A Market Monetarist version of the McCallum rule

McCallum – a inspiration for Market Monetarists

Scott Sumner has often expressed his admiration for Bennett T. McCallum. I share Scott’s view of McCallum and think that his work on especially monetary policy rules has been much underappreciated.

Even though McCallum does not automatically qualify as a Market Monetarist there is no doubt that a lot of his work have similarities with views expressed by the main Market Monetarists.

McCallum is a Market Monetarist in the sense that he stresses the money base rather than interest rates as the key instrument for monetary policy. Furthermore, McCallum argues that the central banks should target nominal GDP (NGDP) rather inflation or the price level. However, contrary to Market Monetarists McCallum stresses the rate of growth of NGDP rather than the level of NGDP.

The McCallum rule – nearly Market Monetarist

McCallum’s work on monetary policy has led him to propose the so-called McCallum rule. Here I am quoting from his 2006 paper “Policy-Rule Retrospective on the Greenspan Era”:

“This rule specifies the growth rate of the monetary base that the Fed should generate, rather than the value of the FF interest rate. Although in fact the Fed does not control growth of the monetary base, it could do so if it chose to and, in any event, we can use this growth rate as an indicator of monetary policy ease or restrictiveness, even if the Fed is not operating so as to exert control of this rate. The rule can be written as

(1)                 ∆bt = ∆x* − ∆vt + 0.5(∆x* − ∆xt-1).

Here the symbols are: ∆bt = rate of growth of the monetary base, percent per year; ∆vt = rate of growth of base velocity, averaged over previous four years; ∆xt = rate of growth of nominal GDP; ∆x* = target rate of growth of nominal GDP. In rule (2) the target value ∆x* is taken to be the sum of π*, the target inflation rate, and the long-run average rate of growth of real GDP (which is presumably unaffected by monetary policy). I take the latter to be 3 percent per year, so with an inflation target of 2 percent, we have ∆x* equal to 5.”

A couple of Market Monetarist modifications 

So far so good. The target for NGDP growth is by the way the same as suggested by Scott Sumner and seems more less to be what the Federal Reserve implicitly was targeting during the Great Moderation and McCallum has in a number of papers demonstrated empirically that the Federal Reserve policies during the Great Moderation more or less were in line with the McCallum rule.

Even though the McCallum rule is rather Market Monetarist in nature it is still not the real thing. I would especially highlight two weaknesses in the McCallum rule that need to be corrected to make it truly Market Monetarist.

First of all, the McCallum rule does not take into account changes in the money multiplier, which obviously is a serious defect in the present situation where the money multiplier has decreased significantly. McCallum implicitly assumes a constant money multiplier. This is obviously problematic, as both traditional monetarists as well as Market Monetarists would argue that is it the development in broader monetary aggregates rather than the money base, which is important for NGDP.

David Beckworth and Josh Hendrickson have both suggested modifying the equation of exchange (MV=PY) to take into account changes in the money multiplier. From Beckworth:

“Note first that since the money supply (M) is a product of the monetary base (B) times the money multiplier (m), MV=PY can be expanded to the following:

BmV = PY

In this form, the equation says (1) the monetary base times (2) the money multiplier times (3) velocity equals (4) nominal GDP or total nominal spending (i.e. aggregate demand). The Fed has complete control over the monetary base, B, which is comprised of bank reserves and currency in circulation.”

As McCallum’s starting point is the traditional equation of exchange it is straightforward to incorporate the Beckworth’s and Hendrickson’s insight. So the first step modification would be to re-write the McCallum rule to:

∆bt = ∆x* − ∆vt − ∆mt + 0.5(∆x* − ∆xt-1).

Note that in this version of the McCallum rule ∆vt is the rate of growth of broad money (for example M2) velocity averaged over the previous four years. In the original version v was velocity of base money rather than of velocity of M2. ∆mt is the rate of change in the money multiplier averaged over the previous four years. This modification therefore mean that we indirectly are targeting broad money rather the money base.

My second Market Monetarist objection to the original McCallum rule is that it fundamentally is backward looking in nature. Market Monetarists stress that market prices provides the best information for the tightness of monetary policy and therefore the best information for forecasting NGDP. Hence, instead of using the lagged development in NGDP the expected growth for NGDP should be used.

My own – quite preliminary – empirical work indicates that NGDP growth one quarter ahead can be forecasted quite successfully based on relatively few financial variables (stock prices, bond yields, the dollar index and commodity prices).

I suggest forecasting the following model for expected NGDP growth model on US data:

E(∆xt+1)=a0+a1∆st+a2∆rt+a3∆et+a4∆ct+a5 ∆xt

Where, ∆st the quarterly change in S&P500, ∆rt is the quarterly change in 30-year or 10-year US Treasury bonds (whatever works best), ∆et the quarterly change in an index for a nominal trade weighted dollar-index and ∆ct is the quarterly change in global commodity prices (for example the CRB index). The expected signs of the coefficients would be a1,a2,a4,a5>0 and a3<0 (higher e is assumed to mean stronger dollar).

