Revisiting the P-star model

I read Milton Friedman’s book “Free to Choose” at an age of 16 years old and ever since then I have been more or less obsessed with monetary theory and particularly the equation of exchange:


My view of the world obviously has developed over the 32 years since I read “Free to Choose”, but I am still fully convinced that monetary policy failure historically has been the main cause of macroeconomic problems – whether it is inflation or recessions and depressions. In fact I am more so than ever.

When I started studying economics at the University of Copenhagen in the early 1990s my obsession with monetary matters continued. That more or less coincided with the publication of a paper, which had quite an impact on my general thinking of how to empirically think about monetary analysis.

The paper “M2 per unit of potential GNP as an anchor for the price level” was written by Jeffrey J. Hallman, Richard D. Porter and David H. Small and first published in 1989.

In the paper the authors introduced the concept of P-star as a measure of where the price level would be in the “long run” (when monetary velocity and GDP was at their long term equilibrium levels). An updated version of the paper was also published in 1991.

Based on the equation of exchange this price level – P-star or P* – was calculated:

P* = M•V*/Y*

Where M is the present level of some monetary aggregate (in Hallman et al.’s paper M2 for the US), V* is the long-term trend level of money-velocity and Y* is potential GDP.

Hallman et al. argued that the actual price level, P, over time should converge towards P*.

Consequently, the gap between P and P* should be a useful indicator of future inflation. Hence, if P*>P then we should expect inflation to accelerate and if P<P* then inflation should decelerate.

This made a lot of sense in the late 1980s when we where in a situation where the Federal Reserve and other central banks had not formulated their nominal targets in any clear fashion and where monetary policy partly still was conducted through money base control.

However, starting from the early 1990s more and more central banks introduced inflation targeting and operationally started being exclusively focused on conducting monetary policy through interest rate controls it became (gradually!) clear to me that the P-star concept might not be useful any more as the price level would be anchored by the inflation target alone and causality in the model would be turned around and velocity would hence become a function of the inflation target.

Therefore, I more or less gave up on the P-star model and only occasionally revisited it when doing analysis of different Emerging and ‘Frontier’ markets, but for some reason my previous post on the McCallum rule made me think it could be fun to have a look at the P-star model using the ‘outside base’ (the money base minus excess reserves) as a measure of M in the model.

So this is what I am going to do in this blog post.

Calculating P-star based on the outside base

To calculate P-star we need an estimate of V* and Y*.

Y* is simply potential GDP and I here I use CBO’s estimate of potential GDP, which I get from St. Louis Fed’s FRED database.

In terms of V* I calculated V based on actual nominal GDP and the outside base (V=NGDP/Outside base).

And then I have de-trended that. I could have used a HP-filter or something similar, but instead I simply estimated a trend based on a linear, squadric and an inverse trend of V.

Velocity trend.jpg

So now I have both V* and Y* and using the outside base as a measure of M I can calculate P*.

The graph below shows my measure of US P* since 1984 and the actual price level P (the GDP deflator).


The orange line, P, is the actual price level while the blue line is P* (P-star). The gap (%) between the two, is the green bars (p-gap).

One can think of the p-gap as a measure of excess liquidity in the economy and when p-gap is positive (negative) monetary conditions are ‘easy’ (‘tight’) and one should expect nominal demand and inflation to pick up (slow down).

It is notable that the p-gap turned negative ahead of the US recessions of ’90-’91, 2000-1 and 2008-9 and in that sense has been a reliable leading indicator of recessions in the US for more than three decades.

But is it also a reliable indicator of inflation?

The simple answer is YES.

To test this I run a simple OLS-regression.

dp(t) = a + b•pE(t-1) + c•p-gap(t-4)

Where dp(t) is the quarterly change in the price level, pE(t) is inflation expectations of consumers (University of Michigan survey) and p-gap(t-4) is the price gap 4 quarters earlier. a, b and c are coefficients.

And here is the model output.

Model p-star.jpg

This isn’t rocket science and will certainly not win me the Nobel Prize, but it is good enough for a blog post. What we see here is that p-gap is statistically significant and hence can be used to predict changes in inflation.

This makes me think that p-star could be a useful indicator for inflation also going forward and maybe it deserves a bit more attention that it has been getting – at for the past 20 year.

In fact I am tempted to say that p-star is no worse (or better) an indicator of the inflationary outlook than i was back in 1989 when it was first suggested.

I will try to do a bit more work on the p-star model going forward and maybe try to model p-star with other money supply measures and maybe for the euro zone as well.

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