## Thursday, April 11, 2024

### My r* Concerns

I recently wrote about r*, which is now the preferred way to refer to the “neutral” or “natural rate” of interest (in real terms). Although my concerns appear hand-wavy, there is a way of expressing them mathematically. I have discussed this in the past, but I hope this version is cleaner.

The first thing to note is that there are multiple ways of estimating r*. I am not too concerned about which one is used, since the ones that I have seen share an important property, which I will shortly describe.

The estimation algorithm is based upon a number of time series inputs. For my purposes, I divide the inputs into two parts: interest rates, and everything else. I will call “everything else” the “real variables” since that is generally what they are.

The property of interest is that if we fix the real variables to match what we saw historically, and replace the interest rate variables with new ones that are x% above the historical interest rates, the r* estimator output would converge to x% the estimate for the historical data. (Depending on the initial conditions, they might start different, but would end up with the x% level shift.)

So what? Imagine that the “true” economic model is that the real economic variables are completely invariant to the level of interest rates. (Note that there is a problem with that assumption, which I will describe later.) We could imagine an infinite number of parallel universes, where the prevailing interest rates were x% different than what we saw historically. What would then happen in those universes is that they would come up an r* estimate that is x% different — but the difference between their real rate and their estimate of r* would be identical to the difference in our universe. The implication is that whatever evidence we have that interest rates affect real variables as conventional logic suggests, they would have the exact same evidence. This is despite the fact that we know that the real variables are unaffected by interest rates under our model assumption.

The interpretation of this is that r* estimators will adapt to the prevailing level of interest rates, and so they appear to offer evidence that interest rate policy works as is conventionally assumed.

For example, go back to a period where estimates of r* were stable at about 2%. Policymakers and market participants acted on the assumption that a real policy rate above 2% would be restrictive, and set rates in that fashion — keeping the policy rate restrictive late in the cycle to fight inflation. We then imagine an alternate scenario where policymakers kept rates 1% higher, and based on the historical experience, would believe that a 3% real rate is needed to be restrictive. However, we are assuming that the business cycle evolves independent of interest rates — and so neither the 2% or the 3% level were significant for future outcomes.

I will then return to a point that I noted: we know that interest rates cannot have no effect on all variables in the economy, since interest rates will determine interest income flows, and so they end up different. The question is: how much does this effect matter for the real variables (e.g., GDP, inflation rates) used in the r* estimators, particularly for relatively small shocks to interest rates (like 200 basis points or less)?

My intuition is that the only way to “break” this adaptation to the prevailing level of interest rates is have the r* estimate be determined by real variables only. The simplest example is an assumption that r* is a constant, like 2%. One popular version was that it would be equal to the long-run real growth rate of the economy (which raises other estimation issues).

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(c) Brian Romanchuk 2024