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Sunday, February 16, 2020

Central Bank Objective Functions

One topic of research that keeps popping up is the question of what the central bank objective function should be. In simpler terms, what is the target of the central bank? (At present, most central banks have an inflation target, possibly with secondary objectives.) This is a preoccupation of many "conventional" economists -- those in the neoclassical tradition, as well as those that are somewhat out of the mainstream (e.g., Market Monetarists are pushing for a Nominal Gross Domestic Product Level Target (NGDPLT). If one sticks to the usual Modern Monetary Theory (MMT) story, this is not really a topic of interest. However, even if I take my MMT hat off, I think these discussions are putting the cart before the horse.


This article is my long form response to a point made by Nick Rowe on Twitter. This is an area that central bank economists are very interested in, and so there is a large literature. I am not interested in pursuing the details of that literature, for reasons that will be made clear.

The basic assumption underlying this field is that the central bank has the power to steer the economy with monetary policy (either interest rates, or changes in the monetary base for the Monetarist wing). This might be supplemented by fiscal policy run by the central bank ("helicopter drops"), although such interventions would require changes to the institutional structure in all of the developed countries I am aware of. There is skepticism among MMTers about the effectiveness of monetary policy in this regard, but we have to put those aside for this discussion to make any sense.

In days of yore, the objective for central banks was maintaining a parity between gold (or gold-backed currencies) and the domestic currency. With a fiat currency, this is modernised to a vaguer commitment to keep the purchasing power of the currency "stable." (This might be supplemented with some commitment to increased employment/output, consistent with stability of the value of the currency.) The belief among conventional economists is that the central bank needs to be independent of democratically elected politicians. However, the central bank needs to have some sort of target that the institution can theoretically be held accountable for.

The usual response is to pick some economic variable, and the central bank's policy is supposed to make that variable act in some "stable" manner. There are three open questions:
  1. What variable is to be targeted?
  2. How is the target to be defined mathematically?
  3. How are we measuring deviations from target to judge performance?
The standard answer to the question of which variable was "consumer price inflation," but interest has shifted to variants. The most popular alternative choice was nominal GDP, but some average wage measure is another possibility.

For the rest of the article, I will pick nominal GDP as the target (to line up with other discussions), but the same principles applies to the CPI index.

How to Define the Target?

Figure: Nominal GDP growth rate and Level

There are two main variants of target definitions: one that works with the rate of change of the target variable, the second with the level. The figure above depicts the two variants for the United States, with a 5% nominal GDP growth rate target.

The top panel shows actual nominal GDP growth, and compared to a 5% target. We can see that GDP growth tended to miss the target somewhat symmetrically before the Financial Crisis, and then tended to be below target. This aligns with what happened with inflation targets in many of the developed countries -- they hit the target with symmetric misses pre-crisis, and then tended to have inflation below target. (This indicates that a 5% target seems to give a similar outcome to 2% inflation targets.)

The usual way of judging whether the central bank hits the target is typically fuzzy. The usual principle is that we want the misses to be symmetrical, without large deviations.We normally do not worry too much about past misses: the objective is to be forward-looking. 

A level target is depicted in the bottom panel. I picked a somewhat arbitrary date (the first quarter of 2002), and set the level target to grow from the current level of nominal GDP by 5% per year. Nominal GDP tracked the target level closely in the initial years, then deviated wildly after the Financial Crisis hit. There has been no tendency for nominal GDP to revert towards the target level.
Figure: Level Target Deviation

The above figure zooms in on the 2002-2014 period, and expresses the miss from target as a percentage of GDP. Up until the Financial Crisis, it stayed near target, but then marched off.

If a level target was how the central bank was being judged, the Fed would have failed. In order to meet objectives, the Fed would have to have somehow induced a period of growth where nominal GDP is greater than 5%. Given that this period was largely a lost decade, this would have presumably been a better outcome for most people.

Does this Matter?

Although one could point to the previous argument and say that a level target provides superior outcomes, one immediately runs into the obvious problem: can the central bank steer the economy with a great amount of precision? The Federal Reserve did throw every piece of stimulus it legally could at the economy in the post-Crisis period -- and still missed its existing mandate, never mind a more ambitious level target mandate.

We immediately run into the whole "zero lower bound" (or "effective lower bound," since mildly negative policy rates are achievable) concern. Was the Fed failure solely due to the inability to cut rates further? I will put that aside, since there is allegedly more interest in the use of fiscal policy to achieve fine-tuning objectives. For the moment, let us assume that government policy is not completely powerless in the face of the bound on interest rates.

We still have a deeper issue to deal with: what power does the central bank have to steer the economy? Almost all the literature on this topic is based on neoclassical models -- where it is assumed that the central bank has the power to fine tune the exact future trajectory of the economy based on expectations (and some hand-waving about equilibrium selection).

The neoclassical community is stuck in the world of 1960s optimal control theory, and has not too much paid attention to post-1980 developments in control theory. Robust control theory is premised on the observation that model errors matter. [Update: Nick Rowe pointed out that the Bank of Canada did a study where the same rule was applied to multiple models, and so that is a strong step in the direction of dealing with model error. I probably should adjust my text, but I have to also see what they did.] Just assuming that the real world matches a mathematical model exactly is what optimal control was premised on -- and caused its failure in most real world applications (other than basic path-planning exercises, such as determining the optimal trajectory for a rocket going to the moon, which is the exercise optimal control theory grew out of). 

For the matter at hand, we cannot evaluate the various objective functions of the central bank without considering how the central bank actually influences the economy. A level target is only better than a rate target if it is possible for the central bank to fine-tune the nominal GDP trajectory very precisely.

