How do you estimate something you cannot observe?
That is the basic problem with r-star, the neutral real interest rate. In the standard monetary-policy framework, it is the real interest rate consistent with the economy operating at potential and inflation stable. Put the real policy rate below r-star and monetary policy should stimulate demand. Put it above r-star and policy should restrain it.
But r-star is not a market price we can look up. It has to be inferred from other things we can observe, using some model of how those things relate to it.
I started trying to estimate an Australian r-star using the standard Holston-Laubach-Williams (HLW) model. That attempt failed. The Australian data did not contain enough information for the model to identify the relationship it needed to recover r-star.
That failure led to a different question. If I cannot see Australian r-star through the HLW model, can I see it through other windows?
I have now tried three. One looks at international and Australian bond markets. One looks at the RBA's own interest-rate decisions. The third imposes the IS relationship that HLW failed to estimate and works backwards to the r-star required to make it hold.
These are not minor variations on the same model. They obtain their information about r-star from quite different places. All three estimate cleanly. More importantly, all three recover a recognisably similar low-frequency history: Australian r-star was much higher, fell substantially over the decades before COVID, and has risen again since the pandemic.
But they disagree materially about exactly where r-star sits.
That combination is the interesting result. I have a much better window on the direction of Australian r-star than I do on its level.
First attempt: let the macroeconomic data find r-star
Holston-Laubach-Williams (HLW) is the natural place to begin because it tries to identify r-star from the macroeconomic relationships in which the concept is supposed to matter.
At its centre is an IS relationship. In simplified form:
\[ \tilde y_t = a_1\tilde y_{t-1} +a_2\tilde y_{t-2} +a_r(r_{t-1}-r^*_{t-1}) +\varepsilon_t \]
In plain English, the output gap for any period is explained by where the output gap has been over the previous two quarters, plus whether the previous period's real interest rate was above or below r-star, with a residual capturing everything else affecting activity.
The intuition is straightforward. If the real interest rate is above neutral, monetary conditions should restrain activity and push output below potential. If it is below neutral, they should stimulate activity and push output above potential. Observing how output behaves at different real interest rates therefore gives the model a way to infer the unobserved neutral rate.
HLW further decomposes r-star into a component related to trend economic growth and another latent component, conventionally called \(z\):
\[ r_t^*=c g_t+z_t \]
where \(g_t\) is trend growth and \(z_t\) captures persistent movements in neutral not explained by growth.
The crucial point is that \(z\) has no observation equation of its own. The model learns about it through the IS relationship. A movement in \(z\) changes r-star; that changes the real-rate gap; and the real-rate gap should help explain the output gap:
\[ z_t \rightarrow r_t^* \rightarrow(r_t-r_t^*) \rightarrow\tilde y_t \]
For Australian data, that last relationship is poorly defined.
The model begins with a prior for the interest-rate coefficient centred around -0.153. The posterior is centred around just -0.021.
That is an important result. The model did learn from the data. It just didn't learn what it needed to identify r-star. Australian output pushes the estimated response to the real-rate gap very close to zero. The rate gap contributes almost no information about output.
At the same time, the unexplained variation in the IS equation remains large.
The prior for the standard deviation of the IS disturbance is centred around 0.27. The posterior is around 0.48.
So the model finds a very small systematic relationship between the real-rate gap and activity sitting alongside a much larger amount of unexplained movement in the output gap. There is very little signal through which it can work backwards from activity to the neutral interest rate.
Without that IS relationship, \(z\) cannot do its job.
There is an additional complication. Trend growth is not an independent anchor sitting outside the model. It is jointly estimated with potential output and the output gap. And because the IS equation is largely an AR(2) model once the real-rate-gap coefficient is pushed close to zero, the decomposition between trend and cycle is itself being determined inside the same weakly identified system.
The resulting r-star looks like this.
The uncertainty bands are enormous. The estimated r-star also has a correlation of 0.998 with estimated trend growth. With the interest-rate term contributing almost no information and \(z\) having no observation equation of its own, the model finds almost no independent movement in r-star beyond the trend-growth component it is co-estimating.
The estimation itself confirms the problem. The model fails several important diagnostics. The maximum \(\hat R\) is 1.04 (the model did not converge), the minimum effective sample size is 72 , BFMI is just 0.03, and 20 per cent of transitions hit maximum tree depth. The worst-behaved parameter is \(\sigma_z\), with a bulk effective sample size of 165 and a tail effective sample size of 72. The next problem areas are concentrated in the latent \(z\) and r-star states themselves.
