Sunday, December 15, 2013

Phillips Curves and Fisher Relations

I think most of us know about Phillips curves. The typical story that goes with the Phillips curve is something like: "When there is a lot of slack in the economy and unemployment is high relative to the 'natural rate,' demand is low, and firms don't want to raise their prices, so inflation is low. When the unemployment rate is low relative to the 'natural rate,' inflation should be high." Of course it's well known that you have to work hard to see that in the data. For example if we plot pce inflation vs. the difference between the unemployment rate and the CBO natural rate of unemployment (quarterly, 1955-2013), we get:
Can't see the Phillips curve there, right? We think there are good reasons for that. There are various shocks hitting the economy from quarter to quarter - some large, some small. Some of those shocks make unemployment and inflation move in oppposite directions, some make unemployment and inflation move in the same direction. That will produce some of the scatter in the chart. Further, in the long run, inflation should have little or no effect on unemployment (Milton Friedman argued for no effect). The modern search theory of unemployment seems to tell us that higher long-run inflation is associated with higher long-run unemployment. Indeed, if we do a reverse regression with the variable on the horizontal axis in the chart serving as the dependent variable, we can fit a long-run Phillips curve to the data, and that's the regression line in the chart. You can see there's not much of a slope to it.

As I mentioned above, if we work hard enough we can find a Phillips curve relationship. For example, if we think there is an episode where monetary factors were important, then we should see the Phillips curve over that period, as monetary shocks tend to move inflation and the unemployment rate in opposite directions in the short run. So, consider the period of time between fourth quarter 1980 and third quarter 1982, when Paul Volcker was using monetary policy to bring the rate of inflation down. For that period, we get:
Of course, economic theorists warned us about the Phillips curve. Even if we think we see a strong relationship in the data, we need theory to understand what it means. It is possible to make serious policy errors if we treat an observed statistical relationship as part of the structure of the economic model which is going to be used to guide economic policy.

One of the more puzzling aspects of New Keynesian economics is the enthusiasm for bringing the Phillips curve back into the mainstream. After the 1970s, the Phillips curve had a low profile in academic work, but has had a resurgence since the 1990s in the form of the "New Keynesian Phillips curve." If you know how the discourse has evolved in central banks over time, you'll also know that, whatever was going on in the economics journals, central bankers never lost their affection for the Phillips curve. That's part of what makes New Keynesian economics attractive for them.

In part, the Phillips curve is used in central banks for forecasting. Indeed, that tendency is not confined to central banks, as the note attached to the CBO natural rate series in FRED indicates:
The short-term natural rate is used to gauge the amount of current and projected slack in labor markets, which is a key input into CBO's projections of inflation.
My best guess is that the Federal Reserve Board's inflation forecasts, which seem to rely on the FRB/US model, or other mechanisms kept secret from those of us on the outside, put significant weight on output gap measures. You can see that in published Fed inflation forecasts, which typically show inflation well below the 2% inflation target in the immediate future, and reverting to 2% over a period of three years or more, because they view the output gap as high and falling.

So, policymakers seem to think that the output gap, say as measured by the deviation of the unemployment rate from the natural rate, is useful in forecasting inflation. But to comment on usefulness, we have to run a horse race. So, let's set the bar really low. Suppose I am a very lazy forecaster, and my forecast for next quarter's inflation rate is this quarter's inflation rate. If the output gap is useful in predicting inflation, then if we deliver output gap measures to our lazy forecaster, he or she should be able to make a better forecast. One indication of what the lazy forecaster has to gain is that, if he or she ignores the output gap measure, he or she should make forecast errors that are correlated with the output gap. If the output gap is high, he or she will tend to miss on the low side, and vice versa if the output gap is low. So, let's check that out.
As you can see, the lazy forecaster's errors are uncorrelated (the correlation coefficient is -.05) with the difference between the unemployment rate and the natural rate. In this sense, a monkey can forecast inflation as well as an economist armed only with a Phillips curve. This is in the spirit of what Atkeson and Ohanian told us long ago.

Perhaps more embarrassing for Phillips curve enthusiasts is what is going on in the recent U.S. data. The last 9 quarters of data (2011Q3 to 2013Q3) looks like this:
So, if I lie on my left side and look at that, it's a nice Phillips curve.

Whether central bankers are embarrassed is not clear. They certainly are puzzled, though, as this WSJ article indicates:
Mr. Bullard has also underscored that even as the economy’s outlook has improved, inflation continued to undershoot the Fed’s 2% target, and central bankers don’t have a good explanation for why this is happening.

