Bayesian approach to estimating NAIRU, potential output, and the output gap
Introduction
Every central banker faces an impossible task: steering an economy using variables they can't directly observe. The "natural" unemployment rate, the economy's "potential" output, whether we're running hot or cold - these are all latent variables, hidden states that must be inferred from noisy data. This post describes a small, exploratory model I built to estimate these unobservable quantities for Australia using Bayesian methods.
The model isn't meant to compete with the suite of sophisticated frameworks used by institutions like the RBA or Treasury. It's a toy - hopefully a useful one for building intuition about how these macroeconomic concepts fit together.
The model
The model jointly estimates two key latent variables:
- The NAIRU (Non-Accelerating Inflation Rate of Unemployment) represents the unemployment rate consistent with stable inflation. The gap between the unemployment rate and the NAIRU is called the unemployment gap. When actual unemployment falls below the NAIRU, we expect upward pressure on wages and prices. When it rises above, we expect disinflation.
- Potential GDP is the economy's maximum sustainable output - what we could produce if all resources were employed at their "natural" rates without generating inflationary pressure. It is often called the speed limit on the economy. The gap between actual and potential GDP (the output gap) tells us whether the economy is running hot or cold.
These two concepts are intimately linked through Okun's Law: when output exceeds potential (positive output gap), unemployment typically falls below its natural rate (negative unemployment gap). The model exploits this relationship to improve estimation of both quantities simultaneously.
The model consists of five interconnected equations, estimated jointly using PyMC's Bayesian framework:
The NAIRU evolves as a driftless random walk:
$$U^{*}_{t} = U^{*}_{t-1} + \epsilon_{U^*}$$
This specification allows the natural rate to change gradually over time in response to structural factors like demographics, labour market institutions, and skill mismatches - without imposing any particular direction or trend on those changes. In short, the NAIRU for any quarter only varies a little from the value it had in the previous quarter.
Potential GDP also follows a random walk, but with an informative time-varying drift:
$$ Y^{*}_{t} = Y^{*}_{t-1} + \alpha \cdot g^{K}_{t} + (1-\alpha) \cdot g^{L}_{t} + g^{MFP}_{t} + \epsilon_{Y^*} $$
This is a growth accounting equation derived from the Cobb-Douglas production function, expressed in log-level (or growth rate) form. Potential output growth equals the weighted contribution of capital accumulation (weighted by capital's income share) plus the weighted contribution of labour growth (weighted by labour's income share) plus growth in multifactor productivity (MFP) - which captures technological progress, efficiency gains, and other factors not explained by simply adding more capital or labour.
The Price Inflation equation (the Phillips Curve)
The expectations and supply-shock augmented Phillips curve in broad terms looks like this:
$$(inflation - inflation\text{-}expectations) = unemployment\text{-}gap + shocks + noise$$
Which I have implemented as follows:
$$ (\pi_{t} - \bar{\pi}) = \gamma_{pi}\frac{(U_t - U^*_t)}{U_t} + \rho_{pi}\Delta_4 \rho^{m}_{t-1} + \xi_{pi}\Xi^2_{t-2} + \theta_{pi}\omega_t + \epsilon_{pi}$$
Rather than use inflation expectations, I used the implicit 2.5 per cent inflation target, and assumed expectations were well anchored for most of the period under consideration (discussed further below). The unemployment gap enters as a ratio to normalise across different unemployment levels. And I have included three data series to account for supply side shocks:
- import prices - capturing exchange rate pass-through (using ABS data),
- COVID-era supply disruptions (via the NY Fed's Global Supply Chain Pressure Index), and
- oil price shocks (using the World Bank oil price data in USD and RBA exchange rate data).
The critical linking equation (Okun's Law):
$$ \Delta U_t = \beta_{okun}(Y_t - Y^{*}_t) + \epsilon_{okun} $$
When output exceeds potential, unemployment falls - and vice versa. I use the change form of the equation rather than levels to avoid spurious correlations. Relating two trending series can cause spurious correlations.
Unit labour cost growth (or wages growth) responds to both the level of the unemployment gap and its rate of change:
$$ \Delta{ulc}_{t} = \alpha_{wg} + \gamma_{wg} \cdot \frac{(U_t - U^*_t)}{U_t} + \lambda_{wg} \cdot \frac{\Delta{U}_{t-1}}{U_t} + \epsilon_{wg}$$
The "speed limit" term (λ) captures how rapid changes in unemployment affect wage pressures beyond what the gap level alone would suggest.
