CBO’s Method for Estimating Potential Output

From A Summary of Alternative Methods for Estimating Potential GDP (PDF):

CBO’s Method for Estimating Potential Output

CBO’s estimate of potential output is based on the framework of a textbook model of long-term economic growth, the Solow growth model... CBO estimates trends —that is, removes the cyclical changes—in the labor and productivity components by using a variant of a relationship known as Okun’s law.

Okun’s law postulates an inverse relationship between the size of the output gap (the percentage difference between GDP and potential GDP) and the size of the unemployment gap (the difference between the unemployment rate and the natural rate of unemployment). According to that relationship, actual output exceeds its potential level when the rate of unemployment is below the “natural” rate of unemployment; actual GDP falls short of potential when the unemployment rate is above its natural rate.

For the natural rate of unemployment, CBO uses its estimate of the nonaccelerating inflation rate of unemployment (NAIRU). That rate corresponds to a particular notion of full employment—the rate of unemployment that is consistent with a stable rate of inflation. The historical estimate of the NAIRU derives from an estimated relationship known as a Phillips curve...

CBO estimates an Okun’s Law relationship for hours worked and total factor productivity (TFP). It uses regression equations that link each variable to the same set of explanatory variables (including the unemployment gap) to capture the effects of fluctuations in the business cycle. It also uses several time trends, which constrain the growth of the potential variables to a constant rate over one or more specified historical periods. CBO then calculates the potential levels of hours worked and TFP from the predictions of the equations when the unemployment gap is set at zero. Those potential levels are combined with the capital input to compute potential GDP.

1. Start with the Phillips curve.
2. Observe the current relation between inflation and unemployment.
3. Use this relation to estimate the NAIRU.
4. Use the NAIRU as "the natural rate of unemployment".
5. Compare the natural and actual rates of unemployment.
6. Call the difference "the unemployment gap".
7. By Okun's law, the output gap is twice the size of the unemployment gap.
8. Calculate the output gap.
9. Add the output gap to actual output, to get potential output.

There's more to it. But this is where the Potential Output number comes from.

The Myth of Robust Analysis

// UPDATE 12 april 2013: See my follow-up post, which invalidates this one.

If you mix red pigment into white paint, the paint gets red.

Yesterday we looked at Chart 2 from The Myth of 'Jobless Recoveries'. Today we consider Chart 1. Laurence Ball, Daniel Leigh and Prakash Loungani write:

Chart 1 illustrates the fit of the estimated Okun’s Law by plotting the unemployment gap (the gap between unemployment and the natural rate) against the output gap (output relative to potential). The relationship is very tight. No year is a major outlier in the graphs.

Graph #2: Ball, Leigh and Loungani's Chart 1.
Okun’s Law, 1948-2011 (Annual US data)

Yesterday, to figure out Chart 2, I reorganized a formula to make it compatible with Okun's law. Chart 2, as I understand it, compares actual unemployment to some calculation of the output gap and the natural rate of unemployment, and finds an amazingly close match for 65 years.

After re-arranging terms, Chart 2 compares the unemployment gap to the output gap. Today's focus, Chart 1, compares the unemployment gap to the output gap directly, one gap per axis. Both charts from the Econbrowser post consider the Okun's Law relation. Well, sure: The topic of that post is the validity of Okun's law.

The Econbrowser post includes both charts under the heading "U.S. evidence: Fit & Stable". To me this means that Laurence Ball of Johns Hopkins University, and Daniel Leigh and Prakash Loungani of the IMF, consider their graphs evidence of the validity of Okun's law. I do not.

Why not? Look again at what they wrote:

Chart 1 illustrates the fit of the estimated Okun’s Law by plotting the unemployment gap (the gap between unemployment and the natural rate) against the output gap (output relative to potential). The relationship is very tight.

They compare the unemployment gap to the output gap and find astonishing validity. That is to say, there is a tight relation between two gaps: the gap between actual and hypothetical unemployment, and the gap between actual and potential output. A very tight relation.

But here's the thing. To figure potential output, the Congressional Budget Office uses the unemployment gap. They take that gap and stretch it to fit over actual output. That gives them the output gap. The output gap looks like the unemployment gap by design.

