They are *still* at it! (Making totally irrelevant arguments about headline vs core.) How to kill this zombie?

Here is a very simple model of inflation. Don't take it too literally. It's just for illustration.

1. H(t) = aH(t-1) + bC(t-1) - cR(t-1) + e(t)

H(t) is headline inflation at time t; C(t) is core inflation; R(t) is the rate of interest set by the central bank; and e(t) is a serially uncorrelated error. Headline inflation depends on lagged headline inflation, lagged core inflation, the lagged interest rate, and an unforecastable shock. (Everything is in deviations from the mean, so I can ignore the constant term.)

Here is a very simple model of the central bank's reaction function.

2. R(t) = dH(t) + fC(t)

The central bank looks at headline and core inflation, and sets the rate of interest accordingly.

Substitute the reaction function 2 into the structural equation 1, to get a reduced form equation for headline inflation.

3. H(t) = [a-cd]H(t-1) + [b-cf]C(t-1) + e(t)

Two econometricians visit this economy. Both want the central bank to keep headline inflation as close to a fixed target as possible. One argues that the central bank should respond only to core inflation, because core inflation is a better predictor of future headline inflation. The second argues that the central bank should respond only to headline inflation, because headline inflation is a better predictor of future headline inflation.

They agree it is an empirical question, and so agree to settle their argument by estimating a regression of headline inflation on lagged headline inflation and lagged core inflation.

They agree to estimate:

4. H(t) = AH(t-1) + BC(t-1) + e(t)

They agree that if they find that A>0 and B=0 then the second econometrician is right, and that the central bank should respond only to headline inflation. And if they find that A=0 and B>0 then the first econometrician is right, and the central bank should respond only to core inflation. And if both A>0 and B>0 they are both partly right, and the central bank should respond to both headline and core inflation.

Comparing equations 3 and 4, it should be obvious that both econometricians are hopelessly wrong.

The econometricians' estimate A is a very bad estimate of the structural parameter a. Instead, it is a good estimate of the reduced form parameter [a-cd]. And the econometricians' estimate B is a very bad estimate of the structural parameter b. Instead, it is a good estimate of the reduced form parameter [b-cf].

If, for example, the central bank knew that equation 1 was a true description of the structure of the economy, knew the exact values of the structural parameters a, b, and c, and wanted to keep headline inflation as close as possible to a fixed target, then it would choose a reaction function in which d=a/c and f=b/c, so that the reduced form would become:

3' H(t) = [0]H(t-1) + [0]C(t-1) + e(t) = e(t)

And the two econometricians would find that A=0 and B=0, so that neither headline nor core inflation predicted future headline inflation. So, if the central bank were reacting perfectly to both headline and core, the two econometricians would conclude from their regression that the central bank should ignore both headline and core, because both are useless as predictors of future headline inflation.

The correct inference to draw from the econometricians' estimates is *not* whether the central bank should react to headline, core, both, or neither. It is whether the central bank should react *more or less strongly* to headline and core *than it has reacted in the past*.

Now can somebody please explain this to the ECB, and stop this irrelevant debate about the predictive power of headline vs core.

(This is just another way of explaining what I have said many times in the past).

Couldn't it be the case that you are wrong, but in a way that reveals a subtle truth?

Perhaps the ECB's reluctance to think about this matter in a way that assumes they are trying to adjust monetary policy optimally to ensure inflation hits a certain target reveals that that is not what they are doing.

Perhaps from the inside it is so obvious that the ECB decisionmaking is based on political appearances and credibility of individuals - rather than actually hitting the inflation target - that your argument simply does not apply. In that case, by being wrong, you might have hit upon a partial explanation for the ECB's inexplicable policy moves and failure to lower the interest rate to zero during the financial crisis...

Posted by: Gregor | June 24, 2011 at 11:36 AM

Gregor: so that all their arguments about core vs headline are just smoke and mirrors? Dunno. I certainly hope you are wrong. I think they could find a way to do smoke and mirrors that isn't so obviously flawed. Obfuscation should be more successful than this!