I therefore don't regard this series as an estimate of Australian r-star.
But the failure is useful. HLW asks the Australian macroeconomic data to supply a great deal of the identifying information itself. In particular, it asks the data to establish the interest-rate relationship through which the unobserved neutral rate can be inferred.
They don't.
So what happens if we supply identifying information from somewhere else?
Second attempt: look at the bond market
The second model starts from a very different proposition: the price of long-term capital is substantially international.
Australian saving and investment do not meet in a sealed domestic market. Australian investors can buy foreign securities, international investors can buy Australian securities, banks borrow internationally and capital moves between jurisdictions in response to relative returns.
So rather than asking Australian output to reveal r-star, this model begins with a global neutral rate and asks how Australian financial-market pricing sits around it.
In simplified form:
\[ r^*_{AU,t}=r^*_{World,t}+w_t \]
The global component is constructed from the New York Fed's HLW estimates for the United States, euro area and Canada. The Australian component, \(w_t\), is a slowly moving country-specific wedge.
But the Australian real bond yield is not itself r-star. It also contains compensation for holding a long-duration bond rather than continuously rolling short-term assets. So, schematically:
\[ y^{10}_{AU,t}=r^*_{AU,t}+TP_t+\varepsilon_t \]
where \(TP_t\) is the term-premium component.
The identifying problem is therefore quite different from HLW's. The model is decomposing an observable Australian market price into a persistent neutral component and a more stationary term-premium component, with the global r-star providing an external anchor.
This is the first additional piece of information. Australian r-star is allowed to depart from the world rate, but not arbitrarily. The model treats the difference as a persistent Australian wedge.
That gives:
Movements in the real bond yield can now be separated into movements in the underlying neutral rate and movements in the term premium.
Unlike the failed HLW model, this system estimates cleanly.
But that does not make the answer assumption-free.
The model can identify an r-star because it has been given substantially more structure. It assumes that there is a global component to neutral, that Australia can have a persistent wedge around it, and that the remaining component of the real long bond yield behaves like a stationary term premium.
In particular, the division between r-star and the term premium matters. A sufficiently persistent movement in the observed bond yield has to go somewhere. How the model distinguishes a persistent change in the Australian neutral rate from a persistent change in the term premium affects the answer.
Conditional on that structure, however, this is a perfectly coherent window onto r-star.
And it produces the first successful Australian r-star path.
Third attempt: ask the RBA
The next model looks nowhere near the bond market.
Instead, it asks what the RBA's own behaviour reveals.
The Reserve Bank has been setting the Australian cash rate under an inflation-targeting regime since the early 1990s. If those decisions reflect a reasonably consistent reaction function, then the observed path of the cash rate contains information about the neutral level around which the RBA thought it was moving policy.
A simplified representation is:
\[ i_t = b_t+\phi_\pi(\pi_t-\pi^*)+\text{policy dynamics}+\varepsilon_t \]
where \(i_t\) is the cash rate, \(\pi_t-\pi^*\) is the inflation gap and \(b_t\) is a slowly moving latent base rate.
The model then allows that neutral base to evolve:
\[ b_t=b_{t-1}+\eta_t \]
The question is not "what does the Australian economy tell us r-star must be?" It is:
If the cash-rate path we actually observed came from a reasonably consistent reaction function, what path for neutral is implied by the RBA's behaviour?
That distinction matters.
The model decomposes observed policy into a slowly moving neutral component and the RBA's response around that neutral level.
This provides another independent source of identifying information. HLW tries to infer neutral from what output subsequently does. The reaction-function model instead infers it from the behaviour of the policymaker setting the interest rate.
And, again, the estimation is extremely well behaved.
But again there is structure.
Most importantly, the model has to decide how quickly neutral itself can move.
If the latent neutral rate is allowed to move quickly, more of the movement in the cash rate can be attributed to neutral itself. Make neutral move more slowly and more of the cash-rate movement has to be explained by the RBA's reaction around it.
The resulting series therefore isn't a clean estimate of the RBA's beliefs about r-star, let alone direct observation of the true economic neutral rate. Persistent influences on rate decisions that aren't captured elsewhere in the reaction function can also be absorbed into the inferred neutral component.
It is better described as a revealed policy-implied neutral rate.
But it is another coherent window. Given the specified reaction function and the permitted movement in neutral, what slow-moving base is revealed by more than three decades of actual RBA decisions?
Fourth attempt: impose the IS relationship and work backwards
The final model returns to the relationship that defeated HLW, but changes the question.
