Wednesday, June 12, 2019

CPI and DIEM inflation forecasts

Apropos of getting into an argument about the quantity theory of money on Twitter, the new CPI data came out today — and it continues to be consistent with the Dynamic Information Equilibrium Model (DIEM — paper, presentation) forecast from 2017. Here's year-over-year inflation:

The red line is the forecast with 90% error bands (I'll get to the dashed curve in a second). The black line is the post-forecast data. The horizontal gray line is the "dynamic equilibrium", i.e. the equilibrium inflation rate of about 2.5% (this is CPI all items), and the vertical one is the center of the "lowflation" shock associated with the fall in the labor force after the Great Recession. Shocks to CPI follow these demographic by about 3.5 years.

Back to that dashed curve — the original model forecast estimated the lowflation shock while it was still ongoing, which ends up being a little off. I re-estimated the parameters a year later and as you can see the result is well within the error bands. The place where it makes more a difference visually (it's still numerically small) is in the CPI level:

Without the revision, the level data would be biased a bit high (i.e. the integrated size of the shock was over-estimated). But again, it's all within the error bands. For reference, here's a look at what would have looked like to estimate a bigger shock in real time — unemployment during the Great Recession.


PS/Update +10 minutes: Here's the log-derivative CPI inflation (continuously compounded annual rate of change):


  1. Any thoughts on implications given that the equilibrium inflation rate in your model is 2.5%, while the Fed remains preoccupied with a 2% target?

    My sense is that the more dovish members might have succeeded in shifting the board a bit toward treating the 2% target more "symmetrically," and there's even some talk about letting the economy generate "catch-up inflation" following periods of inflation below the target, but if the equilibrium inflation rate is actually 2.5%, does this mean that the Fed will remains systematically biased towards periodically contracting the economy when they notice inflation persistently closer to a level that is 50 bps above their target?

    1. This is interesting because it depends on which measure of inflation they look at. The dynamic equilibrium for the GDP deflator is about 1.4% and the one for core PCE inflation is 1.7%. Core CPI inflation looks like it has a rate much closer to 2% (about 2.2%).

      Most of the briefing materials point to core PCE (which would have the opposite effect making policy too accommodative on average going forward**), but then there are statements that mention other measures.

      **The demographic shock ended sometime in the late 90s, so there hasn't been a very long period where we've actually seen the dynamic equilibrium. A question I have: did the on average too accommodative policy (due to low core PCE inflation dynamic equilibrium) produce the two asset bubbles in the late 90s and mid-2000s?

    2. Sorry about temporarily forgetting the important distinction between inflation measures. And it is interesting that your estimate for equilibrium core PCE is, as you say, BELOW the Fed's target. I feel like you've commented on this before – including the extent to which it has contributed to the development of asset bubbles – so I apologize if I made you repeat yourself. Thanks for the explanation.

    3. No worries. I'm actually not sure if I've ever talked about the effect of 1.7% core PCE inflation equilibrium on monetary policy before. I do think that I've had a conversation in comments about the question of whether that is actually due to an asymmetric inflation target (i.e. on average missing too low, per David Beckworth's figure).

      I'm also not really sure which measure the Fed actually uses. They put core PCE in the briefings, but often comment about headline and core CPI. I can imagine that an inflation hawk on the FOMC would start citing headline CPI inflation to make their point!

    4. Not disputing your hypothetical regarding an inflation hawk, but was pretty sure I understood the Fed's target to reference the PCE measure specifically. Still not sure the best way to confirm that, but I was able to find this quote within a couple minutes in a Neel Kashkari blog post from last year (

      "The FOMC has defined its price stability mandate as inflation of 2 percent, using the personal consumption expenditures (PCE) measurement. Importantly, we have said that 2 percent is a target, not a ceiling, so if we are under or over 2 percent, it should be of equal concern. We look at where inflation is heading, not just where it has been. Core inflation, which excludes volatile food and energy prices, is one of the best predictors we have of future headline inflation, our ultimate goal. For that reason, I pay attention to the current readings of core inflation."

