## Wednesday, June 4, 2014

### Taylor rules

Last week I attended a conference at Hoover, "Frameworks for Central Banking in the Next Century." It was very interesting for its mix of academics, Fed people, and media. The Wall Street Journal had an interesting article Monday morning, "BOE's Carney may need to play a fourth card" on BOE governor Mark Carney's struggles with rules. I am left with more questions than answers, which is good.

Rules

What do we really  mean by "rules?" The clearest version would be mechanical, the Federal Funds rate shall be $i_t = 2\% + 1.5 \times (\pi_t - 2\%) + 0.5 \times (y_t-y^*_t )$ say, with $$i$$ = interest rate, $$\pi$$ = inflation $$y - y^*$$ = output gap. The numbers come in,  the Fed mechanically borrows and lends at that rate. This is something like an idealized gold standard.

That is not what anybody has in mind, obviously.  So what do we really mean by "rules?"

One of the biggest problems is what goes in to the output gap part. If the Fed is going to respond to economic conditions, how do we measure those conditions? Unemployment? The Fed got in a bit of a mess first saying 6.5%, then rethinking whether maybe employment vs. unemployment matters, and then worrying about long-term unemployed. Once you get to "labor market conditions," the line between rule, judgment, and discretion gets muddy.  Output gap? Then relative to whose "potential?" Just how much of current slow growth is "supply" vs. "demand" possibly fixable by monetary policy is at the center of the current policy debate.  It's easy to say "we're really following a rule, we just think the output gap is bigger than you think." Athanasios Orphanides famously pointed out that contemporary views of the output gap in the 1970s justified a lot of loose policy that to later eyes looked like violations of a rule. I don't mean to say it's impossible, or that many people haven't thought long and hard about it, just to point that this is a tough question.

Moreover, I think even the ardent rules supporters have in mind some flexibility to deal with temporary exigencies. The rule is sort of a long-run commitment, not something mechanical. After all, much of the point is to "anchor long run expectations." But  my diet also seems to have a daily temporary exigency, and once again rule vs. discretion gets muddy.

David Papell's presentation and Monika Piazzesi's comments were very thought-provoking in this regard. David set out to measure the extent of rules-based vs. discretionary policy.  This is deep. Fundamentally, if we can't measure something, it becomes a much muddier concept. I don't think David succeeded, but he did the obvious first step and leaves me with a much clearer view of the problem.

David estimated rules with OLS regressions, roughly $i_t = r^* + \phi_{\pi} (\pi_t - \pi^*) + \phi_y (y - y^*) + \varepsilon_t$ He sensibly measured the amount of rule-following vs. discretion by the volatility of the error term, and correlated that volatility with economic performance to try to measure the contribution of rules-based policy to economic stability.

But the Fed can surely answer, "We're following a rule, but you're using the wrong measure of u. Our measure of u becomes your error term."  The Fed can also answer "that's a ridiculously simplified textbook rule. We follow a rule, but it includes a lot of other right hand variables like financial stability, long-term unemployment, housing bubbles and 10 different measures of output gaps. Variation in those omitted right-hand variables is showing up in your error term, not deviations from a rule."

Those replies would also answer the economic performance correlation. The Fed could go on and say "in times of high economic instability, the other components of our rule move around a lot, so there is more omitted-variable volatility. Economic volatility causes estimated Taylor Rule residuals, not the other way around."

More deeply, Mike Woodford's book recommends that the Fed respond directly to shocks to other parts of the economy, or shocks to the "natural rate," and then add Taylor rule responses, $i_t = r_t^* + \phi_{\pi} (\pi_t - \pi^*) + \phi_y (y_t - y^*)$ (There is now a t subscript on $$r^*$$). So optimal rule-based policy has this character of apparent "discretionary" residuals from regressions.

So really where is the line between rule, a guideline (Captain Barbossa),  a general indication of intent, "forward guidance," communication, principled discretion and willy-nilly discretion? Where is the line between law, commitment, promise, pie-crust promise (Mary Poppins, made to be broken), and the golden-retriever approach to life? Is the issue about rules vs. discretion, or is it just about simple and transparent rules vs. complex and obscure rules; about communication rather than commitment?

At a deep level, we social scientists think of the Fed like every other actor as always following "rules," some function from environment to action that describes behavior. Optimization always results in such a rule.  Genuine randomness (the quantum mechanics of behavior?) is't really part of the framework; unpredictable behavior is the result of simplified models and agent's better information, not genuine randomness. (There is an exception for mixed strategies of course, but I don't think that's relevant here.)  So is there anything but rules based policy? This question has long bugged me in interpreting impulse-response functions. The Fed never says "and we added 25 basis points for the fun of it." They always describe all actions as reactions to the environment -- a rule.