For example, in the post-Financial Crisis period, it would have needed to quickly steer nominal GDP growth to a level greater than 5% -- even though forced deleveraging was crushing over-extended industries. It would have needed to keep the greater-than-5% growth running for a period of time to eliminate the deviation from target, then swiftly transition to 5% growth rate to resume growth at the target rate.
  • This requires the ability to quickly change growth rates, which runs counter to the received wisdom that policy works with a lag.
  • This assumes that growth can be snugged lower, without requiring a recession to achieve this.
  • Since it seems somewhat implausible that real GDP growth can be directly controlled by interest rate policy, the implication is that inflation will have to follow an erratic path: rapidly rising, sticking at a high level, then dropping to a "normal" level. Most conventional economists point to the importance of "anchored" inflation expectations: would actors in the economy believe that such a unusual inflation trajectory counts as "anchored"?

How Important Are Beliefs About Central Banks?

One of the arguments in favour of the level target is that a big deviation from target will make stimulus efforts more credible. That is, the mere existence of  a level target would have prevented the slowdown in growth after the Financial Crisis, since stimulus efforts would have appeared more credible than they would if they Fed just wanted to bump up the inflation rate slightly.

The entire evidence for this theory is based (once again) on assuming that highly controversial models are accurate descriptions of the economy. These models are designed to produce that outcome, and generally do not take into account real-world concerns like industrial over-capacity and the need to deleverage balance sheets. In other words, highly vulnerable to model error.

If "Output Gaps" Exist, Level Targets are Problematic

Although I am skeptical about the usual ways in which output gaps are calculated, we can accept that there is a hidden state variable that offers a summary of whether the economy is "overheating" or not. I refer to this concept as a "generalised output gap" (to distinguish it from particular candidate series that have been proposed, all of which have empirical problems). 

Imagine that there is a generalised output gap. What would have to happen to that variable in the aftermath of the Financial Crisis?
  1. It would have fallen to a deep negative level, courtesy of the collapse in activity.
  2. The Fed would have had to (somehow) stimulate the economy to reverse the variable to a positive level quickly.
  3. The generalised output gap would then have to be modulated so that inflation sticks at a high level for an extended period of time to allow the deviation to be erased.
  4. The generalised output gap would then have to moved to a slightly negative level to allow the return to a steady 5% nominal GDP growth rate.
If the generalised output gap exhibits any kind of "momentum" -- hard to change levels or sign quickly -- this hypothetical exercise is largely impossible. Since most attempts to infer the level of candidate output gap variables have to assume some form of momentum to make estimation of the output gap feasible, we see an immediate problem.

Although neoclassicals can point to inflation coming in close to 2% in the post-1990 era in most of the developed countries, this anchoring does not imply that the central bank can hit arbitrarily complex trajectories. (In particular, critics could point to the widespread reforms aimed at neutering the bargaining power of labour that were implemented in most of the those countries at that time, which can also explain observed data.)

If the generalised output gap does have momentum, the least disruptive strategy is to ignore past errors, and just try to keep forward growth/inflation near some target level -- which is the pre-existing strategy for inflation-targeting central banks.

Average Growth: Worse than Level Targets

The problem with ignoring past errors is that the central bank is largely unaccountable for errors: they missed, whoops, won't do that again. We cannot set a quantifiable metric for success.

One way to create a metric is to average inflation over a past period, and see if it was near target. Why not make that the official target? The reason is straightforward: it is even more destabilising than a level target.

As was pointed out by Nick Rowe on Twitter, imagine a country where the central bank wants to hit a 2% annual average inflation target, and the target is defined by averaging 2% over the past 3 years. We then assume that it hit 2% over previous years, then some shock drops the inflation rate to 1% in a year. In order to hit target over the 3-year window, it needs to hit 3% in the following year (and we assume that the target is always hit thereafter). The following year, it needs 2% inflation. The problem is that in the next year, the 1% drops out of the 3-year window, and so the previous 2 years are 3% and 2% -- so it needs 1% to hit the average.

The sequence of inflation rates looks like: 2, 2, 1, 3, 2, 1, 2, 3, ...

This is due to the use of a moving average; in the frequency domain, a moving average creates a "ringing" effect, and so its use is avoided in engineering applications.

We could try to lengthen the moving average period to reduce this ringing effect. However, this effectively turns the objective function into a level target: the deviation in level terms can be approximated by the sums of the annual deviations (so long as growth rates are much lower than 100%, which is the case for modern developed countries).  (There appears to be a bizarre distinction in the literature between geometric and arithmetic averages, which completely ignores the reality that the "random" deviations in these series swamps the gap between those averages.)

Alternatively, there needs to be a weighted average, to reduce the ringing. This is mathematically equivalent to a level target with the target being nudged towards the observed level every period. This will do very little to deal with the fundamental problem -- that the central bank likely lacks the ability to fine tune growth trajectories in an arbitrary fashion in practice.

Concluding Remarks

Until model error is taken seriously, the debate about the mathematical definition of the central bank target is very much akin to debating the number of angels that can dance on the head of a pin.

(c) Brian Romanchuk 2020


  1. Thanks Brian. Interesting read.

    By the way: One way to think about "model error" is to try to find a monetary policy rule that works reasonably well (even if not optimally) in a variety of different economic models. Looking for "robustness". The Bank of Canada did some experiments like that, a few years ago.

    1. Thanks for the feedback. I have to run, but I adjusted my article text slightly to account for the BoC work. From a technical perspective, robust control looks at stabilising entire families of models and not just ones chosen by the modeler, but the BoC work should be a strong step in the right direction.

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