of equations seems unconvincing, since in general it is possible to have fewer equations than unknowns and yet still have some of the unknowns being uniquely solvable. But you give a very nice link! +1 $\endgroup$ – Brent Kerby Oct 7 '15 at 20:47
  • $\begingroup$ I re-read your comment and I think I understand it better now. It is true that you could change individual MoMs and keep the YoY the same, but I do not see how that fact proves that this cannot be done. $\endgroup$ – Maddenker Oct 8 '15 at 18:45
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Unfortunately, this cannot be done. To see this, imagine that we were to multiply every February entry in idx by some constant, say, 2. Then the MoM values will change (for February and March), but none of the YoY values will change. Therefore, from the YoY data alone the MoM data cannot be uniquely reconstructed -- not even the last 12 months.

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