One of the oddest things to happen in the past year has been the fight between John Taylor and his Taylor Rule.

It started on April 27, 2009, when Krishna Guha wrote an article: Fed study puts ideal interest rate at -5% "> :

The ideal interest rate for the US economy in current conditions would be minus 5 per cent, according to internal analysis prepared for the Federal Reserve's last policy meeting. The analysis was based on a so-called Taylor-rule approach that estimates an appropriate interest rate based on unemployment and inflation. A central bank cannot cut interest rates below zero. However, the staff research suggests the Fed should maintain unconventional policies that provide stimulus roughly equivalent to an interest rate of minus 5 per cent.... The assessment that the US central bank needs to provide stimulus equivalent to a substantially negative interest rate is unlikely to have changed ahead of this week's policy meeting.... [M]any Fed officials expect they may well keep rates near zero for another 18 months to two years and some might see value in making this more explicit.... The last meeting saw the Fed buy long-term treasuries for the first time in decades. The large initial impact of the move on markets is no longer visible, but officials think the policy was reasonably successful. Previous staff analysis suggested the $300bn purchase would reduce the yield on 10-year treasuries by 25-35 basis points, and officials think the rate today is about this much lower than it would have been if they had not started buying. Further purchases are possible, particularly if the Fed again downgrades its economic forecasts. The staff analysis comparing unconventional operations to interest rate cuts suggests more might be needed anyway...

I believe that Krishna Guha was reporting on a study like that of Glenn Rudebusch's San Francisco Fed "The Fed's Monetary Policy Response to the Current Crisis" http://www.frbsf.org/publications/economics/letter/2009/el2009-17.html with this chart:

which is in turn based on a 2006 Rudebusch paper in the International Journal of Central Banking http://www.ijcb.org/journal/ijcb06q4a4.htm containing Rudebusch's estimated Taylor Rule:

Then two weeks later John Taylor, in his speech "Systemic Risk and the Role of Government" http://www.frbatlanta.org/news/CONFEREN/09fmc/taylor.pdf, fires back, saying not so:

According to a widely cited article4 appearing in the Financial Times two weeks ago, the Fed’s Taylor rule calculations show that the interest rate should be -5 percent. The article was based on a leaked report from the Fed.... [T]he calculations are way off. The Taylor rule specifically says that the interest rate should be one and a half times the inflation rate plus a half times the GDP gap plus one. Whether you average a broad based GDP inflation index over the past year, as I originally suggested, or whether you use core inflation rates, the inflation rate is not less than 1 percent at this time; it is closer to 2 percent, but let’s suppose the Fed takes it as 1 percent. The GDP gap seems to be around minus 4 percent. Now, if we put those numbers into the rule, we get 11⁄2 times 1, plus 1⁄2 times -4, plus 1, which equals .5 percent not -5 percent. The Fed’s calculation reported in the Financial Times has both the sign and the decimal point wrong. In contrast my calculation implies that we may not have as much time before the Fed has to... raise the [federal funds] rate...

At the time it seemed to me that there were two things wrong with Taylor's speech:

With a May unemployment rate of 9.4% and an Okun's Law coefficient of 2, an output gap of 4% would imply a NAIRU of 7.4%--and yet every estimated NAIRU I know of is somewhere close to 5-6%, implying an output gap not of 4% but of 7% to 9%.

There is a key ambiguity in what a "Taylor Rule" is: is it the original rule with the original parameters set out by John Taylor in 1993, or is it a rule of that form that central banks happen to follow?

When Glenn Rudebusch writes:

A rough guideline for setting the federal funds rate that captures the Fed's behavior over the past two decades is provided by a simple equation that relates the funds rate to the inflation and unemployment rates. This equation is obtained by a statistical regression of the funds rate on the inflation rate and on the gap between the unemployment rate and the Congressional Budget Office's estimate of the natural, or normal, rate of unemployment. The resulting empirical policy rule of thumb—a so-called Taylor rule—recommends lowering the funds rate by 1.3 percentage points if core inflation falls by one percentage point and by almost two percentage points if the unemployment rate rises by one percentage point...

This is the rule:

i = 2.4 + 1.39π + 0.92ygap

that Glenn found gives the best fit to actual Fed policy between 1987 and 2005. He calls this a Taylor Rule--and it is this rule that says that the federal funds rate right now should be -5%.

John Taylor, by contrast, says that the Taylor Rule is the rule:

i = 1 + 1.5π + 0.5ygap

back in 1993 that he found fitted Fed behavior from 1987-1992

So we have two different questions and answers.

Glenn Rudebusch asks: "If the Federal Reserve could set the Fed Funds rate negative and were following the same reaction function it follows in the late 1980s, 1990s, and early 2000s, where would it want to set the Federal Funds rate?" And the answer is: "-5%.":

John Taylor asks: "If the Federal Reserve were to set the Fed Funds rate according to the particular reaction function that in 1993 I fittd over 1987-1992, where would it want to set the Federal Funds rate?" And the answer is: "0.5%."

Seems to me that John Taylor in the long run wants "Taylor Rule" to mean any statistically-fitted reaction function in which interest rates respond to inflation and the output gap and not to the one rule he fitted over 1987-1992, Abd us seens to me that a rule fitted over 18 years should probably be preferred over one estimated over 6 years in a time span that ended more than 15 years ago. But chacon a son gout...

This is worth revisiting because recently John Taylor has renewed the argument that the Federal Reserve should be massively cutting back on reserves and raising interest rates, if not now then at least soon:

John Hilsenrath: Q&A: John Taylor on His Rule: Hilsenrath: There’s been a lot of debate about how to use the Taylor rule in this kind of environment. What is your view of what the Taylor rule is saying given where the economy is? Taylor: There are disagreements, but if you look at average estimates of inflation and the GDP gap you get a fed funds rate of close to zero. That’s not based on a forecast of inflation or an excessive GDP gap. It is not perfect, but that is where we are. Some people are saying it should be minus five. Hilsenrath: Using your rule. Taylor: Not really. As far as I’ve been able to tell you have to change the rule to get that. For example, you have to have a higher coefficient for the output gap to get that. And some people use forecasts. They don’t use the current level. They stick forecasts of inflation or the output gap into their models. They might extrapolate out what the forecast is saying. Some of them are forecasting declining inflation. They put that in and that will lower the rate too. In terms of the actual Taylor rule, I don’t see how anyone can get minus 5 from that. Hilsenrath: One could argue that using a forecast is right because you want policy to be forward looking. Taylor: No I disagree. In fact the original way the Taylor rule was formulated was specifically to say that you should get the best estimate of where you are right now and then you react to it this way. It already builds in forecasts. After all policy makers have to use current information, whether it’s commodity markets or current GDP. They can’t see the future. They have to project out from current observed things. You put the current observed things into the rule. That’s all you have anyway. And then you see what the best coefficients are to get the fed funds rate. I don’t think it’s correct to say you have to put forecasts in...

It seems to me that the best way to avoid confusion is to redefine "Taylor Rule": henceforth "Taylor Rule" should mean only i=1.5π+0.5ygap, and we need to find some other name for reaction functions fitted to central-bank behavior.