Does anybody know how to measure the correlation between monthly and quarterly data?

I would like to calculate the correlation between ISM PMI (monthly) and Real GDP growth (quarterly). So, the 2 time series have different frequencies, and different number of datapoints.

The correlation between PMI and US Real GDP Growth with a 12 month time lag, is supposedly around 85%. How can this be calculated?

If your answer should include an example, then preferably in R or Excel.

Source data can be viewed and downloaded here:
PMI : https://www.quandl.com/data/FRED/NAPM-ISM-Manufacturing-PMI-Composite-Index
GDP : https://www.quandl.com/data/FRED/A191RO1Q156NBEA-Real-Gross-Domestic-Product

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    $\begingroup$ down sample PMI by averaging or some other mean $\endgroup$ – Aksakal Apr 5 '16 at 19:57
  • $\begingroup$ You could just replicate each quarterly measurement in one series 3 times to compare with monthly data in the other. But if you correlate then two things are mixed up, the underlying correlation that you would have got if you had had monthly data not quarterly, and the effects of the coarsening to quarters (which might push the correlation either way, so far as I can see). There might be some scope for investigating the effects of the coarsening. But you aren't the first person to have this problem with economic time series, so there should be economic or econometric literature to consult. $\endgroup$ – Nick Cox Apr 5 '16 at 20:19
  • $\begingroup$ To make this answerable, could you explain what such a correlation would mean? "Correlation" is a concept that applies, strictly speaking, to paired data, which these are not. It's conceivable that you are viewing one (or both) of these time series as being temporal aggregates of more refined time series, which themselves are not directly observed, and perhaps you are asking how to estimate some underlying correlation between those unobserved series? It may be worth noting that questions like this one are asked and answered in geostatistics: search "change of support." $\endgroup$ – whuber Apr 5 '16 at 22:36

Go back to basics and ask, "What is a correlation?" The right answer is that there are many measures of "correlation" when broadly considered as pairwise metrics for demonstrating association between variables. The naive answer is that it's a Pearson correlation since that's the most commonly taught and known form. But Pearson correlations only measure pairwise, linear association and, moreover, have a rigorous requirement for interval or ratio scaled data. Spearman correlations, on the other hand, measure monotonic association, e.g., between ordinally scaled variables. For financial data which don't always meet the requirements of a Pearson, Spearman correlations are a much more sensible metric.

There are also many, many measures of association for categorically scaled data, as well as nonlinear measures of association such as "distance" correlations, and so on.

In addition, both Pearson and Spearman correlations range between $-1$ and $1$. Given that (and Street vernacular notwithstanding), it's completely erroneous to speak of them in percentage terms.

Not knowing where you got this rule of thumb of a 12 period lag resulting in an "85%" correlation between GDP and PMI, what is your goal? In other words, why are you even interested in replicating such a hoary convention? Moreover, you have such a long time series from Quandl -- back to 1950 -- what is it that makes you so concerned about the difference in the units of time?

What would I do to address your question? I would merge the two series with 3 monthly periods per quarter and do a whole lot of exploratory analytics: scatterplots, time lines, etc. Based on that, next I would run both Pearson and Spearman correlations using different lags, e.g., 1 quarter up to whatever you think a reasonable maximum # of lags is. Then, I would examine where the association was maximized. That's it.

Of course there are more rigorous ways to answer the question that go beyond simple measures of pairwise association, but that's not what you're asking for.

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    $\begingroup$ +1. Provocative as always, but the sharp distinction between Pearson as requiring interval scale and Spearman as suitable for ordinal scale is quite overdone, as Spearman correlation is nothing but Pearson correlation applied to ranks. So, you are condemning on the one hand what you encourage on the other. But that's one detail only. $\endgroup$ – Nick Cox Apr 5 '16 at 20:15
  • $\begingroup$ @NickCox But that difference in scaling is the point, not their calculation. $\endgroup$ – Mike Hunter Apr 5 '16 at 20:16
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    $\begingroup$ I don't see what extra point you are making there. My point is that Pearson correlation can be, and is, applied to ordinal, interval and ratio data; it's just conventional to call it Spearman when the data are ordinal. (The inference is in detail different.) $\endgroup$ – Nick Cox Apr 5 '16 at 20:22
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    $\begingroup$ On Spearman and Pearson, I don't see that you have addressed my point directly. But it's a matter of algebra only that pushing ranks through Pearson correlation gives the same result as Spearman correlation. The controversial question of what is appropriate for what scale type is alongside, but not at all the same, and I am not raising any of that here. I made the same point in discussion of Hand, D.J. 1996. Statistics and the theory of measurement. Journal of the Royal Statistical Society. Series A 159: 445–492 doi.org/10.2307/2983326 (see pp.481-2), but it's quite standard. $\endgroup$ – Nick Cox Apr 5 '16 at 20:48
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    $\begingroup$ Bjorn: Although DJohnson has raised some very interesting issues, I suspect that almost everyone in your field would use Pearson correlation for your data, and I've commented above to that effect. Sorry, but I don't use R beyond very elementary level and can't offer code. But there will (surely) be a way to convert quarterly series to monthly series in R. For that question, use R help or Stack Overflow, but a reproducible example would help any such question. $\endgroup$ – Nick Cox Apr 5 '16 at 21:04

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