Evan Soltas
May 30, 2012

If These Seasons Could Talk

...they'd tell us that seasonally periodic trends in any economic variable consistent over long time spans may be deeply informative as to the structural factors driving secular changes in said variable.This post applies this insight to civilian labor force participation among 16- to 19-year-olds (see FRED's graph here), but one could theoretically reapply my method to any other seasonal variable which changes its seasonal shape.

Admittedly, I don't know very much yet about econometrics and am trying to learn -- and towards that end, what I show in this post is that by using several discrete-time Fourier transforms to decompose seasonal variation at different points in a time series, we can discern "regimes" in the seasonal variations. That is, the seasonal pattern of variation in a variable tends to be relatively consistent for long periods of time and then break abruptly. These regimes tend to correspond to identifiable economic or social time periods or levels of the economic variable, suggesting the interrelation of seasonal and secular change in the variable.

(Note: This post is easier understood some knowledge of Fourier series and analysis, but suffice it to say that I am analyzing the periodic behavior of a time series, and noticing that the periodic behavior tends to be strikingly and abruptly different at different times, and these switches seem to be related to the de-seasonalized variable itself.)

Let's also define what I mean by regime: a section of the time series which is substantially different from other sections of that time series. (See here and here for more on regimes in econometrics.) Also, I pause to note here that it is my understanding that econometric tools for seasonal de-trending are significantly more sophisticated than merely Fourier analysis -- I'm not quite there yet. In fact, a few weeks ago I employed the first tool discussed in the just-linked presentation, seasonal averaging, to crudely de-trend a hiring index; see here. But the point is that a creative reapplication of this more basic form of seasonal de-trending has eome untapped power to reveal regime changes.

Step one is to fit a Fourier series to a given year.Then one applies this Fourier series to all of the years in the time series to remove all seasonally periodic variation. This, however, fails to remove actually all of it, as the pattern of periodic variation -- the Fourier series -- is not the same for the entire time series. To determine when a new fit is needed, one tests the one-year rolling correlation of the Fourier-detrended data series with the original. A low correlation indicates that the time series has been successfully de-trended at that point in time. When one looks at the graphs of these correlations over time, one observes that the the non-correlation exists for a distinct window of time, beyond which there is a sharp reversion to high correlation between the "de-trended" and the original, implying that the de-trending beyond that point is not successful, and by extension that the seasonal trend fit from one year does not hold true beyond that point.

I assume that this variable is seasonally periodic, i.e. a period of 12 months. To find the distinct regimes, I picked a year, found the Fourier series for that year, and then when I completed the correlation fit test described above, I picked another year well outside the de-trended window of time. I repeated this process until the coefficient of determination for rolling correlation was always below 0.5, picking an arbitrary threshold -- i.e. such that the seasonal fit explained at least half of the variation. Too low of a threshold will create excessive numbers of regimes where the regimes are not significantly different; too high of a threshold will fail to distinguish sufficiently between regimes.Through this process, I ended up with 5 regimes which collectively characterize the seasonal variation of the time series, which goes from January 1948 to April 2012. My 5 regimes were fit from the years 1957, 1969, 1997, 2008, and 2011.In the graph above, which I reproduced at the beginning, you see the ultimate point of this line of inquiry -- not merely to determine the regimes, but rather the meaning of these regimes. As a note, when the regimes flutter, as they do a few times, especially when the variable is experiencing fast secular change, you're best off assuming that the regime is the dominant one in that section of the time series -- at such points, the fit of any Fourier series will get swamped by the secular change, and it will be "hard" for the correlation to determine which regime it is.

But notice that when youth labor force participation is high in the 70s, 80s, and 90s, my program determines also a specific and solid seasonal regime (blue) for most of that window of time. Most of the 60s is its own regime (yellow). As participation began to decline in the 2000s, the regime shifted (red); now that participation has appeared to stabilize at extraordinary low levels, the regime has shifted again (purple). I'm not quite sure what green is, although it was necessary because the seasonal behavior in the late 90s is otherwise unusual and above the threshold, as if it is some transitional phase before the more enduring red, but the green regime also seems to pick up the most errors, the flickering, which suggests to me that the green regime is faulty and probably negligible in all other locations.

What this reveals is the interrelation of seasonal and secular, i.e. structural, change in this variable, suggesting that economic and social influences generate a regime which determines both. Hopefully, this opens up new avenues for analysis of seasonally-volatile phenomena which have underlying secular change -- what I think could be an excellent new tool to understand the behavior time series.