      Not sure that's dispositive or helpful, but thought I'd pass it along given how appreciative I am of your thoughts and insights.

    5. Yes, I think I forgot what my

      I went back and found this footnote on my blog ...

      [1] Somewhere, I don't remember where (possibly Tim Duy?) noticed that the Fed seems to actually be looking at average headline inflation of 2% ...

      It's in a post about headline PCE inflation (which is what the Fed uses, not core PCE inflation which is just used as an indicator per Kashkari). It's ambiguous what the "headline" means in the footnote. I was only vaguely recalling this post, and confused it my own head.

    6. Thanks for the clarification.

      Just out of curiosity, is it both worth it and possible to estimate an equilibrium for "headline PCE inflation," which I am assuming is simply the measure derived from the PCEPI series rather than that derived from PCEPILFE, where these abbreviations are the ones from the St. Louis Fed's FRED website?

      I'd assume its virtually the same as for the Core PCE measurement, but I'd be prepared to hear a counter-intuitive result as well (assuming, again, that you think it's worth your time to compute and that there is not some flaw in trying to compute it for PCE that did not pose a problem for your core PCE computation of 1.7%).

      Also, I have gotten the impression over time that you would place less emphasis on the Fed and monetary policy based on your research, so I'm sorry to get hung up on questions that come from a Fed-/monetary-policy-centric perspective.

    7. Actually, headline PCE seems to be about 1.6-1.7% as well. There is a really big effect from 2000-2009 (Iraq War on gas prices, dot-com and housing booms) that makes it harder to estimate than the other series. Maybe that's what led to the mis-management/inflation hawkery in the midst of the financial crisis?

      And no worries. I do think the Fed has an effect on coordinating beliefs, which can trigger a recession or make one worse. E.g. I think Fed actions made 2008 worse per here:

  2. Replies
    1. Jason, perhaps I can pick your brain about one last thing, but I'll understand if it's something that might have to wait for a response.

      I've been looking over some of your models that give forecasts into the future, and in at least one case, you linked to a FRED model in which you provided the fit parameters, which was your model for the 10-year treasury rate. From the FRED model, it was easy to see how any given pair of NGDP and MB quantities could be used to estimate the corresponding 10-year rate, but I haven't yet caught on to how you project estimates into the future to make a forecast. Does it entail making projections of the independent variables, which in this case seem to be NGDP and MBCURRIC, or at a minimum, making a projection of what their ratio will be? If so, how do you generate those projections? If not, then are you directly generating future instances of the dependent variable, in which case is your method some kind of simple extrapolation?

      Presumably there is something fundamental I don't yet understand about the full DIEM, even if I think I get the idea of the framework and how it constrains the model to one consistent with "information transfer" where parameters are then estimated based on past data. Thanks.

    2. The 10-year rate is what I call an information equilibrium (IE) model between A and B, while the dynamic information equilibrium models (DIEM) are a bit more specific — they assume two variables A ~ exp(a t) and B ~ exp(b t).

      In the case of the 10-year rate model, it's actually two IE relationships: between the "price of money" p and the interest rate and between NGDP and M0 (MB minus reserves).

      r ⇄ p
      p : NGDP ⇄ M0

      The former means

      (dr/dp) = k1 r/p

      The latter means

      p = (dNGDP/dM0) = k2 NGDP/M0

      To do a forecast, I had to forecast NGDP and M0 (approx log linear growth) and take into account the variation of the rate r around the path with an AR process. End result is:

      log(r) = k log(NGDP/M0) + c + AR(n)

    3. Thanks for the explanation, and yeah, I had to look back at your posts a few times to get the two-step model building process from "price of money" to interest rate. The problem is that I could go on asking questions indefinitely, so I suggest you cut me off, but I did have several final thoughts/questions that I'll leave here anyway, without expecting anything further from you, which are these:

      (1) There must be an interesting interplay between these variables and some of your other work (e.g., labor force participation driving inflation, inflation expectations impacting nominal interest rates, etc.)