Framed that way, I think one answer is before us. If we go back to Kydland and Prescott rules vs. discretion, or Odysseus, the key to a "rule" is precommitment.  You're following a rule (and a rule is beneficial) when you commit to an action ex-ante that you would prefer not to take ex-post, and that commitment has benefits to your overall objective.

"Forward guidance" or "communication" say "here is what we think we will feel like doing in the future." (But we retain the right to change our mind.) A rule says "here is what we will do in the future," maybe describing a state-contingent set of actions, "even if we will not feel like it at the time." ("And here is a set of costs we impose on ourselves so that we will choose to follow through" helps a lot to make it credible.)

It's pretty clear that the Fed has been doing the former, not the latter. The WSJ article on the BOE makes a similar point. Three rules in a year is not a lot of commitment.

This difference is where my scepticism of stimulative promises came from. If the Fed promised to keep rates low in the future, in order to stimulate today, that promise can only have effect if people imagine the Fed chair going to Congress when inflation has hit 5% and saying "no, I promised to keep rates low in order to boost the economy in the recession, and now I have to do that though we all know it's time to raise rates." Nobody believes the Fed chair will do such a thing.  The "guidance" is a "forecast of how the Fed will feel," not a commitment, not a promise with a self-imposed cost, some way of binding itself to the mast.

The intricate legal structure surrounding the Fed, and many of its traditions,  do constitute a lot of "rules," by the way. The Fed might dearly like to drop money from helicopters, buy Treasury debt directly, or lend directly to under-"stimulated" businesses. Legal restrictions against such actions are regretted ex-post, and admired as producing overall better outcomes. At best, forward guidance amounts to a set of promises that the Fed will feel it somewhat costly to renege on.

Now, I think we are ready to start thinking about measurement. I don't think that can be a purely empirical exercise. We need to write down some sort of objective, and find promised behavior ex ante that is regretted ex post, but nonetheless beneficial overall. I'm not sure how to do it, but at least the concept has some potentially measurable content.

Models

My second thought prompted by the conference overall, and made concrete by thinking about David's paper is: What is the model of the economy in which the rule is supposed to work?

In David's regression, we can ask the question: Embed the rule in a model. Suppose that the Fed follows the rule perfectly, and we generate artificial time series from the model, and run the regression. Does the regression reveal the Taylor rule that the Fed is following?

In the new-Keynesian model, the answer is no.  Bob King pointed out  long ago that we can write the Taylor rule in such models as $i_t = i_t^* + \phi_{\pi} (\pi_t - \pi_t^*) + \phi_y (y_t - y^*_t )$ where we now interpret the * variables as equilibrium values, and the non-starred values as deviations from equilibrium. When the Fed follows such a rule, in that model, we observe $$i_t = i_t^*$$ , $$\pi_t = \pi^*_t$$ and $$y_t = y_t^*$$. There is no variation in the right hand variables on which to estimate the Taylor rule. The Taylor rule is not identified when placed in a new Keynesian model. In a new-Keynesian model, the "Taylor rule" becomes the "Taylor Principle," a set of off-equilibrium threats not seen in equilibrium. The Fed introduces instabilty to gain determinacy, rather than introduce stability as it does in old-Keynesian models. (This is a not so subtle plug for "Determinacy and Identification With Taylor Rules")

More generally, the point of monetary policy is to stabilize output and inflation, so simple regressions of interest rates on output and inflation no more measure the policy rule, than simple regressions of inflation and output on interest rates measure the effect of monetary policy. This is a point James Tobin made about 50 years ago (post hoc ergo propter hoc). Chris Sims got a Nobel Prize for VARs to address the problem.

Most simply, monetary policy shocks affect output and inflation, so the right hand variable is correlated with the error term.  The new-Keynesian model is an extreme case of this behavior, in which the right hand variable and error terms are perfectly correlated.

These questions were not really on anyone's mind. They are hard questions, they are old questions, and they don't have easy answers.

The larger question is, what model of the economy do policy people use to think about how monetary policy affects the economy?  The clear answer at this conference is, some unwritten mixture of old Keynesianism and old Monetarism. Old Keynesianism: higher rates reduce "demand" which reduce output which through a Philips curve reduces inflation. Old Monetarism: higher interest rates reduce some quantity of money which works its way through to prices. Neither can be written down or spoken aloud without provoking chuckles. But we had a whole conference on "rules" without an explicit mention of "transmission mechanism" (i.e. "model"), and surely the verbal reasoning conformed more to those 40 year old stories than anything written since.

