From Toy Model to Working Framework

Extending the NAIRU and Output Gap Model for Australia

Introduction

In my earlier post, I described a "toy" Bayesian model that jointly estimates NAIRU and potential output for Australia. That model had five equations: a random walk for NAIRU, a Cobb-Douglas production function for potential output, a Phillips Curve, Okun's Law, and a wage equation.

This post documents an extended version. The model now has nine equations, adding an IS curve, a participation rate equation, an exchange rate equation, and import price pass-through. The additions create a more complete transmission mechanism from monetary policy through to inflation, and provide additional identifying information for the latent states.

The model still isn't meant to compete with institutional frameworks. But it's no longer just a toy - it's a working tool for thinking through how Australia's macroeconomy fits together. A complete set of charts follows at the end of this blog post.

What's New

The original model estimated NAIRU and potential output using inflation, wages, and Okun's Law as identifying equations. The extended model adds:

1. An IS Curve linking the output gap to monetary policy via real interest rates, with a fiscal impulse term (capturing government spending effects)
2. A participation rate equation capturing the discouraged worker effect
3. An exchange rate equation based on (modified) uncovered interest parity
4. Import price pass-through linking the TWI to import prices, which feed into the Phillips Curve

These additions matter because they complete the monetary transmission mechanism. In the original model, interest rates appeared nowhere - the output gap was identified purely through Okun's Law and its consistency with the unemployment gap. Now the model can trace: policy rate → real rate gap → output gap → unemployment gap → inflation. Each link is estimated, and each provides additional identifying information.

I've also refined the inflation anchor. The original model used the 2.5% target throughout and assumed well-anchored expectations. The extended model uses a phased approach:

• Pre-1993: Inflation expectations
• 1993–1998: Gradual transition toward the target as credibility was established
• Post-1998: The 2.5% target midpoint

This means the NAIRU estimate has a cleaner interpretation after 1998: it's the unemployment rate needed to hit the inflation target.

The Nine Equations

State Equations

NAIRU remains a Gaussian random walk without drift:

$$U^*_t = U^*_{t-1} + \varepsilon_t, \quad \varepsilon_t \sim \mathcal{N}(0, \sigma^2_U)$$

The innovation standard deviation is fixed at σ_U = 0.25.

Potential output follows the same Cobb-Douglas structure:

$$Y^*_t = Y^*_{t-1} + \alpha \cdot g_K + (1-\alpha) \cdot g_L + g_{MFP} + \eta_t$$

with α ≈ 0.3 and σ_Y = 0.3.

Observation Equations

1. Okun's Law (unchanged in form):

$$\Delta U_t = \beta_{Okun} \cdot (Y_t - Y^*_t) + \varepsilon_t$$

Prior: β_Okun ~ N(−0.2, 0.15)

2. Price Phillips Curve (refined anchor):

$$\pi_t = \pi^{anchor}_t + \gamma_\pi \cdot \frac{U_t - U^*_t}{U_t} + \rho_\pi \cdot \Delta_4 \rho^m_{t-1} + \xi \cdot \text{GSCPI}_t^2 \cdot \text{sign}(\text{GSCPI}_t) + \varepsilon_t$$

The unemployment gap enters as a ratio to normalise across different unemployment levels. The GSCPI captures COVID-era supply disruptions with a non-linear (squared) specification.

Prior: γ_π ~ N(−0.5, 0.3)

3. Wage Phillips Curve (unchanged):

$$\Delta ULC_t = \alpha + \gamma_{wg} \cdot \frac{U_t - U^*_t}{U_t} + \lambda \cdot \frac{\Delta U_{t-1}}{U_t} + \varepsilon_t$$

The "speed limit" term (λ) captures how rapid changes in unemployment affect wage pressures beyond what the gap level alone would suggest.

Prior: γ_wg ~ TruncatedNormal(−1.5, 1.0, upper=0) - constrained negative

4. IS Curve (new):

$$\tilde{y}_t = \rho \cdot \tilde{y}_{t-1} - \beta_{IS} \cdot (r_{t-2} - r^*) + \gamma_{FI} \cdot \text{fiscal}_{t-1} + \varepsilon_t$$

where:

• ỹ = Y − Y* = output gap
• r − r* = real interest rate gap (cash rate − inflation anchor − r*)
• Interest rate enters with a 2-quarter lag (transmission delay)
• Fiscal impulse = government spending growth minus GDP growth

Priors: ρ ~ N(0.85, 0.1), β_IS ~ TruncatedNormal(0.2, 0.1, lower=0)

The truncated prior enforces the theoretical sign: higher real rates reduce the output gap.