What about inflation expectations
A traditional Phillips curve would include inflation expectations as a key driver. I've omitted them here, for two reasons.
First, since the RBA adopted inflation targeting in 1993, expectations have been remarkably well-anchored around the 2-3% target band. When a variable barely moves, it adds little identifying information to the model - you're essentially trying to explain variation in inflation with something that doesn't vary. The unemployment gap and supply shocks do most of the work in the post-1993 sample.
Second, finding compelling expectations data for Australia is genuinely difficult. Survey-based measures (like the Melbourne Institute's consumer expectations) are noisy and may not reflect the expectations that actually matter for price-setting. Market-based measures from inflation-linked bonds are only available from the mid-2000s and come with their own liquidity and risk premium issues. Bond-market measures of inflation expectations arguably reflect inflation risk rather than expected inflation. Rather than include a poorly-measured variable that theory says should matter, I chose to let the well-anchored expectations assumption do implicit work in the background.
This is a defensible choice for the inflation-targeting era, but it does mean the model would be less suitable for periods when expectations were unanchored - like the 1970s and 1980s, or potentially if credibility were to erode in the future.
The Bayesian approach
The Bayesian approach offers several advantages for this problem. It naturally handles uncertainty: rather than point estimates, we get full posterior distributions for every parameter and latent state. This is crucial when policymakers need to understand not just their best guess, but how wrong they might be.
The model uses weakly informative priors based on economic theory and previous research. For instance, the Okun coefficient is given a prior centred on -0.2 (consistent with the literature for quarterly data), the Phillips curve slope is expected to be negative, and the productivity/labour force pass-through coefficients are expected to be around 1.
The model is implemented in PyMC with NumPyro as the sampling backend. Data comes from the ABS (unemployment, GDP, prices, unit labour costs, import prices) and the RBA (cash rates, exchange rates). The Global Supply Chain Pressure Index comes from the NY Fed, and oil prices from the World Bank.
The estimation uses the NUTS sampler via NumPyro, running 100,000 samples across 6 chains with 5,000 tuning iterations each. This extensive sampling helps ensure robust inference despite the model's complexity.
The code is available in a Jupyter notebook on my Github repository for those interested in the implementation details or in extending the model. It comes with an explanatory markdown document.
Key findings
The NAIRU has declined from around 15% in the mid-1980s (reflecting the aftermath of the 1980s recession and high structural unemployment) to roughly 4.9% today. The decline was particularly steep through the 1990s as labour market reforms took effect. Since 2000, the NAIRU has been relatively stable in the 5-6% range, with recent estimates suggesting it may have edged lower.
The confidence bands are narrow enough to be useful but wide enough to be honest - there's genuine uncertainty about these estimates.
The trend line shows potential growth declining at about 0.04 percentage points per year over the sample. From around 3.5-4% in the mid-1980s, potential growth has fallen to roughly 2% today. This reflects both slower productivity growth (the well-documented productivity puzzle) and changing demographics.
This matters for monetary policy: lower potential growth implies a lower neutral interest rate (r*). If the economy can only sustainably grow at 2% rather than 4%, the interest rate that neither stimulates nor restrains has also fallen. For context, current policy debate in Australia centres on whether potential growth is closer to 2.0% or 2.2%. This model comes in at the lower end of that debate.
The output gap captures Australia's business cycle nicely. Note the sharp swings during COVID - the economy collapsed well below potential in 2020 (the largest negative gap on record), then overshot during the reopening boom. The current estimate of -0.07% suggests the economy is now roughly at potential.
The recession of the early 1990s shows clearly as deep negative gaps. The mining boom period (2000s) appears as a sustained positive gap.
This chart shows the model's estimate of potential GDP (green) against actual GDP (black). The series move together over the long run, with actual GDP oscillating around its potential. The COVID drop appears as a sharp deviation that has since largely recovered.
The unemployment gap (U - U*) is the mirror image of the output gap, via Okun's Law. Negative values indicate a tight labour market (unemployment below NAIRU, inflationary pressure); positive values indicate slack (disinflation). The current estimate of -0.66 percentage points suggests the labour market remains somewhat tight.
The extreme negative readings in the late 1980s reflect genuine overheating before the early 1990s recession. The post-COVID period shows the rapid swing from elevated unemployment to historically tight conditions.
A caveat: the model assumes well-anchored inflation expectations throughout, which fits the post-1993 inflation-targeting era but not the earlier period. The elevated NAIRU estimates and the unemployment gap in the 1980s may partly reflect this regime mismatch - the model attributes inflation variation to labour market slack that might more properly be explained by unanchored inflation expectations.
