All Road Lead To r*

One of the major theoretical tasks facing neoclassical economists is the estimation of the variable r*, which is similar to what was termed the natural rate of interest. This concept has a long history, and as will be discussed here, this will continue so long as mainstream economics bears any resemblance to the current consensus theory. The reason is straightforward: if one assumes that the real rate of interest is a major driver of economic outcomes, we need to estimate what the neutral position of the control variable is.

My analysis here is based on a linear model framework. This matches the (old) standard practice of using linearisations of dynamic stochastic general equilibrium (DSGE) models. I expect two immediate responses to this statement, both of which miss the point of what I am trying to make. The first are statements about the uselessness of linear models from critics of DSGE. The second will be condescending statements from DSGE defenders about the new generations of nonlinear models that are being ignored.

I am ignoring the nonlinear models for one key reason: I view them to be teaching models, not empirical models (a distinction I discussed earlier). I see no value in having arguments about the capabilities of teaching models; there is an infinite number of them, and they can be used to demonstrate practically any desired behaviour. My concern is the application of the models to the real world, and linear models -- and close cousins -- are far more tractable. I will return to nonlinear DSGE models later, but their complexity is a distraction from what I want to discuss here.

Conventional Linear Models

Rather than chase after a particular model, I want to discuss any model in an extremely wide class of models, which I term conventional linear models. My argument is that this class would capture the linearisation of most DSGE models, as well as other models that might be used by central bankers.

The models are assumed to have the following properties.

1. The model is discrete time (not continuous). Since my interest is models that are fitted to data, and observed macroeconomic data are discrete time (e.g., monthly, quarterly), this is not restrictive.
2. The model is linear. This is restrictive, but nonlinear models will be discussed elsewhere.
3. The real short-term policy rate is a variable that can be set to control the economy. It does not matter whether the inflation rate used is expectations or historical inflation rates, but the assumption is that it would be measurable. (This assumption is what classifies the model as being "conventional," as it it follows the conventional thinking about interest rate policy.) The exact mechanism of this control is somewhat open, but I will discuss the effect as being an "acceleration", with the variable denoted a(t). (The linearity assumption does not pin down the sign of the effect on the acceleration. If the sign is flipped, interest rate policy works backwards from the usual assumption -- which might be termed an "anti-conventional model.")
4. We assume that the central bank can (in some manner) set the real interest rate. This means that it is not possible for the inflation rate to move in parallel with the nominal interest rate. (E.g., the inflation rate is an expected rate that is assumed to be the nominal policy rate less 2%, which implies a constant real rate.) This assumption is justifiable on the argument that no observed inflation rate (historical or measured expectations) moves one-for-one with the policy rate. Note that this would exclude some fairly deranged DSGE models.

Under these assumptions, if we fix a time point, we see that the acceleration is a linear function (plus constant) of the real interest rate if all other variables are fixed. This implies that there is a value of the real rate (r(t)) where the acceleration is zero.

I will now deal with some technical digressions that allow me to justify that this is a general property of any such model.

Interest Rates: Set Each Period, or a Reaction Function?

One of the immediate pedantic objections one might make is that the interest rate policy in a DSGE model is the result of a reaction function; it is not the central bank setting the interest rate each month. This objection does not pose any substantive issues for what I am discussing, and for simplicity, I will just refer to the real rate being set. There are a number of justifications for not explicitly dealing with reaction functions.

• Not every model specifies the interest rate as a reaction function.
• If the model specifies the interest rate as a reaction function, we can just see the effect of replacing the old reaction function with a new one, shifted by a constant. So long as we avoid feedback loops cancelling out nominal rate changes completely (which is implicitly covered by the last assumption), the real rate of interest is a policy variable that can be changed.
• If one insists that interest rate policy is a reaction function, one needs to embed the entire yield curve within each time step of a historical analysis. I am not discussing forward time, I am discussing calendar time. It is unclear whether such a model would end up qualifying as a linear model.
• (Alternative phrasing of the previous point.) In practice, we are calibrating a model against an observed real policy rate. I am interested in this calibrated model, not the theoretical properties of a model that cannot be fit to data directly.