HLW asks the Australian data to estimate how strongly the output gap responds to the real-rate gap while simultaneously using that relationship to identify r-star. In my Australian estimation, the first part of that exercise largely fails: the estimated interest-rate effect is pushed close to zero, leaving too little information to identify r-star.
The inversion model deliberately stops asking the data to do both things at once. Instead, it starts by asserting that higher real interest rates relative to r-star reduce output, and then asks what path for r-star would make that relationship consistent with the output gap we actually observe.
The relationship is also much more delayed than a simple one-period IS equation. In stripped-down form, the model is:
\[ \tilde y_t = \beta\left[ w(r-r^*)_{t-4} + (1-w)(r-r^*)_{t-8} \right] +\varepsilon_t, \qquad \beta<0 \]
The output gap today is therefore related to the real-rate gap roughly one to two years earlier. The slope \(\beta\) is estimated, but constrained to be negative, so the data set its magnitude while the model imposes its sign.The model estimates how much weight to put on the four-quarter and eight-quarter lags. In the headline specification the effective lag is a little over six quarters.
That long lag matters. Monetary policy normally works with delays, while the RBA also changes interest rates in response to what is happening in the economy. Looking several quarters back helps separate the effect the model is asserting for monetary policy from the central bank's contemporaneous response to economic weakness or strength.
The starting point is still the same empirical problem. There is no stable downward-sloping Australian IS relationship sitting plainly in the data for the model to recover.
So this model supplies part of the relationship rather than asking the data to recover it from scratch. It imposes a negative sign and a long lag structure, then lets the data estimate the strength of that relationship and how much weight to place on the four-quarter and eight-quarter lags.
Once the negative relationship is imposed, the model can work backwards. For any observed output gap and history of real interest rates, there is a corresponding history of r-star that would make the asserted IS relationship fit.
But that does not uniquely determine r-star. The model also has to decide how quickly neutral itself is allowed to move. It does that by imposing a random-walk process:
\[ r_t^*-r_{t-1}^* \sim N(0,\sigma_{r^*}) \]
The parameter \(\sigma_{r^*}\) controls how quickly r-star is allowed to move. A small value forces a very smooth neutral rate; a larger value allows r-star to adjust much more rapidly. In the headline specification, \(\sigma_{r^*}\) is fixed rather than estimated.
This assumption turns out to matter enormously.
When \(\sigma_{r^*}\) is small, the model forces r-star to move slowly and the residual has to absorb more of the mismatch between the asserted IS relationship and the observed output gap. As \(\sigma_{r^*}\) rises, r-star itself is allowed to account for more of that variation.
The estimated IS slope matters as well. A stronger response of output to the rate gap requires a smaller interest-rate gap to explain a given output gap; a weaker response requires a larger one. The inferred r-star therefore depends both on the imposed structure of the relationship and on the speed at which neutral is allowed to move.
This makes the interpretation of the model unusually transparent. It is not claiming that Australian data have revealed a stable IS curve and then delivered r-star as an empirical by-product.
It asks a deliberately conditional question:
If the output gap responds negatively to the real-rate gap with a lag of roughly one to two years, and if r-star moves at the speed I allow it to move, what path for r-star is required to reconcile that structure with the Australian data?
Conditional on those assumptions, the model estimates cleanly and produces a well-defined r-star path. One final qualification is that the latest four quarters are not informed by the data. Because the model identifies r-star through rate gaps lagged four and eight quarters, the final likelihood observation only reaches r-star four quarters earlier. The remaining states are the random walk carried forward: their median stays near the last informed estimate while uncertainty widens. This is a consequence of identifying neutral through a lagged monetary-policy channel, not an estimation failure.
That makes it almost the mirror image of HLW.
HLW says: tell me the IS relationship and r-star.
The Australian data cannot do that reliably.
The inversion says: I'll give you the IS relationship and the permitted speed of r-star. Now tell me the r-star path those assumptions imply.
And with that additional structure, the model can.
Three successful models, three different sources of information
We now have three models that successfully produce an Australian r-star.
They do so for quite different reasons.
The bond-market model gets its identifying information from international and Australian financial-market prices. Its r-star is particularly conditional on the decomposition between the persistent Australian neutral-rate wedge and the term premium.
The RBA reaction-function model gets its identifying information from the RBA's observed behaviour. Its neutral rate is particularly conditional on the assumed policy reaction function and how quickly that latent neutral is allowed to move.
