# Do U1 to U6 Tell Us Anything?

It's been a while since I've written here. Sorry about that. There's a lot going on at school for me right now, and you might have noticed that it's been a particularly busy few months for Wonkbook, the public-policy newsletter I write with Ezra Klein. End of self-promotion.

Thankfully, I have a bit of time tonight for a thought about labor markets. The first Friday of every month -- or, at least months when the government doesn't shut down -- the Bureau of Labor Statistics releases what's called the "Employment Situation Report." It contains the normal unemployment rate, at 7.3 percent in October, and a host of variant unemployment rates that either include or exclude various parts of the labor force. The most expansive definitions of unemployment include part-time workers who say they would like full-time work; the most narrow define unemployment as involuntary joblessness in excess of 15 weeks. They go with the names "U1" through "U6."

Economists and the news media really love to dissect the report. One of the ways that's done is by looking at how these variant rates move with or against the normal rate of unemployment. U5 and U6 rose even though U3 fell, and so on.

This could be useful if U5 told us something different than U3. That question is never really answered. To a skeptic, we might just be comparing the realizations of some random process.

So here's the question: Is there really any information at all in the variant unemployment rates? My answer: less than you think.

U1 through U6 don't collapse to one single measure of unemployment -- more on what I mean by that in a moment -- but the six measures don't tell us six different things. In fact, we can reduce most of what's going on in unemployment to just two dimensions.

First, take a look at U1 through U6.

See how they move together? To an economist, this suggests that the unemployment rates could be linear combinations of each other. For instance: The U4 rate, which includes so-called "discouraged" workers, could just be twice the normal (U3) rate of unemployment.

This particular example isn't true, but the general concept is. It turns out we can think of each measure of unemployment as equal to a constant plus another constant times a global measure of unemployment.

Written out for the math people: *u _{i,t} = a_{i,t} + b_{i,t}k_{i}u_{t}^{*}*, where

*u*is the specific measured rate of unemployment,

_{i}*a*,

_{i,t}*b*and

_{i,t}*k*are regression parameters, and

_{i }*u*is the global measure of unemployment. A note on the subscripts: "i,t" means specific to that measure and that time, "t" means specific to that time but constant across measures, and "i" means specific to that measure but constant across times. For people who read this blog too much, you'll notice that this is the same kind of analysis I did for corporate-bond spreads.

_{t}^{*}If that math was gibberish to you, that's okay, and all you need to know is that we're using linear regression to see how the unemployment rates relate to each other.

The first thing we do is take the average of each measured unemployment rate over the sample window, which happens to be 1994 through 2013. In case you're curious, the average normal unemployment rate over that period was 6.0 percent. With the averages in hand, I divide them all by the average U1 rate of 2.4 percent.

We can think of the result as the average relative rates of unemployment -- for instance, if you want to guess the U6 rate, your best guess is 4.4 times the U1 rate. If you were to graph the unemployment rates at any particular time as points, with the actual rates on the vertical axis and the average relatives on the horizontal, this method will give you a line.

Now it gets pretty simple, and you just fit equations of a line at each time *t,* giving you the values for *a *(the intercept) and *b* (the slope). When I said I could collapse the unemployment rates down to two dimensions, those are the two I meant.

Here's the picture of *a* and *b*:

You can think of *a* as reflecting labor-market conditions at the narrow side and *b *as reflecting the farther-out reaches of unemployment. The picture above, with *a *falling faster than *b*, implies that narrower definitions of unemployment are improving more quickly than wider definitions of unemployment. It's a signal, in other words, that labor markets are probably looser than the normal measures suggest -- and that various degrees of underemployment are a real, serious concern.

This model of the unemployment rates, it turns out, fits well enough as to explain almost all of the behavior of unemployment.

A word on that, however: The errors are autocorrelated. For the econometricians in the audience, the average Durbin-Watson statistic across the measures of unemployment is, yikes, 0.12. For those who aren't econometricians, what I mean is that when the model misses, it's likely to miss persistently, which is why I won't quote some sort of measure of fit here.

This post is a bit longer than intended, so here are some conclusions to take home: (1) It's pretty pointless to be discussing U1 through U6, as most of the information is contained in any two measures, and (2) this analysis shows that labor markets are improving for the most employable much more quickly than they are for marginal workers, which is not surprising but is just a sad reality of life in 2013.

The next Employment Situation report comes out in seven days. It might help to keep this all in mind.