That wide gulf is worth pondering from both sides.

(There were a lot of really interesting papers and discussions. I especially recommend Marvin Goodfriend's paper, which I'll try to blog at some point in the future.)

1. I am the only one who thinks that "forward guidance" is an incredible waste of time? The commitment is not binding and I cannot imagine any information that the fed has unique possession of. The whole song and dance feels far too much like the Wizard of OZ .

2. “The Fed got in a bit of a mess first saying 6.5%, then rethinking whether maybe employment vs. unemployment matters, and then worrying about long-term unemployed. Once you get to "labor market conditions," the line between rule, judgment, and discretion gets muddy. Output gap? Then relative to whose "potential?" Just how much of current slow growth is "supply" vs. "demand" possibly fixable by monetary policy is certainly open to debate.”

And about predictive economics, “willy-nilly discretion” and the overall abysmal track record regarding discretion related to predicting:

[Paraphrasing] Forecasting people's activity, who are they, themselves, forecasting, with those forecasting the forecasters, in turn, having to forecast the forecasters' forecast of what politicos do or don't do.…….can lead to poor forecasting. - Alex Tabarrok

3. Were any market monetarists invited to the conference, or was it mostly a meeting of like minds? Many have argued that unemployment rates are a poor benchmark for stabilization. On the other hand, they praise the performance of Stanley Fischer during his tenure as central banker in Israel. Even someone like me (without a Ph.D.) can look at a graph of Israeli nominal GDP during that time and conclude that a rule was being followed.

With so much demand for its assets, the U.S. stuck at the zero lower-bound does face a more difficult stabilization problem than Israel. I think this is a good time to ask if we should move from stabilization via policy towards more automatic stabilization - by having debt indexed to nominal GDP. Kevin Sheedy provides theory that supports this proposal, though he does not advocate such a radical change himself.

1. Anwer,

Bob Shiller and Mark Kamstra make the same case here:

http://cowles.econ.yale.edu/P/cd/d17a/d1717.pdf

"This new debt instrument should be of great interest to the Government for its stabilizing influence on the budget (as coupon payments fall in a recession with declining tax revenues) and for its yield, based on our valuation."

Very true, but Keynes had different ideas about the role of government - namely that it should act in a countercyclical fashion. Coupon payments (as a government expenditure) based on GDP would fall when GDP falls and rise when GDP rises, meaning government would be acting in a pro-cyclical fashion.

That may be okay for investors who trade on economic volatility and boom / bust cycles, but it would ultimately destabilize GDP growth. Felix Simon describes some other problems with them here:

Better would be securities that operate in a countercyclical fashion - high return / potential return when growth is below potential and low return / potential return when growth is above potential (ala Keynes).

Coupon debt indexed to the output gap would make more sense.

Equity in the form of tax burden reducing securities would make even more sense. In that case the owner of equity would have the ability to positively affect the return on his / her investment rather than relying entirely on a macro variable like GDP or output gap. In short the owner of the equity would have an individual incentive to seek employment / income source, to generate a taxable income, to realize the return on government equity that is owned.

With tax burden reducing securities you get both the stabilizing influence on budget - as employment falls, fewer people can immediately realize the return on investment, and you countercyclical action by the federal government when it raises the potential return on investment in response to a high output gap.

2. Frank: This is a great comment. The revised version of the paper (which I will post when done) will include a much deeper set of thoughts on inflation-stabilizing government debt, and will reflect this.
John

3. Frank I do remember Shiller making that argument in favour of his proposed debt instrument, and I think you are right that this focus on government budget stabilization is misplaced. The US federal government has great access to credit and should use it to help us smooth out shocks with counter-cyclical policies of the type that you suggest.

I'm thinking of the debts of people like you and me. Atif Mian and Amir Sufi (the latter a colleague of Cochrane's at Booth) have been writing and speaking about the effects of debt overhang on consumers, and how it amplifies macroeconomic cycles. Their policy proposals advocate some degree of debt forgiveness during downturns. They want to index home loans to some measure of local home prices. In cases where there is no natural benchmark, I think that it's good to index debt to nominal GDP. Kevin Sheedy makes an argument for market monetarism by noting that well-timed inflation has that effect on the real value of debts, which makes it stabilizing.

Ultimately the principle I'm using is that risk should be borne by the least risk-averse party, which should be those with greater wealth. Generally speaking, it should be efficient for lenders to insure borrowers from macroeconomic risk. One way to make that the default is to use Shiller's Trills as the asset that is loaned to consumers.