Posted by: Nick Rowe | June 24, 2011 at 11:49 AM

Yeah OK you've made your point. As I read the data, while their premise is slightly off, they have still reached what I see the proper conclusion, that while they may (not will) do slightly better by reacting more to headline, they don't have to.

Posted by: jesse | June 24, 2011 at 12:13 PM

Does the Taylor-rule approach make more econometric sense? As a practical matter, I think Gregor is probably right. The ECB is more concerned with its image, in Germany especially, than it is with getting the right inflation rate.

Posted by: Kevin Donoghue | June 24, 2011 at 12:14 PM

jesse: I'm still very unsure about the best way to try to explain my point. Did you find this way clearer?

Kevin: My equation 2 is sort of similar to a Taylor Rule, in that I think of the Taylor Rule as just one example of a central bank reaction function that is supposed to help it keep inflation on target. (I've left out the unemployment rate/output gap of course, but only to keep it simple.) But the Taylor Rule leaves open whether it should be headline or core inflation, or both, that belong in the Rule.

Posted by: Nick Rowe | June 24, 2011 at 12:31 PM

What if the CB decided (or was told) that it would loose its credibility unless it reacted to headline inflation?

Posted by: Patrick | June 24, 2011 at 12:38 PM

Patrick: Hmmm. If it *were* correct that the CB should react only to core, and ignore headline, then what the CB needs to say is this: "Look, in the past we have been responding to core and ignoring headline (here's our estimate of equation 2 to back this up), and yet neither headline nor core predicted future headline (here's our estimate of equation 3 to back this up), so we must have been doing it right".

Posted by: Nick Rowe | June 24, 2011 at 12:45 PM

Does it matter to this analysis that H and C are not independent of one another? The claim underlying the use of core inflation is (crudely) that H(t) = C(t) + g(t), where g, like e, is largely unpredictable. If that's true (and whether it is is, of course, debatable), then reacting to H at all is a mistake because in so doing you assign a nonzero weight to g, which by assumption is uncorrelated with H(t+1).

Posted by: Rplzzz | June 24, 2011 at 12:49 PM

"Did you find this way clearer?"

I can't say it was clearer but I'm not versed in the language of economic maths. I understand your point after including the weight of previous comments, including this one, that you have generously awarded this blog. I'll try to decipher and translate the latest angle of your argument into something I can understand better. ;)

Posted by: jesse | June 24, 2011 at 01:24 PM

At the risk of creating a too contrived scenario: I guess I'm wondering if there may be a pathological dynamic where people are experiencing headline inflation due to some kind of slow moving shock (e.g. increasing scarcity of some vital resource). The price of some goods excluded from core is heading to permanently higher levels. The CB keeps saying "we are doing it right; this too shall pass". In the meantime, people are experiencing inflation. They start to set expectations according to what they are experiencing, and ignore the CB. The CB now starts to worry that its credibility will be shot by the time the shock has run its course, and perhaps an inflationary spiral will have taken hold.

Maybe too contrived.

Posted by: Patrick | June 24, 2011 at 01:27 PM

A real shock is recessionnary and there is nothing a CB can and so should do about it. Using monetary restraint will merely add another shock to the system.

Funny that monetarist always seems to confuse real and money phenomena once they are in charge of a CB

Posted by: Jacques René Giguère | June 24, 2011 at 04:42 PM

Isn't it obvious the answer is that both should be used? The right way to think of this is that the CB has a target for 'r', a hidden variable representing the true inflation rate of the economy. No one knows for sure what 'r' is. The CB has a model of how economy operates that turns control input 'i' the overnight rate into the hidden variable 'r'. They also have observations of r, which you've denoted as H and C. We believe H and C to decompose into r + eh and r + ec respectively. That is, they are observations of r plus some random noise. Suppose we assume that random noise is Gaussian, and we want to compute i based H, C, and our Model.

This problem has an optimal solution, and it's known as a Kalman Filter.

Posted by: Jon | June 24, 2011 at 11:20 PM

Rplzzz: the claim being made by people who say that the central banks should look *only* at core (and ignore headline) is that a=0 in 1. H(t) = aH(t-1) + bC(t-1) - cR(t-1) + e(t). And if so, the bank should set d=0 in 2. R(t) = dH(t) + fC(t), because that will results in [a-cd]=0 in 3. H(t) = [a-cd]H(t-1) + [b-cf]C(t-1) + e(t).