5. Participation Rate (new):

$$\Delta PR_t = \beta_{PR} \cdot (U_{t-1} - U^*_{t-1}) + \varepsilon_t$$

This captures the discouraged worker effect: when unemployment exceeds NAIRU, some workers exit the labour force.

Prior: β_PR ~ TruncatedNormal(−0.05, 0.03, upper=0) - constrained negative

6. Exchange Rate (new):

$$\Delta e_t = \rho \cdot \Delta e_{t-1} + \beta_{ER} \cdot (r_{t-1} - r^*) + \varepsilon_t$$

A modified UIP equation. Higher Australian real rates should attract capital and appreciate the currency.

Prior: β_ER ~ TruncatedNormal(0.3, 0.2, lower=0)

The large error variance (~3.0) reflects exchange rate volatility and the well-documented UIP puzzle—interest differentials have weak predictive power for exchange rates. Finding near-zero coefficients is expected, not a specification failure.

7. Import Price Pass-Through (new):

$$\Delta_4 \rho^m_t = \beta_{PT} \cdot \Delta_4 TWI_{t-1} + \beta_{oil} \cdot \Delta_4 Oil_{t-1} + \rho \cdot \Delta_4 \rho^m_{t-1} + \varepsilon_t$$

Currency appreciation reduces import prices; oil prices add a direct supply-side driver.

Priors: β_PT ~ TruncatedNormal(−0.3, 0.15, upper=0), β_oil ~ TruncatedNormal(0.1, 0.05, lower=0)

This equation closes the loop: monetary policy → exchange rate → import prices → inflation.

How the Equations Reinforce Each Other

The original model identified NAIRU primarily through the Phillips Curve and wage equation, with Okun's Law providing the link to potential output. The extended model adds more cross-equation discipline:

1. The IS Curve constrains the output gap: If rate hikes don't reduce the output gap, either the rate gap measurement or the output gap estimate is wrong.
2. The participation equation constrains NAIRU: The discouraged worker effect provides an independent signal about labour market slack.
3. The open economy equations close the transmission mechanism: Policy rate → exchange rate → import prices → inflation gives another channel for monetary policy to affect prices.
4. Supply vs demand attribution becomes cleaner: With import prices modelled explicitly, the Phillips Curve residual is more clearly "demand" (the unemployment gap) versus "supply" (import prices, GSCPI).

The web of cross-equation restrictions means implausible estimates in one dimension are penalised by poor fit in others.

The Transmission Mechanism

One way to summarise the model is as a complete (if stylised) monetary transmission mechanism:

$$\text{Cash rate} \xrightarrow{\text{IS}} \text{Output gap} \xrightarrow{\text{Okun}} \text{Unemployment gap} \xrightarrow{\text{Phillips}} \text{Inflation}$$

Plus the exchange rate channel:

$$\text{Cash rate} \xrightarrow{\text{UIP}} \text{TWI} \xrightarrow{\text{Pass-through}} \text{Import prices} \xrightarrow{\text{Phillips}} \text{Inflation}$$

Each arrow is an estimated equation with the theoretically expected sign. When someone asks "does demand-side inflation respond to rate hikes?" - the answer is embedded in the model structure. The IS curve shows rate hikes reduce the output gap. Okun's Law links that to unemployment. The Phillips Curve links unemployment to inflation. Each coefficient is estimated with the expected sign. That's the transmission mechanism - not one chart, but the whole system.

Inflation Decomposition

The Phillips Curve can be decomposed term by term:

Demand contribution: γ_π · (U − U*)/U

Supply contribution: Import price pass-through + GSCPI

Baseline: Inflation anchor (expectations or target)

The "demand" component is really demand as manifested in the labour market. The chain is:

$$\text{spending} \rightarrow \text{output gap} \rightarrow \text{hiring} \rightarrow \text{U falls below NAIRU} \rightarrow \text{wage pressure} \rightarrow \text{prices}$$

The Phillips Curve captures the last link. So "demand-side inflation" means inflation attributable to labour market tightness - demand filtered through the jobs market, not demand measured directly. The model doesn't decompose demand by source (households vs government vs exports) - for that you'd need national accounts, sectoral data, and household surveys.