A Conundrum
Looking at the current estimates together reveals a somewhat awkward position. The output gap has closed to roughly zero - the economy is producing at near its potential, which sounds healthy. But the unemployment gap remains negative at -0.66 percentage points, meaning the labour market is still tighter than its sustainable level.
This combination - output at potential but unemployment below NAIRU - suggests we may be just one supply shock away from stagflation. If a shock hits (energy prices, supply chain disruptions, geopolitical events), the economy has limited capacity to absorb it without either inflation rising or unemployment spiking. There's no "slack" buffer in the labour market to cushion the blow.
The benign interpretation is that the NAIRU estimate is too high, and the labour market can sustainably operate at current unemployment rates. The less benign interpretation is that we're in a fragile equilibrium where any adverse shock forces an unpleasant choice between accepting higher inflation or engineering a rise in unemployment. The confidence bands remind us that we can't be certain which interpretation is correct.
One way to evaluate these latent variable estimates is to see if they produce sensible policy prescriptions. I computed a Taylor Rule rate using the model's output gap and potential growth (as a proxy for r*):
$$i_t = r^* + \pi_{coef} \cdot \pi_t - 0.5 \cdot \pi^* + 0.5 \cdot y_{gap}$$
The Taylor Rule tracks actual RBA policy reasonably well through the 1990s and 2000s. The persistent gap since the GFC reflects the structural shift in neutral rates and the RBA's risk-management approach in a low-inflation environment.
The current Taylor Rule prescription of about 4.5% exceeds the actual cash rate of 3.6% by about 0.9 percentage points - suggesting that on this mechanical benchmark, policy is somewhat accommodative given current inflation and output conditions. But Taylor Rules are guides not gospels. They can illuminate whether policy is broadly tight or loose relative to historical norms, but they cannot capture the full information set, non-linearities, or judgment calls that shape real-world decisions.
Model validation
The model passes several sanity checks:
Theoretical consistency: All key parameters have the expected signs with high probability. The Okun coefficient is negative (100% probability), the Phillips curve slopes are negative (100% probability), and the capital growth accounts for about 30% of the Labour/Capital growth shares.
Coefficient magnitudes: The Okun coefficient of about -0.1 implies that 1% excess output reduces unemployment by 0.1 percentage points per quarter - consistent with standard estimates. The Phillips curve slope suggests meaningful but not explosive inflation responses to labour market slack.
Sampling diagnostics: Typically there are zero divergences during sampling. The sampling diagnostics show good effective sample sizes across chains, with solid R-hat statistics, all of which suggest the model converged properly.
Nonetheless, this is explicitly a toy model - useful for building intuition but not for serious policy analysis. Several limitations are worth noting:
Sample period: The model is estimated from 1984Q3 onwards. The pre-1993 period reflects a different monetary policy regime (before inflation targeting), which may affect coefficient stability. However, excluding this data caused convergence problems, suggesting it provides important identifying information.
Model uncertainty: The confidence bands show meaningful uncertainty, particularly for real-time estimates at the end of the sample. The latest readings should be treated with appropriate scepticism.
Simplifications: The model omits many factors that could matter - expectations formation, financial conditions, global factors, supply-side policies. It's a reduced-form approach that abstracts from structural relationships.
COVID period: The pandemic created unprecedented data patterns that may not be well-captured by a model estimated on pre-COVID dynamics. The inclusion of the Global Supply Chain Pressure Index helps, but is unlikely to fully address this. Also, some of the worst spikes around COVID, were assumed away in the input data (for example, labour force growth was smoothed during the COVID period).
Model stability: Like many state-space models, the estimates can be sensitive to small specification changes-adjusting priors, adding variables, or changing the sample period can shift results. The broad picture remains stable (NAIRU has fallen, potential growth has slowed), but precise numbers move around. Treat point estimates as indicative rather than definitive. Remember that the RBA and Treasury use a suite of models to protect against individual model bias.
What's next
This is a work in progress. I'll continue developing and testing this toy model - experimenting with alternative specifications, incorporating new data as it arrives, and seeing how the estimates evolve. If you have suggestions or spot errors, I'd welcome the feedback.
[Note: I updated this page when I extended the potential GDP equation to its full Cobb-Douglas form.]
Final note
This analysis draws on approaches from RBA working papers, Treasury research, and academic literature on NAIRU and output gap estimation. It is provided for educational and exploratory purposes only and should not be used for investment or policy decisions.
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