But my assertion is that you can't tell whether a=0 by estimating whether A=0 in 4. H(t) = AH(t-1) + BC(t-1) + e(t). That's because, even if they estimate 4 perfectly (with an infinitely big data set), A will equal [a-cd], not a.

Patrick: that's not really a too contrived example. It might be right. But I think your example assumes that the central bank is getting it wrong, because it is ignoring headline when headline actually matters.

Jacques: if an adverse supply-side shock hits, then there is a trade-off between keeping inflation on target and keeping output at the natural rate. This is one argument against inflation targeting. But that's outside the scope of this post.

Jon: headline is the sum of core plus non-core. Someone who says "look at core only" is saying that non-core contains zero information, which sounds implausible. Someone who says "look at headline only" is saying that the bank should respond in exactly the same way to core and non-core, which also sounds implausible, since core and non-core behave differently. So yes, prime facie, both those positions sound very implausible, and it seems very plausible that the bank should look at both headline and core. But ultimately it's still an empirical question. Some implausible things do turn out to be true, sometimes.

You may be right about the Kalman filter. I tried to get my head around it, but failed. I'm not very good at econometrics. (But Jeeez, I'm still amazed that people who are much better econometricians than I am keep making the simple mistake I am pointing out here!)

Posted by: Nick Rowe | June 25, 2011 at 06:09 AM

Part of the argument is that non-core inflation is volatile: lots of noise, not much signal. I'm not sure whether that means it should be ignored. It's a bit like taking exchange-rate movements into account, which doesn't seem like an especially good idea.

Posted by: Kevin Donoghue | June 25, 2011 at 07:18 AM

Nick: I agree that equation (1) with a=0 is

amodel in which you would ignore H, but I'm not sure it's the only one. In particular, if H and C are not independent of one another, then it's a mistake to enter them both as independent variables in the model.Also, you wrote in reply to Jon, "Someone who says "look at headline only" is saying that the bank should respond in exactly the same way to core and non-core, which also sounds implausible, since core and non-core behave differently. So yes, prime facie, both those positions sound very implausible, and it seems very plausible that the bank should look at both headline and core." Isn't this equally an argument for including every other measure of inflation that could be constructed by subtracting out some component of headline inflation? Thus, we could have "CPI ex. housing and rent", "CPI ex. consumer electronics", "CPI ex. fruits with edible peels," and so on. It's hard to imagine that you could construct a better predictor of future inflation using all those non-independent variables than you could get by picking just a well-chosen subset, perhaps even a subset comprising just the variable we call "core inflation".

I agree, however, that estimating the parameters in equation 4 won't resolve the empirical question.

Posted by: Rplzzz | June 25, 2011 at 01:46 PM

Rplzzz: "In particular, if H and C are not independent of one another, then it's a mistake to enter them both as independent variables in the model."

Not quite. If H and C were perfectly correlated, it wouldn't matter which you included, and which you looked at. But if they are imperfectly correlated, each contains *some* information that is not captured by the other.

Your second point (about if two subsets is good, why not 3, or 4 or...) is a good one.

Posted by: Nick Rowe | June 25, 2011 at 06:12 PM

Nick:

The argument the "do not use headline inflation" camp is making is very simple in its essence: a CB should not use the simplistic strategy of "raise rates when headline inflation goes up and drop rates when headline inflation goes down".

They are making this argument because this is precisely what CBs like the ECB have been doing for years and are doing today.

Do you accept that characterisation?

If you accept that characterisation then their core-inflation point is even simpler: "use core inflation instead of headline inflation, because it results in a provably better policy".

Again, do you accept that this is their point?

if yes then no equation has to enter the picture because the argument is about real policy variants - not about an "ideal CB" which could in theory implement an "ideal policy" based on just about any inflation metric that essentially includes all core components (just with different weightings) ...

If both of your answers were "yes" then we have solved the riddle. If any of your answer was "no" then the question you framed does not apply and we need to discuss the step where you say "no" first :-)

Posted by: White Rabbit | June 26, 2011 at 03:13 PM