Then they use actual output and the output gap to estimate potential output. So when you look at potential output, part of what you're seeing is the unemployment gap.

Just as Milton Friedman worked inflation into his "money relative to output" (by using inflation-adjusted output in the denominator), the CBO works the unemployment number into the calculation of potential output. Just as Milton Friedman's results are skewed to look like inflation, so is the CBO calculation of potential output skewed to look like unemployment.

In the case of the CBO, this is not a problem. CBO is modeling potential output, based on the natural rate of unemployment. There's nothing wrong with that. But for people who want to use potential output in other calculations, it is important to remember how potential output is figured. Otherwise you can end up doing bad arithmetic, just like Friedman did.

Laurence Ball, Daniel Leigh and Prakash Loungani take potential output and work it backwards. They show similarity between the output gap and the employment gap. But you should expect to see this similarity, just as you should expect to see similarity to inflation in Milton Friedman's numbers. But the similarity does not mean what Ball, Leigh and Loungani say it means. They say it is evidence Okun's law is valid.

It is not evidence. The output gap looks similar to the unemployment gap because the output gap has the unemployment gap figured into it: The paint is red because it has red pigment mixed in.

All they have done, really, Ball and Leigh and Loungani, is check the CBO's numbers. CBO starts with unemployment and ends up with potential output. Ball and all start with potential output and end up with unemployment numbers. If Ball's final numbers match the numbers CBO started with, it only means that nobody made a mistake in their arithmetic. It doesn't mean Okun's law is valid.

I suppose if they use Potential Output from some other source, calculated some other way, then my objection may be the thing that's not valid. That could be. But if they identified their data sources in the post, I didn't see it. And since their charts show an unbelievable validity, I cannot trust their charts.

// Update 6:26 AM 17 March 2013

In response to an email, Laurence Ball replies:

In our main results, we calculate potential output and output gaps separately from unemployment gaps and the natural rate, using the Hodrick-Prescott filter in each case. You're right that using CBO output gaps is circular--we mention that in the paper.

The Myth of Jobless Recoveries

Okun's law asserts a relation between the rate of unemployment and the rate of economic growth. The higher the unemployment, the lower the growth; and the lower the unemployment, the higher the growth. Makes sense. But Okun's law is stronger than I let on: It associates a one-point change in unemployment with a two-point change in output. That's pretty specific.

A recent guest post at Econbrowser reported the testing of Okun's Law, and found that the two-to-one relation holds good. In The Myth of 'Jobless Recoveries', Laurence Ball, Daniel Leigh and Prakash Loungani write: "Fifty years after Okun’s paper, we find that this relationship fits very well, including during the Great Recession." Okun's Law, they report, is "strong and stable". They offer this graph:

Graph #1: Ball, Leigh and Loungani's Chart 2.
Actual and Fitted Unemployment Rate, US, 1948Q2-2011Q4.

There are two lines on the graph. One of the lines is the actual unemployment rate. The other is an estimate of the unemployment rate which they got by following Okun's law backward from output. The two lines are so close together they look like one line. Ball, Leigh and Loungani offer this as evidence Okun's law is strong and stable.


Ball, Leigh and Loungani say that their chart "shows the tight fit between actual unemployment and the estimate based on Okun’s Law." Statements like that catch my eye. I like to see if I can duplicate the results. Duplication tests both the selection of data and the calculation of values. It helps me understand the calc.

The "actual" unemployment rate is a time series I can find at FRED. That part is straightforward. What I have to work out is "the estimate based on Okun’s Law." Ball, Leigh and Loungani state the law clearly:

The textbook version states when U.S. output dips one percent below its potential, unemployment rises above its natural rate by about half a percentage point.

In other words,

Formula 1.

But they say their graph shows "actual" unemployment (the UNRATE, I presume) and the "fitted unemployment rate from Okun specification". Plus, their graph looks like the UNRATE graph, with the high peak around 11 and the last peak around 10.

To get "actual" by itself, I added the Natural Rate of Unemployment to both sides:

Formula 2.