Acceleration: Vague for a Reason

Since I want to hit every model possible, I have to leave the notion of "acceleration" unspecified. When we are looking at an attempt to fitting the model against data, it is the variable that is affected by the real rate. 

The equation is assumed to collapse to:

a(t) = b(t) r(t) + c(t),

where:

• t refers to a point in calendar time that is being isolated. (It does not refer to arbitrary time points on a forward time axis, which could potentially cause confusion.)
• a(t) is the acceleration associated with the real interest rate at time t.
• r(t) is the real policy rate at time t.
• b(t), c(t) are constants, which may vary over time. Under the assumption that the model is linear, c(t) is capturing the effects of all other variables.

We then have some technical points.

• The acceleration variable may not be an economic variable within the model, rather it might be a re-labelling of a variable of interest. The most important case may be time lags -- it could be the change of an economic variable at a future time point. In this case a(t) is time-shifted with respect to the model.
• If the effect of an interest rate change shows up with multiple lags, the variable a(t) would be some form of average effect.
• I use acceleration as that is fairly conventional to assume that interest rate changes provide a shock to either real GDP growth rates, or inflation rates (or both). It is possible that a model could  have the variables of interest move parallel with the interest rate change. Since this case is somewhat far-fetched, I prefer not to use a more ambiguous term than "acceleration."
• The interactions of the variables might be complex, so a(t) might be a control input to the variables of interest (normally GDP, inflation).

At Any Given Time Point, There is a No-Acceleration Rate

Since we know that a linear model has to be of the following form (if all variables other than r(t) is fixed):

a(t) = b(t) r(t) + c(t),

we can re-write it as:

a(t) = b(t) (r(t)  + c(t)/b(t)).

That is, the acceleration is a scaling (by b(t)) of the difference between the real policy rate and the current time point's "neutral" level (-c(t)/b(t)). (Why can we divide through by b(t)? If b(t) is zero, then interest rate policy has no effect on the economy, which we preclude by the conventional economic model assumption.) That is, if r(t) is set to match the "neutral" level, the acceleration associated with time t is zero.

Note that the "neutral rate" at a time period is not r*; it will rise and fall in line with the other variables in the model. It is the level of rates needed to cancel out the other variables at a time point. (I return to r* later.)

Why Do We Care?

Central bankers exemplify conventional thinking. Their main job is to steer the economy with interest rate policy, and so that is what they focus on.

• When a policy meeting comes up, they need to know what rate of interest to set now, Although there might be New Keynesian central bankers who love bantering about reaction functions, they need to announce a new target interest rate at the end of the meeting.
• They do not want to hear that interest rate policies are ineffective.
• They do not want to hear blah-blah about multiple equilibria, or whatever. Even if the policy rate is a range, it is normally an interval, not a set of intervals.

In other words, they need a "neutral" baseline for the policy rate, so that they can set the policy rate relative to that based on the desired policy stance.

Since central banks are a major source of economics research funding (and prestige appointments), the implication is that there will always be a flow of conventional models to be calibrated.

Relation to r*

To get from the time period neutral rate to r*, we need to add an assumption (that puts it in line with most DSGE models). We have to assume that there is a long-run steady state in the economic model. (In a steady state, there is no acceleration, but variables can grow at fixed rates.)

Under the uniqueness assumption, we can look at all state variables versus their steady-state level. We denote these steady-state variables with stars, and can re-write the linear system equations in terms of deviations from their "star value." That is, r* is the steady state value of r.

The relationship between r* and the one-period "neutral" rate described earlier is that the neutral rate will coincide with r* if we are are in steady state -- or all other deviations cancel out exactly (presumably a low probability event). However, it gives a baseline interest rate, that is modified by other deviations from steady state. We need to estimate all the variables (and their star values) simultaneously given this interaction.

Concluding Remarks

The above logic explains why r* is going to be a central concept in any mainstream attempts to discuss policy. The addition of nonlinearities just makes empirical work harder, but most likely ends up trapped chasing something that resembles r*. 

Unfortunately, the theoretical inevitability of r* appearing in mainstream models does not imply that the concept is useful. I obviously have strong doubts. I will return to this topic in upcoming articles. (I have discussed my complaints previously, but I am re-writing them in a cleaner fashion.)

(c) Brian Romanchuk 2020