I've read Felix Salmon's analyses of Trills, and what I remember of them is that he doesn't know what discount factor to use when pricing Trills. Salmon tries to conclude that asset pricing fails when applied to the aggregate economy. That's clearly absurd, but I'll leave the proof as an exercise for someone who has taken Prof. Cochrane's MOOC.

4. Anwer,

"I think you are right that this focus on government budget stabilization is misplaced.."

I don't think I said that. I said that government policy should be countercyclical rather than pro-cyclical (at least according to Keynes). Also, I don't think you should necessarily look at the nominal credit markets to determine what government policy should be.

First, if I loan you money at 3% to buy a gun and shoot your neighbor would you? How about 2%?

Second, who should regulate the credit markets - the central bank, the federal government, or some combination of the two? Saying that the federal government should base its spending decisions on the cost of its own borrowing misses the point that part of the federal government's job is to regulate the cost of credit. You can try to argue that the central bank is independent of the federal government to which I would say - when the bullets start flying, does the central bank dare to contradict the Congress? You can try to argue that too much government debt would bankrupt a country to which I would say - a government can always rollover the principle and extend duration.

There are plenty of good reasons for the federal government to limit it's borrowing capacity (trade balance, productivity, etc.). Bankruptcy / insolvency risk is not one of them.

There are plenty of reasons the federal government should avoid buying guns and shooting its neighbors regardless of whether it is paying 1% on borrowed money or 10% on borrowed money.

See Irving Fisher's Separation Theorem

http://en.wikipedia.org/wiki/Fisher_separation_theorem

- For monopoly enterprises (like the federal government) spending / investment decision is independent of financing decision.

5. There is a different sense in which NGDP-linked debt can be counter-cyclical and stabilizing, which is how I thought market monetarists envisioned it. The Fed would commit to buying and selling such debt at a price such that implied NGDP equaled the Fed's target for NGDP. Implied NGDP here is analogous to breakeven inflation in TIPS. When market expected NGDP is below target, people will want to sell NGDP-linked securities, so the Fed will be a net buyer --- automatic QE. Conversely, when market expected NGDP is above target, the Fed will be a net seller: automatic contractionary policy. Pegging implied NGDP would seem to be a transparent rule.

What are the arguments for why this wouldn't work? Perhaps, there are some difficulties in setting up such a market initially. However, we already have a TIPS market. Could the Fed peg TIPS breakeven inflation for rules-based inflation targeting?

To be fair, there may also be an issue in relating the pegged implied NGDP level to market expected NGDP. NGDP seems to be correlated to the stock market (positive beta) so, presumably, there will be a risk premium between expected NGDP and "breakeven NGDP". If risk premiums were stable, so would be the gap between expected and breakeven NGDP, but risk premiums may not be stable.

6. This resembles Scott Sumner's proposals for NGDP futures targeting, that are sometimes discussed on his blog. A good critique was posted yesterday by Steve Cecchetti and Kim Schoenholtz. I'm not sure if I can post links here but their post is titled "Monetary policy target regimes: inflation, price level, nominal GDP, etc."

They note that NGDP targeting does protect debtors, but creates unpredictable inflation as a byproduct. This is why I advocate innovation in debt contracts; it allows us to attain multiple goals simultaneously - and price stability is an important one.

But my preferred innovation is a radical one, and I think that what you, Sumner and others suggest is intended as a second-best option that is more feasible politically.

7. BC,

Presumably NGDP linked debt is a federal government liability. Presumably also that central bank operations must occur on the open market. How do fiscal and monetary policy interplay?

When NGDP (actual) is below the Fed's target for NGDP, the Fed will want to be a net buyer. People (including the federal government) may be net sellers or buyers.

Conversely, when NGDP (actual) is above the Fed's target, the Fed will want to be a net seller. Again people (including the federal government) may be net buyers or sellers.

The biggest problem with open market operations as countercyclical policy is that it takes two to tango - for every asset the central bank wants to buy / sell, someone else must take the opposite position.

Government / people may not want to take that opposite position.