Conclusion: If I want to find a Phillips curve relationship in the data, then through some process of specification searches, Bayesian estimation with alternative priors, or whatever, I will be able to find it. If I'm looking hard for the Virgin Mary, I can find her in a grilled cheese sandwich. But Phillips curve thinking isn't helping monetary policy right now. It just makes people puzzled.

So, my attempt in what follows is to reduce puzzlement. While the Phillips curve can be hard to find in the data, the Fisher relation is not. From the same data set that produced the first chart, we get:
So, over the long run, there's a clear positive correlation between the nominal fed funds rate and the pce inflation rate. Irving Fisher taught us that, in credit markets, borrowers and lenders care about real rates of return. Thus, there should be an inflation premium built into the observed nominal interest rate - if the inflation rate is higher, the nominal interest rate should be higher. This just compensates lenders for the decline in purchasing power they experience between the time a loan is extended and when it is paid back. Indeed, some mainstream models, including New Keynesian models (which are basically neoclassical growth models with sticky prices and wages) have the feature that the long-run real interest rate is a constant, determined by the subjective rate of time preference of the people who live in the model.

So, if we suppose that the real rate of interest is constant in the long run, we might fit a straight line to the points in the chart above (as I have done), and then think of the deviations from that straight line as short-run deviations from a "natural real rate of interest." One factor that may cause the real interest rate to fluctuate in the short run is monetary policy. In particular, some models of the short-run nonneutrality of money tell us, and central bankers certainly have told us for a long time, that in the short run monetary tightening - an increase in the nominal interest rate - makes the real rate of interest go up and the inflation rate go down. That's a feature of New Keynesian models, and it also comes out of other traditions. For example, segmented markets models exhibit a liquidity effect (real interest rate goes down in response to monetary stimulus in the short run), one example of which is this Alvarez/Atkeson/Kehoe model.

You can see the liquidity effect in the data from the Volcker disinflation period.
In the picture the trend runs from northeast to southwest - that's the Fisher effect at work (lower nominal interest rate and lower inflation). But there are periods of time when inflation is falling and the nominal interest rate is rising - that's the liquidity effect. An important observation from this period is that the liquidity effect is relatively short lived. That is, the Fisher relation may be a long-run phenomenon, but this isn't an in-the-long-run-we-are-dead long run.

Here's a heuristic approach to this. This, I think, is roughly what you would get if you took some of the models I have been working with, and put in aggregate shocks and short-run liquidity effects. First, suppose that the long-run real rate of interest is a constant. Then, in the next chart, LRFR is the long-run Fisher relation.
So, suppose I am Paul Volcker, and I'm faced with a situation at point A where the inflation rate is high and the nominal interest rate is high. The curve SRLE1 is the short-run tradeoff I face. I can reduce inflation in the short run by increasing the nominal interest rate, thus moving to B. But that won't work to reduce inflation in the long run, so after increasing the nominal interest rate, I have to begin reducing it. In that process, the short run tradeoff shifts down. Ultimately the economy comes to rest at point D where inflation and the nominal interest rate are both lower, and the short run tradeoff is SRLE2.

Next notice, in the Fisher relation data chart, that a nonlinear long-run Fisher relation would fit better than a straight line. That is, at low rates inflation, the real interest rate tends to be low. I get an effect like that in some of my work, and it comes essentially from an effect on the liquidity premium associated with interest-bearing safe assets. So, in the next chart, the curve LRFR1 denotes the long-run Fisher relation, which is concave.
So, one long run equilibrium is at point A, at the zero lower bound. Of course, the nominal interest rate can't go below zero. But, our central bankers argue, QE (quantitative easing) works just like a reduction in the fed funds rate. In some of my work, that's close to being true, so suppose we accept that. Then, effectively we have a short-run tradeoff SRLE, and it's as if we can get more inflation by moving down that curve to point B. But, the lesson from the Volcker era is that short-run liquidity effects are short-lived. Further, my work shows that there is another liquidity effect, associated with the interest bearing liquid assets, that causes the long run real rate to increase as a result of QE. So the long-run Fisher relation shifts up to LRFR2, and the long-run equilibrium is at E, as a result of QE, which implies lower inflation. And remember we don't have to wait long for this long run.

Further, there are other forces in play currently that will tend to move point E to the left if the nominal interest rate stays at zero. The destruction of private sources of collateral and the shaky state of sovereign governments in parts of the world gave U.S. government debt a large liquidity premium - i.e. those things reduced real interest rates. As those effects go away over time, real rates of return will rise, shifting up the long-run Fisher relation, and reducing inflation if the Fed keeps the nominal interest rate at the zero lower bound.

