Thread by @jasonfurman

A thread on the perils of interpreting moving averages that will be of interest to almost no one. Also has implications for annual averages which is a slightly more important topic. This fleshes out one aspect of my error with the Atlanta Fed wage data.

https://twitter.com/jasonfurman/status/1492200678487437319?s=20&t=J8NQYCERNmMVcE4E0l_Xkg

I'll use a made up hypothetical of the following wage data. What would you say happened to wages in the pandemic period (which for expositional simplicity I'm drawing as starting in 2020)?

I think you would say wages fell in the pandemic.

The right calculation for what happened in wages in the pandemic would be:

Wage growth in pandemic = (wage level Dec 2021 / wage level Dec 2019).

In the picture above this is -1% if you annualize (take the square root to get growth per year).

You could also add up all the monthly growth rates during the pandemic (ignoring a small compounding term) & get the same answer:

Wage growth in pandemic = wage growth in Jan2020 + wage growth in Feb2020 + ... + wage growth in Dec2021

This is a moving average of the monthlies.

IF you used the same type of moving average in the Atlanta Fed data for 2020 and 2021 then you would actually say that the wages grew 0.1% over the pandemic period of 2020 and 2021.

That is because it is taking a moving average, basically averaging 12 month growth rates going back 24 months so it reflects not just growth rates after January 2020 but before.

You can express the 2 yr moving average that I took from the Atlanta Fed as a showing the growth rate as a function of the individual monthly growth rates (ignoring compounding):

Feb19 + 2*Mar19 + 3*Apr19 + ... + 12*Jan20 + 12*Feb20 + ... + 12*Jan21 + 11*Feb21 + ... + 1*Dec21

If this formula looks weird and unintuitive to you it is because it is weird and unintuitive. It says that what one calls "2020 and 2021" is actually more affected by 6X more affected by what happened in July 2019 (which shouldn't matter) than December 2021.

In fact, 22% of the data used to compute the 2020 and 2021 growth comes from 2019 (should be 0%) and only 7% comes from the last six months (should be 25%).

When applied to actual data this matters a lot because real wages up a lot in 2019 and down a lot in the last six months.

This is why I said that this corrected figure was internally consistent (the wage timing matches the inflation timing) but didn't answer a question anyone should care about (was more about real wages in 2019 than in the last six months).

https://twitter.com/jasonfurman/status/1492200714742939654?s=20&t=zh9kVeFofQweQYgv8iwUjQ

Note that all of the above is about the way I combined two years of Atlanta Fed data. But if you're using their data (or any other moving average) the same applies: the Jan 2022 "12 month" growth rate is heavily affected by what happened during 2020.

Using moving averages is a common & reasonable way to deal with data noise. It is a good way to understand the general direction of travel over a longer period for a slowly evolving series. But is not suitable for answering questions about what happened in a particular period.

When there is no data noise, I try to avoid moving avgs & the closely related annual average because of all of the weirdness. For GDP, for example, I always prefer to look at growth from Q4/Q4, just like inflation for 2021 is generally reported as Dec/Dec.

https://twitter.com/jasonfurman/status/956891837293309953?s=20&t=zh9kVeFofQweQYgv8iwUjQ

(Small caveat: Even w/o data noise you sometimes might care about annual averages if you're interested in a cumulative, like GDP down 2020-Q2, even if it rebounded by 2020-Q4 you might want to show the losses for the year.)

I've seen better threads/pictures explaining the annual average vs. Q4/Q4 growth rate issues but I can't remember where, on the off chance you got this far and wrote one LMK and I'll add it here.

Otherwise, this is now FIN.