I plotted that at FRED:

Graph #2:Actual Unemployment (blue) and the "Fitted" Number (red)

Pretty good match. I don't like it though, because to make it match I had to use two vertical axes with different scales. Maybe that's because of the "two-to-one" thing, which I didn't consider in this graph. Couldn't figure out how to get FRED to do that. (The output values are log values. At FRED I took log values as the last step. So there was no extra step where I could multiply or divide by two.)

(Oops. I left the "times 2" out of my formulas, too.)


I thought it might work better if I left NROU with UNRATE, as in the first formula above. Went back to FRED and came up with this:

Graph #3: The Employment Gap (blue) and the Output Gap (red)
Click Graph for FRED Source Page

For the blue line I started with the actual rate of unemployment and subtracted the natural rate of unemployment. (Thus, where actual unemployment is above the natural rate, the blue line is above zero. Where actual is below the natural rate, the blue line is below zero.)

Because both unemployment series are given in percent values where 5% is presented as 5.0 rather than 0.05, I divided the result of subtraction by 100 to get the decimal value.

Finally, I multiplied by 2 because Okun's law says there is a two-to-one relation between unemployment and output.

For the red line I took potential output and divided it by "real" GDP. (Thus, where real output is below potential, the red line is high. Where real output is above potential, the red line is low.) Then I applied FRED's "natural log" transformation.

You see the result. The two lines are a good match to each other, and the graph is similar to the "Chart 2" graph. My numbers are lower, because I'm subtracting the natural rate from the actual rate. But both sets of numbers are lower on this graph, and the two sets still match up well.

Is the similarity "evidence" that Okun's Law holds good? Come back tomorrow...

Imagine That!

Both potential output and the natural rate of unemployment are imaginary numbers. Estimates. But not like estimates of actual unemployment and actual output, which are based on actual conditions. As Arthur Okun wrote in 1962:

The quantification of potential output -- and the accompanying measure of the "gap" between actual and potential -- is at best an uncertain estimate...

According to an old FRBSF Economic Letter,

The natural rate is the unemployment rate that would be observed once short-run cyclical factors have played themselves out.

(Emphasis added.)

Of the NAIRU, Bill Mitchell says simply, "it is not observed."

The NAIRU. That's the number CBO uses to calculate the natural rate of unemployment:

For the natural rate of unemployment, CBO uses its estimate of the nonaccelerating inflation rate of unemployment (NAIRU).

We're dealing with hypothetical quantities here. The unobserved NAIRU is used to calculate the natural rate of unemployment. The natural rate of unemployment is used to calculate potential output.

Oh... and potential output? According to William Gavin, potential output is "a theoretical concept that means different things to different people."

An Interesting Comparison

Via Random Eyes: Corporate Profits and Personal Income, pinned at 1960:

FRED Graph#4kX

Morrisey lines

After looking again at yesterday's graph, it occurred to me to ask: How much of the revision to Potential GDP is in the past, really, and how much is in the future? Despite Altig and Gavin's graphs, and the 1977 ERP, it looks like most of the revision is in the future.

So I copied the graph from yesterday and changed it by deleting all the data from after the date-of-estimate. For each date in the legend, I deleted all the numbers after that date for that series. I stripped away the future. What's left does show a bit of revision, but not a lot.

Graph #1

William Gavin is right: Estimating the past is a lot easier than estimating the future.

Still, the graph reminds me of high school math class. Mrs. Morrisey would draw a diagram on the chalkboard and then, when her lines didn't intersect like she wanted, she would go back-and-forth with the chalk and make a good thick line to show the intersection she was trying to show. My buddy called them Morrisey lines.

// Google Drive Spreadsheet

Cooperating with Alfred

I finally figured out where to get the "vintage" data that ALFRED offers. After you find the series you want (in my case, Potential GDP) and after you get a graph on the screen, that's when you click download.

ALFRED then gives you a selection window with several old versions of the data -- all the versions they have, I guess. You just highlight the vintages you want (in my case, all of them) and the thing is ready to send you a file (in my case, a zipped Excel file).

I wanted to look at Potential GDP because, as you know, it has been revised down.

