### A Word on Taylor Rules

**Matthew Martin**7/30/2013 10:03:00 AM

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"an equation economists and Fed watchers like to use to see what the central bank’s federal funds rate ought to be given broad conditions in the macroeconomy."Bernstein plugs some numbers and finds that the current federal funds rate "ought" to be negative, which it obviously can't do because of the zero-lower-bound--hence, the Fed needs more "quantitative easing". In what I can only describe as a "teachable moment," John Taylor (inventor of the Taylor Rule concept), plugs the same data into a different Taylor rule and exclaims that the fed funds rate "ought" to be above zero--and hence that the fed needs to raise the rate immediately.

That's not all. Commenter Bob Wyman points us to Mankiw and Krugman (update: links have been corrected), who each plugged the data into yet two more different Taylor Rules and unsurprisingly produced two more different results. We now have a total of four Taylor Rules, and four different predictions for what the fed funds rate "ought" to be. Oh, and here's my favorite part--none of these Taylor rules actually predict the Fed funds rate:

Mankiw is the red line, while Krugman is purple (actual fed funds rate is in green). I suspect that if I poll a dozen more economists, I will get a dozen more Taylor Rules and a dozen more predictions of what the fed funds rate "ought" to be. The whole concept of the Taylor Rule is fundamentally flawed in a variety of different ways.

For one, the Taylor Rule doesn't provide an adequate description of the Fed's actual policy function. As the graph above shows, all of the versions of the Taylor Rule come with a pretty hefty margin of error--in fact, Taylor Rules consistently under-predict the Fed's reaction to recessions.

Second and more importantly, the Taylor Rule functional is not the result of an optimization procedure. John Taylor originally justified his version of the rule on the basis that it provided a close approximation of actual Fed behaviors. As the four examples above demonstrate, however, there's no One True Taylor Rule that can be used to determine the optimal interest rate policy. Here's a generalized version of the Taylor Rule: $$R_{ff}=\pi+\beta_{\pi}\left(\pi-\bar{\pi}\right)+\beta_u\left(\bar{u}-u\right)+2$$ where $R_{ff}$ is the Fed funds rate, $\pi$ is the actual inflation rate, $\bar{\pi}$ is the target inflation rate, $\bar{u}$ is the long-run "nairu" unemployment rate, and $u$, is the actual unemployment rate, while $\beta_{\pi}$ and $\beta_u$ are constants.

It turns out that this Taylor Rule functional tells us almost nothing useful. We know from the data the values of $u$ and $\pi$, and maybe we can estimate $\bar{u}$ though even that is a dubious proposition. Using a proof that involves eigen values and linear algebra that I won't go into, you can show that the optimal Taylor rule has $\beta_{\pi}>0$--this result is famously known as the "Taylor Principle." That is in fact all we can show. The Taylor Rule does not tell us the optimal inflation target $\bar{\pi}$, nor does it tell us the optimal weight $\beta_{u}$ to put on the output gap. And aside from the constraint in the Taylor Principle, the rule doesn't even tell us what weight to put on inflation. Neither Taylor, nor Bernstein, nor Krugman, nor Mankiw derived any optimality conditions to get their Taylor rules. These choices were all completely, utterly arbitrary.

To my knowledge, no one has actually derived a socially optimal interest rate policy function, so we really don't know what one would look like. It probably doesn't resemble the quaint linear function Taylor proposed. What we do know, however, is that the Taylor Rule functional does not maximize price and output stability. Future Federal Rerve Chairwoman Janet Yellen derived an optimal control function which she outlined in this speech.This technique derives an interest rate policy function that minimizes a the output gap and inflation weighted by a "quadratic loss function," which is to say it minimizes the variance of inflation and output--and the results are not consistent with the Taylor Rule prediction. What remains to be done in future research is to apply these optimal control techniques to derive a interest rate policy function that maximizes social welfare, rather than minimize an abstract loss function.

Here's the lesson of this teachable moment: we now have the mathematical techniques and computational power to derive optimal non-linear policy functions from the data. We should do so, and stop whining about the Taylor Rule, which is now nothing more than obsolete curiosity in the history of economic thought.

Max7/31/2013 02:29:00 AM"For the purposes of the exercise, the loss function is equal to the cumulative discounted sum from 2012:Q2 through 2025:Q4 of three factors--the squared deviation of the unemployment rate from 5-1/2 percent, the squared deviation of overall PCE inflation from 2 percent, and the squared quarterly change in the federal funds rate. The third term is added to damp quarter-to-quarter movements in interest rates."

This is a pretty frightening "loss function"! Where did 5-1/2% come from, and who decided that the quarterly change in the Fed Funds rate was something important?

Matthew Martin7/31/2013 07:08:00 AMAs for why they included the federal funds rate in the loss function, I agree that this is odd. I suppose they think that too much interest rate volatility would spook investors and result in lower growth, but I don't think that this is justified by any science. I will say, however, that the entire concept of the loss function is somewhat arbitrary--ideally we would be using optimal control methods to derive the interest rate policy function that maximizes the social welfare function (that is, summed utility across households), not merely some made-up loss function. However, it is worth noting that the Fed charter includes a "dual mandate" about price stability and maximum employment, but nothing about maximizing social welfare.