      (2) If you had a strong view that the relative growth rates and ratio of NGDP and M0 were going to change, and you might imagine any number of reasons for this, then you could use that forecast in the model as well (e.g., I'm thinking regime change type scenarios more than temporary shocks)

      (3) I feel like you wrote a post once which stepped through how your model anticipated the higher rates in the 70s, and then their flattening, and then their long decline, but then there must be something in how you're forecasting NGDP and M0 that is fairly successful. Perhaps it is only using trends in recent results, or to put it naively, something like second derivatives to get the trends and turning points right. (I don't know enough about autoregressions, which is what I'm assuming AR refers to but perhaps that's the step I don't get).

      (4) Did you have to play around with these concepts of "price of money" in IE with "interest rates" in order to get a model that gives a good empirical fit or did it just seem logical to relate them in that way on an a priori basis (basically a question of how the model ended up being composed of two IE processes and the extent to which good empirical fit does or doesn't help you work back to the underlying IE relationships, such that the learning and model-building happens forwards, backwards, or in a give and take, so to speak)?

      Anyway, you can see that I'm quite taken with your models, as they seem to be working very well to date and might ultimately be able to explain lots of other interesting phenomena (e.g., one might consider why, in recent years, M0 might need to be higher relative to NGDP, along with further implications for asset bubbles, which you've already cited, as well as higher wealth concentrations/inequality, and then theoretical justification for possible remedies, such as higher tax rates on the top incomes in all its forms). All in all, many thanks for your work and help.

    4. (1) One thing: you can treat the NGDP and M0 separately as DIEMs and if you look at the shocks ("economic seismogram") the shocks to NGDP (e.g. big one in the 70s) precede the shocks to M0 (e.g. big one in the 80s) — meaning causality goes from NGDP to M0, not the other way around. Which is why I tend not to focus on interest rates very much anymore.

      (2) Yes, you could. However, the interest rate relationship (information equilibrium IE) isn't as accurate as the various dynamic equilibrium relationships (DIEMs). We might think of the interest rate equation

      log r ~ k log NGDP/M0 + c

      as simply a leading order approximation over the long run, while DIEMs are higher order and effective over the short run.

      (3) I did, and all I was doing was log-linear extrapolation of the variables along with modeling deviations around that trend using an estimated autoregressive (AR) process. An AR process is basically like a kind of correlated noise — AR(0) is just white noise, while AR(n) depends on the n previous time periods.

      It effectively does take into account second derivatives when you use an AR(2) or greater process (which uses three lags). But primarily the curvature came from log-linear extrapolation of NGDP and M0. Since M0 lags NGDP, NGDP started to increase slower than M0 at one point, which would cause the log-linear extrapolation to produce a lower interest rate in the future using log(r)=k log(NGDP/M0)+c.

      (4) Yes. I noticed the relationship between NGDP and the monetary base including reserves (MB) and that log(NGDP/MB) has almost exactly the same basic shape as the effective funds rate mostly from just playing around with the variables. Unfortunately, it's not perfect without a "fudge factor" — which I actually called c in this post:

      where I wrote

      c log(r) = log(NGDP/(k MB))

      which can be transformed back into the form

      log(r) = a log(NGDP/M0) + b

      with a = 1/c. It was just a fudge factor — I rationalized it by saying the interest rate wasn't exactly the same as the "price of money" (or technically, the money price of output in the IE language — the "price of money" in terms of output would be the inverse). Eventually I noticed you could get that fudge factor by saying the price was in information equilibrium with the interest rate:

      But overall, it's been a trial an error process. I think I now have a better overall understanding of how and why everything fits together (e.g. variables with similar shock structures in their DIEMs will have IEs relating them).

    5. This was great, as were your previous responses. I particularly appreciated the color on some of the differences between IE relationships and dynamic equilibrium relationships, which I will try to review further from some of your previous posts, presentations, etc. Also, thanks again for the original post and thoughts on inflation, which I hope I haven't detracted from.


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