8. BC,

Here are the possibilities:

Situation #1 (Revenge of the mattress stuffer)
NGDP (actual) is below Fed's target for NGDP
Fed wants to be net buyer
Government wants to be net buyer
People want to be net buyer

Situation #2
NGDP (actual) is below Fed's target for NGDP
Fed wants to be net buyer
Government wants to be net seller
People want to be net seller

Situation #3
NGDP (actual) is below Fed's target for NGDP
Fed wants to be net buyer
Government wants to be net buyer
People want to be net seller

Situation #4
NGDP (actual) is below Fed's target for NGDP
Fed wants to be net buyer
Government wants to be net seller
People want to be net buyer

Situation #5 (Let the good times roll)
NGDP (actual) is above Fed's target for NGDP
Fed wants to be net seller
Government wants to be net seller
People want to be net seller

Situation #6
NGDP (actual) is above Fed's target for NGDP
Fed wants to be net seller
Government wants to be net buyer
People want to be net buyer

Situation #7
NGDP (actual) is above Fed's target for NGDP
Fed wants to be net seller
Government wants to be net buyer
People want to be net seller

Situation #8
NGDP (actual) is above Fed's target for NGDP
Fed wants to be net seller
Government wants to be net seller
People want to be net buyer

Under situations #1 and #5, government and people are competing for the same securities. Central bank makes things worse by trying to take same position as people and government.

Under situations #2 and #6, government and people are competing for the same securities. Central bank makes things better by taking opposite position as people and government.

Under situations #3, #4, #7, and #8, government and people are taking opposite positions. Central bank's best option is to do nothing.

Notice that countercyclical open market operations by the central bank is not dependent on whether NGDP is above or below target. It is dependent on whether people and government are net buyers / sellers of NGDP linked securities.

9. Frank I think it's useful to look carefully at processes of adjustment. If everyone wants to buy a security but no one wants to sell it, the price must rise until there is equilibrium. The whole point of policy is that we want to adjust to shocks in the best possible way (with reductions in output and employment as the worst way to adjust).

NGDP targeting uses the price level to adjust to shocks, and this makes it unpopular with those who believe strongly in price stability. One theory used to justify this type of targeting is that ultimately we need adjustments in the real value of debts, and this is attainable through control of the price level. If the best way to stabilize really is by adjusting balance sheets, then we should just write that into debt contracts and avoid disturbing prices unnecessarily. With less uncertainty, we'll have more economic output.

10. Anwer,

"Frank I think it's useful to look carefully at processes of adjustment. If everyone wants to buy a security but no one wants to sell it, the price must rise until there is equilibrium."

Assuming rational profit seeking agents. Two of those agents (central bank, government) are not for profit enterprises (and they may not even be rational).

And so central bank and government may both want to buy a security price be damned. The individual person can simply wait to see who offers the highest price. Price is determined by the patience of the individual not by any sort of equilibrium.

Likewise a central bank and government may both want to sell a security price be damned. The individual person can simply wait to see who is willing to take the lowest price. Again, price is determined by the patience of the individual not by any sort of equilibrium.

"The whole point of policy is that we want to adjust to shocks in the best possible way (with reductions in output and employment as the worst way to adjust)."

Which policy - central bank or government? Central bank open market purchases are all about liquidity. Individual / government has security to buy / sell, central bank takes opposite side of the trade - very simple. Of course the government places limitations on what the central bank can buy / sell, but I am presuming that NGDP linked securities would be okayed.

If central bank tries to get on the same side of the trade as everyone else because NGDP is higher / lower than target, then it will defeat the whole point of open market operations. Fed will be pulling liquidity when liquidity is demanded and pushing liquidity when it is not.

Yes I understand what policy tries to do. But policy by the central bank that makes market failures more likely seems counterintuitive.

4. Good post.

There is a simple way to test whether a central bank has been following a rule to target (say) 2% inflation at a horizon of (say) 2 years. (Strictly, what you are testing is the joint hypothesis that the central bank is following that rule AND that the central bank has rational expectations.)

You run a simple regression.

On the left hand side you have deviations of inflation from 2%.

On the right hand side you have any variables that were in the central bank's information set with a 2 year lag. (Strictly, you need to use real-time data.)

If any of the RHS variables can forecast deviations of inflation from the 2% target at a 2 year horizon, you reject the joint hypothesis. (Because forecast errors should be orthogonal with respect to the information set under rational expectations.)

My old paper, with James Yetman, testing this on the Bank of Canada: http://www.jstor.org/discover/10.2307/3131945?uid=3739464&uid=2129&uid=2&uid=70&uid=3737720&uid=4&sid=21104114209267

5. One solution to the rules vs. discretion problem is to follow a level targeting regime (e.g. for NGDP or a price index) but allow intentional deviations in the short run. Then the implicit rule is, "If the target variable gets far off the target path, we will do something extreme to get it back on and keep doing that extreme thing until it does get back on." And of course then the central bank always has to consider that, when it allows intentional (or unintentional) deviations from the target path, it is making a potential commitment to do something extreme in the future. So if it has objectives may cause it to want to deviate, it has to weigh the advantage of deviation with the risk of needing extreme measures in the future.

Comments are welcome. Keep it short, polite, and on topic.

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