If the Fed actually wants to increase the inflation rate over the medium term, the short-term nominal interest rate has to go up. We need to be at a point like D. There used to be a worry (maybe still is) of "turning into Japan." I think what people meant when they said that, is that low inflation, or deflation, was a causal factor in Japan's poor average economic performance over the last 20 years. In fact, I think that "turning into Japan" means getting into a state where the central bank sees poor real economic performance as something it can cure with low nominal interest rates. Low nominal interest rates ultimately produce low inflation, and as long as economic stagnation persists (for reasons that have nothing to do with monetary policy), the central bank persists in keeping nominal interest rates low, and inflation continues to be low. Thus, we associate stagnation with low inflation, or deflation.

So, here's the policy advice for our friends on the FOMC. Continuing to engage in short-run monetary stimulus, through QE, will have little or no effect on real economic activity. The short run stimulative effects of monetary policy have pretty much played themselves out, and the real effects get smaller the more you do it. If there's any tendency for inflation to change over time, it's in a negative direction, as long as the Fed keeps the interest rate on reserves at 0.25%. Forget about forward guidance. You've pretty much blown that, by moving from "extended period" language, to calendar dates, to thresholds, and then effectively back to extended periods. That's cheap talk, and everyone sees it that way. So, as long as the interest rate on reserves stays at 0.25%, there are essentially no benefits in terms of more real economic activity. But you're losing by falling short of the 2% inflation target, which apparently you think is important. And you'll keep losing. So, what you should do is Volcker in reverse, except you don't have to move the inflation rate up much. For good measure, do one short, large QE intervention. Then, either simultaneously or shortly after, increase the policy rate. Under current conditions, the overnight nominal rate does not have to go up much to get 2% inflation over the medium term. Otherwise, you're just stuck in a rut, which would be too bad.


  1. Typo: "So, in the next chart, the curve LRFR1 denotes the long-run Phillips relation, which is concave."

    Should be Fisher.

    Reading well so far, but I think I'm about to get to the bits I may disagree with!

  2. This is a nuanced and important post. The Fed, at this point, has no idea what they are doing. However, the conventional wisdom is so entrenched that these policy makers refuse to consider any alternative points of view. Keep at it. Thanks

  3. Any thoughts on Daly and Hobijn? They have some data on rigid wages e.g. figures 1 and 2.

    1. Sorry, I meant to post this under "Finger on the Pulse?" I had too many windows open.

    2. This is related to the NBER working paper someone else sighted. Not sure what to make of this. I want to understand what people think this has to do with recent history.

    3. I need to read it closer, but if my quick look at it is correct, the claim is that during large contractions rigid wages are more binding, so more adjustment is through
      unemployment, which isn't too surprising.

      What is harder to follow without a closer read is the long run outcomes. The relevant quote I think is

      "Second, deep recessions in which downward nominal wage rigidities are particularly binding result
      in substantial pent up wage deflation. As a consequence, the recovery of such a recession involves
      continued wage cuts by firms, alleviating the wage rigidities constraint, resulting in more hiring, and lowering the unemployment rate. During the recovery from such a recession this leads to a
      simultaneous deceleration of wage inflation and decline in the unemployment rate."

      Why that would be the case I would need to think about. My intuition is that really low inflation decreases the speed real wages adjust, causing slower adjustments back to the long run Phillips Curve. The long run could be longer in coming. I don't understand quite yet the logic why the long run Philips curve would be non-vertical as one of their figures implies.

    4. If you put a lower bound on nominal wage changes, then if you have a long period of low inflation and the constraint binds, the distortion will be cumulative, and over time you'll have a real wage that is way too high relative to what is efficient. But currently the rate of nominal wage increase is going up, and the inflation rate is going down. What do you make of that?

    5. That would be problematic, wouldn't it. But what measure of nominal wages do you have in mind? Daly and Hobijn use a composite index of 4 measures including BLS microdata. The only one on I could quickly find on FRED is the Employment Cost Index (linked below). I don't see acceleration of wage inflation in that index, but I am not an expert when it comes to wage data, so I am not going to argue it is the right one to look at.

    6. For what ever reason that link reverted the graph from annualized percent change to levels. FRED beta...

    7. I was just looking at the series Krugman used in his blog post (see
      I had no good reason to think that was a good measure of the aggregate wage rate. In any case, I was thinking that we don't have to be looking at wages. The theory says that when there is low price inflation we should observe high unemployment. I don't see that in the first chart above.

    8. Yep that is a bit interesting. Krugman goes on to link to the Daly and Hobijn paper which theoretically implies something different than what the data now shows. Earlier in recession the correlation works out (at least with regards to wages), but not anymore since about a year ago, though who knows what the Daly Hobijn index would show.

      If you meant the last year by "I want to understand what people think this has to do with recent history." that is a good point.