Estimating this model should be straightforward even though there obviously could be problems in terms of multiple-correlation, but those challenges should be relatively easy to overcome.

Obviously it would not be necessary to estimate an equation for expected NGDP if there was a tradable NGDP future, but as we all know such a things does not exist in the real world.

Using the estimated equation for expected NGDP growth one quarter ahead we get the following modified McCallum rule:

∆bt = ∆x* − ∆vt − ∆mt + 0.5(∆x* − E(∆xt+1))

Now we are pretty close to having a Market Monetarist version of the McCallum rule, which takes into account changes in the money multiplier and is forward-looking in nature.

The McCallum-Christensen rule

The only issue that we have not discussed is McCallum’s focus on the growth of NGDP rather than the level of NGDP. However, it should be noted that the McCallum rule is a feedback rule and if in NGDP growth (lagged/forecasted) falls below ∆x* then the money base will be expanded. This should ensure that NGDP returns to a stationary path, but it is not given that it would return to the pre-shock trend if a shock where to hit the economy.

The size of the coefficient on the feedback part of the modified McCallum rule does therefore not necessarily have to be 0.5. It could be bigger or smaller.

Therefore, I suggest writing the rule in a more generalized form:

∆bt = ∆x* − ∆vt − ∆mt + beta(∆x* − E(∆xt+1))

Where beta is a coefficient bigger than zero. Further empirical work and simulations of the rule will have to show what would be the appropriate size for beta. I hope some of my readers will take of the challenge of the further empirical work on the McCallum-Christensen rule.

Leave a comment


  1. Wouldn’t that expected x formula get us into the circularity problem? And shouldn’t we try to get the level of NGDP in there somehow? How about:

    Dbt = Dx* – Dvt – beta * DxL(t-1)

    D = delta. DxL(t) is NGDP deviations from trend.

    I think this should be enough since the market will know that NGDP will eventually get back to trend. So there’s no need for the market to worry that the central bank will screw up in the long run. Not as good as Scott’s NGDP futures idea, but far better than what we have now. Someone should get McCallum to start blogging!

    I don’t see the need to split up base velocity. The money multiplier and money velocity is part of the base velocity.

  2. Peter,

    I don’t think there will be a problem with circularity. In fact with a market based forwarded looking like this the Federal Reserve would likely drastically have increased the money money base well before the collapse of Lehman Brothers and not after as was actually the case.

    I guess when you refer to circularity you think that the markets would start to price in recovery once the rule was announced and that therefore would be no increase in the money base. That might be it, but that is exactly the point – money policy works with long and variable LEADS.

    In terms of money multiplier I think it is an important element. In fact if the Federal Reserve had followed a normal McCallum the response to the Great Recession would have been significantly weaker than it has been in reality.

  3. I was thinking of a circularity problem like Scott explained here: http://www.themoneyillusion.com/?p=7210 . Maybe that wont be a problem for a rule based approach. I was just concerned about the calculations for expected future NGDP.

    I still don’t get the money multiplier. Bv = BmV. So v = mV. ln(v) = ln(m) + ln(V). So McCallum’s rule should be equal to your version in that regard. What am I missing?

  4. Peter, now I get it. You should note that I replace the money base in the model with broad money (M2). But have a look at Beckworth comments – that will make it more clear.

  5. Benjamin Cole

     /  October 9, 2011

    Fascinating post. I leave the fine points to others—I think as long as the public understands the Fed is targeting nominal GDP, and will use all of its guns to get there, we will be fine.

    Like I say, the Fed ought to release a picture of Bernanke with his hand on the lever of a printing press. The caption: “Think I cannot get nominal GDP up? Make my day.” (Any photoshopppers out there? We need this photo to make our point.)

    Go M-Cubed! The Market Monetarism Movement.

  6. Benjamin, something like this? http://www.quickmeme.com/meme/355w4z

  7. Benjamin Cole

     /  October 10, 2011


    That’s fun, but I would prefer a photo of a Department of Treasury printing press, perhaps with a workman tending it, except the workman has his face replaced by Bernanke’s. If Bernanke’s hand was on a lever of some sort all the better.

  8. Ultimately, the only thing that central banks control is their own balance sheet. All else is commentary (i.e. part of the Bank’s communications strategy). Interest rates are an intermediate target. The Bank of Canada for example makes a commitment every 6 weeks to keep that intermediate target (the overnight rate of interest) constant for 6 weeks.

    The funny thing is, New Keynesians say that intermediate targets are unnecessary. And they use this to argue that monetary aggregate should play no role in targeting inflation. [Can anyone remember whose paper I’m trying to recall here??]. But by the same token, interest rates are unnecessary, because they are just another intermediate target.

    Damn. Whose paper am I thinking of??.

  9. Alex Salter

     /  October 10, 2011

    Solid primer. I’m a little worried your NGDP forecasting equation is massively sensitive to endogenous offsetting behavior, but the rest is a good exercise to market monetarist logic.

  1. Horwitz, McCallum and Markets (and nothing about Rush) « The Market Monetarist

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