The IS inversion model gets its identifying information from imposing a negative, lagged relationship between the real-rate gap and the output gap. Its r-star is particularly conditional on that imposed structure and on the permitted speed of movement in neutral.
These aren't three noisy runs of the same statistical model. They are three answers to three different questions that coincide as r-star under a sufficiently strong version of the standard framework.
That makes their comparison interesting.
The first thing to notice is how much they agree.
All three recover a large historical decline in Australian neutral interest rates. All three put neutral substantially lower in the late 2010s and around the pandemic than it had been in earlier decades. And all three show a material rise since.
That agreement is not mechanical. The bond model is looking at capital-market pricing. The reaction-function model is looking at central-bank behaviour. The inversion is looking at the combination of real rates and the output gap after imposing an IS relationship.
Yet the broad low-frequency shape survives.
I think that tells us something.
It does not mean the historical direction of r-star is identified independently of a model. Each series still exists only because of its identifying assumptions. But the decline and subsequent rise appear robust across the three very different structures I have tried.
Now look at what they disagree about.
The bond-market and IS-inversion models produce real rates, while the reaction-function model produces a nominal base for the cash rate. To compare them, I express all three in nominal terms by adding the 2.5 per cent midpoint of the RBA's inflation target to the two real series.
On that basis, the estimates for the June quarter of 2026 are 3.58 per cent from the bond-market model, 2.99 per cent from the RBA reaction function and 4.03 per cent from the IS inversion.That is a spread of just over one percentage point.
And historically the disagreement has sometimes been considerably larger.
The shaded area here is not an uncertainty interval. It is the range between the three structural answers. The line through its middle is simply their mean, included to make their common movement easier to see. It is not an estimate of r-star.
At its widest, in late 2005, the models were 2.12 percentage points apart. The latest spread is 1.03 percentage points.
So while the broad direction looks surprisingly robust, the level does not.
What exactly have we found?
There is a tempting next step: put all three models into a larger state-space system, posit a single hidden Australian r-star, and treat the bond market, the RBA reaction function and the IS inversion as three noisy measurements of it.
I haven't done that.
It would be elegant, but it would also assume something I am trying to learn.
To construct that model I would have to assume that all three are measuring exactly the same latent object. I would then have to specify how noisy each measurement is, whether their errors are correlated, whether any has a systematic bias, whether those biases can drift, and how quickly the supposedly common r-star itself can move.
The resulting precision could easily be precision supplied by those assumptions.
There is also a deeper question raised by the exercise.
The three models may be looking at the same equilibrium concept through different institutional mechanisms. If so, their differences tell us about measurement.
But it is also possible that the relevant neutral rate does not map identically across very different parts of a modern financial system.
The rate that equilibrates international capital pricing, the rate around which a central bank behaves, and the rate that leaves domestic aggregate demand unchanged need not empirically coincide at every point in time. Bank funding margins, fixed versus variable mortgage structures, housing collateral, exchange rates, corporate capital-market access and cross-border flows can all sit between a policy interest rate and the economic behaviour we ultimately care about.
I don't have an answer to that question. But combining the series into one latent variable would make the answer an assumption.
For now I prefer to leave the windows separate.
Through a glass, darkly
I began this exercise by asking whether Australian r-star could be seen through windows other than HLW.
It can.
That changes how I interpret my original failure.
The canonical HLW model asks the Australian macroeconomic data to identify r-star through an IS relationship. In my estimation, that relationship is too weak to provide the latent \(z\) component with the information it needs. The model fails to recover a usable r-star.
But that does not mean there is no information about Australian r-star elsewhere.
Give the problem the structure of international capital pricing and the bond market provides one answer.
Give it the structure of the RBA's reaction function and three decades of monetary-policy decisions provide another.
Assert the negative IS relationship that HLW could not estimate and invert it, and the combination of real rates and the output gap provides a third.
All three models work. All three identify their latent states cleanly conditional on their respective structures. And all three tell broadly the same historical story.
But they don't agree closely enough on the level for me to pretend we have a precise measure of Australian r-star.
That is where I have ended up after four attempts at the problem.
We don't have a good window on where Australian r-star is. We have a better window on where it has been going.
The large historical decline looks increasingly difficult to dismiss as an artefact of any one particular model. So does at least some post-pandemic rise. But whether nominal neutral today is 3 per cent, 3½ per cent, 4 per cent or somewhere else remains substantially dependent on what structure is used to look for it.
And I am not yet convinced that every window is necessarily looking at precisely the same object.
I can see Australian r-star better than when I started.
But only through a glass, darkly.
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