      Another point with the Krugman graph, I don't know if the decreasing rate of wage inflation is all that different post 2001 recession and the last one. I would have expected a bigger difference based on the quote I posted a few comments above given the different magnitude of two recessions.

  4. I think what everyone trips on is the idea that moving the FF rate up would actually increase inflation. To be sure Market Monetarists like Sumner always quote Friedman as saying that low nominal interest rates are a sign of tight money but what they mean is that in the longer run nonminal rates will go up with higher inflation and more growth.

    You're heterodoxy from their standpoint is a kind of reverse causality. That it's' not that higher inflation will lead t higher nominal rates but that nominal rates will actually cause higher inflation.

    I give you credit at least for shaking things up-I tend to think they've gotten a little complacent anyway.

    1. Causality isn't really an important part of the story. I'm just talking about relationships that have to hold in equilibrium. But, I think it's clear that, if the central bank can control anything, it can control the nominal overnight interest rate. Thus, it's useful to think of policy in terms of the overnight interest rate as a policy instrument. In general we want to think about alternative policy rules - how the overnight rate should respond to various endogenous variables - when we evaluate policy. In this case, though, my starting point is a very simple question: What happens if the central bank pegs the nominal interest rate at zero for a long time? And the answer is: no one should be surprised if this produces low inflation.

  5. Mr. Williamson let me just be straightforward: are you suggesting that higher public debt by the U.S. government would now be stimulative?

    So is the best thing the U.S. Congress can do is pass the President's jobs bill?

    1. Those are two different questions.Yes, I think part of our problem is insufficient government debt. There of course many ways to get more government debt. You can have a temporary tax cut. That's straightforward. But I haven't studied the jobs bill, and have no idea what's in it.

    2. Then I have nothing bad to say about you. See you and Krugman don't agree on the Phillips Curve-what its shape is, whether it's useful or not.

      My feeling about it is-who cares? You both agree there's a liquidity trap-ok you seem to differ with him on why there's one-and you both agree that what we need right now is an increase in public debt-again for maybe very different reasons.

      I agree, As far as I'm concerned that's good enough for me. So I'm a Krugman fan and a Stephen Williamson fan.

      Its not that as a layperson I don't have respect for theoretical disputes-I'm aware they are often very important and fertile ground for furthering our economic knowledge-one of the most important things in the world for human society.

      Still I'm also a pragmatist. If we do fiscal stimulus for the right or wrong reason doesn't matter to me. If policymakers do it with the wrong understanding of the phillips curve it will help just as much.

    3. "If we do fiscal stimulus for the right or wrong reason doesn't matter to me."

      I disagree. I think it should matter. Policymakers can be doing the right things for the wrong reasons. That might work for a while, but eventually they'll get into a situation where they do the wrong things for the wrong reasons.

      Where I part ways with Krugman is mainly in his lack of respect for the majority of people who are actively engaged in macroeconomic research, and for some of the people who made key contributions to the profession in the last 40 years. That lack of respect is both uninformed and unnecessary.

    4. Let me suggest that there can be wrong reasons to bash Krugman as well. Let's assume you're right in your criticism of him-the truth is I can admit I don't know. I don't hang in the circles you and he do so I don't know the peiople you believe he disrespects and so I can't have an informed opinion about it.

      What I do know is that most of the people who hate Krugman do so for the wrong reasons-they oppose increasing government debt by hook or by crook. I believe your a Canadian by birht? If so you may not have a sense of just how nasty politics is down here even if you now liver here-it depends how much you personally follow politics.

      On the other hand I'm waiting with bated breath for your answer to Sumner's reply to your comment over at MI. I'm sure you already saw it but just in case:

      "Steve, I claim that action would lower TIPS spreads. The best way to raise inflation rates is to raise the NGDP target path, and do level targeting. The expectations of faster NGDP growth (if sufficiently large) will tend to raise TIPS spreads and probably long term rates."

      Yep, the answer is-NGDP targeting preferably with a NGDP futures market sort of modeled on the TIPS market.

      How did I guess that would be his answer?

  6. Nick I don't think I have any trouble guessing where SW lost you.

    Guess where SW loses NIck Rowe

  7. Wouldn't having the real interest rate constant in the long run result in non-neutral money. Say the real interest rate is stuck at r* and the nominal interest rate is pegged at i*, so inflation (π) is i*-r*. Since nominal GDP growth (m) is equal to real GDP growth (g) plus inflation, permanent increases in nominal GDP growth must result in permanently higher real GDP (non-neutral money). Either the real interest rate doesn't converge to r* over time, the central bank can't hold i at i* forever, there is some constraint on nominal GDP that disallows the CB from increasing the money supply, or money ceases to be neutral.