Turns out, the "real" PGDP data from ALFRED comes in not only several vintages, but also several different "base year" series. To whit:

Data Values	Start Year
Billions of 1982 Dollars	1991-01-30
Billions of 1987 Dollars	1992-01-22
Billions of Chained 1992 Dollars	1997-01-28
Billions of Chained 1996 Dollars	2000-01-27
Billions of Chained 2000 Dollars	2004-09-13
Billions of Chained 2005 Dollars	2010-01-26

If you select one year's data from all the different vintages and make a graph of it, the graph shows an upward-stepping pattern. Upward-stepping, because every time they pick a different base year and adjust for inflation, the number gets bigger. So, if I want to compare vintages across base years, I'll have to adjust for base years. I didn't get into any of that today.

Just for the base-year 2000, ALFRED offers eleven different vintages. The oldest is dated September 13, 2004, and the newest August 27, 2009. That period includes the time of the financial crisis, so I figured I'd look at that.

Graph #1

Move your mouse up and down over the legend to highlight different lines on the graph. You will see that as the mouse moves down toward newer vintages, the highlighted Potential GDP line on the graph is most often lower also.

My Google Drive Spreadsheet is available.

8 + 2 = 10

Graph #1 Source: CBO PDF February 2013

Graph #2 Source: Wikipedia (CBO 2009)

Pretty easy to find graphs that looks like these, graphs that show GDP climbing back to its potential and closing the gap.

You know why the graphs show the gap closing? It's because the people who come up with the numbers just assume the gap will close within ten years! As a CBO background paper observes:

CBO assumes that any gap between actual GDP and potential GDP that remains at the end of the short-term (two-year) forecast will close during the following eight years.

Unfortunately, while the graphs show GDP recovering, there are people who say that GDP is *NOT* going to climb back up to potential. Jim Bullard says it:

I want to now turn to argue that the large output gap view may be conceptually inappropriate in the current situation. We may do better to replace it with the notion of a permanent, one-time shock to wealth...

The wealth shock view puts a different expectation in play. The negative wealth shock lowers consumption and output. But after the recession ends, the economy simply grows from that point at an ordinary rate, neither faster nor slower than in ordinary times. It is more like an earthquake which has left one part of the land higher than another part. There is no expectation of a “bounce back” to a higher level of output after the recession ends. This is closer to what has actually happened since mid-2009. Output has grown at a moderate rate, but not a rapid rate, since the recession ended.

Thomas Hoenig seems to have said it even before Bullard.

And Scott Sumner is clearing his throat:

In 2009 I advocated going all the way back to the old trend line. I currently favor going about 1/3 of the way back. If we keep on the same track for a few more years I’ll through in the towel and advocate starting a new 5% trend line from where we are.

Stranger than fiction, people who are not saying what Bullard says, people like William Gavin and David Altig, are showing graphs where instead of GDP moving up to trend, the trend of potential output keeps getting lower!

Those graphs that show GDP will recover soon, well, "soon" is always just a little out of reach, and even the people who present those graphs seem not to believe them.

The Consequences of Denial

From mine of 13 Feb 2012:

Anyway, Bullard. He says if we overestimate potential output and set policy by it, we will encourage excessive demand and we will get inflation like we got in the 1970s. (And, he says, inflation is already above target.)

Everybody else says the economy is not growing enough, and we don't have jobs enough, and unemployment is too high, and demand is insufficient, not excessive.

How can there be such a difference in views? I think the trouble arises from the way we explain inflation. Here's Mitchell again:

Inflation is driven by nominal aggregate demand growth that exceeds the capacity of the economy to respond in real terms – that is, to increase output.

Too much money chasing too few goods. For Billy, as for Milton and Anna, inflation is caused by excessive demand -- by demand "that exceeds the capacity of the economy to respond". Demand being excessive relative to potential output is the cause of inflation, they say. Exactly what Jim Bullard says.

If you think of inflation along those lines, and you admit we're getting inflation already, then you end up thinking that "the capacity of the economy to respond" must somehow have been crippled. You end up thinking that there must have been a sudden drop in potential output. Exactly what Jim Bullard says.


But all we need -- if we wish to undermine Jim Bullard's argument -- is to realize that demand-pull isn't the only inflation story there is. There is also a cost-push inflation.