Current Estimates

The extended model produces estimates broadly consistent with the original:

Variable	Original	Extended
NAIRU (current)	~4.9%	~4.8%
Output gap	~0%	~0%
Unemployment gap	−0.66pp	−0.5pp
Potential growth	~2%	~1.7%

The slightly lower NAIRU in the extended model may reflect additional identifying information from the participation and IS equations. The tighter credible intervals suggest the extra equations are adding discipline.

The Taylor Rule comparison shows the current prescription (~4.5%) above the actual cash rate (~3.6%), consistent with the original finding that policy is somewhat accommodative on this mechanical benchmark.

What Remains a Toy

Despite the extensions, this is still a simplified framework:

Model uncertainty: The equations are standard but contestable. Time-varying Phillips Curve slopes, non-linear Okun coefficients, or regime-switching could matter.

End-point uncertainty: Real-time estimates at sample end carry wider uncertainty than mid-sample.

The UIP puzzle: The exchange rate equation acknowledges that UIP fails empirically. Weak coefficients are expected.

COVID: The 2020–2022 period required special handling (GSCPI, labour force smoothing). Interpret with caution.

No forward-looking behaviour: Expectations enter only as an exogenous anchor, not as model-consistent rational expectations.

The RBA and Treasury use suites of models to protect against individual model bias. This is one model. Treat point estimates as indicative.

Theoretical Lineage

For those asking where these equations come from:

Component	Attribution
Phillips Curve	A.W. Phillips, 1958
Expectations augmentation	Friedman (1968), Phelps (1967)
Supply shock augmentation	Gordon (1977), Bruno & Sachs (1985)
Okun's Law	Arthur Okun, 1962
IS Curve	John Hicks, 1937 (formalising Keynes)
Cobb-Douglas	Cobb and Douglas, 1928
Taylor Rule	John Taylor, 1993
NAIRU concept	Friedman (1968), Modigliani & Papademos (1975)
UIP	Early 20th century arbitrage condition

Conclusion

The extended model provides a more complete framework for thinking about Australia's macroeconomy. The additional equations - IS curve, participation rate, exchange rate, import price pass-through - create a full (if stylised) monetary transmission mechanism and provide additional identifying information for NAIRU and potential output.

Current estimates suggest:

• NAIRU around 4.8% - the labour market remains somewhat tight
• Output gap close to zero - the economy is near potential
• r* around 2% - consistent with the global decline in neutral rates
• Inflation increasingly demand-driven as supply disruptions unwind

The model's value lies not in any single point estimate but in its internally consistent framework. When actual outcomes deviate from predictions, it signals measurement issues, model misspecification, or genuine structural change - each warranting different responses.

Charts

NAIRU Estimate for Australia

The NAIRU has declined from around 6–7% in the mid-1980s to approximately 4.79% today. The shaded pre-1998 period indicates when NAIRU was estimated relative to inflation expectations rather than the fully anchored inflation target. Post-1998, NAIRU represents the unemployment rate needed to hit the 2.5% target.

Unemployment Gap Estimate for Australia

The unemployment gap (U − U*) shows labour market slack. Positive = disinflationary slack; negative = inflationary tightness. The current estimate of −0.56pp suggests the labour market remains somewhat tight, though less so than the extreme tightness of 2022.

Output Gap Estimate for Australia

The output gap shows the economy's position relative to potential. The COVID period dominates with an unprecedented negative spike. The current estimate of −0.04% suggests the economy is essentially at potential.

Actual vs Potential GDP

Actual GDP (black) tracks potential GDP (green) over the long run, with cyclical deviations. The COVID drop appears as a sharp deviation that has since recovered.

Potential GDP Growth Rate (proxy for r*)

Potential growth has declined from ~4% in the 1980s to ~1.7% today (trend: −0.04pp/year). This secular decline implies a lower neutral interest rate and reduced policy space during downturns. It is largely driven by a decline in productivity. 

Natural Rate of Interest (r*) — Comparison

Three estimates of r*: raw model output (volatile), pure trend (smooth), and hybrid (75% trend, 25% raw). The hybrid estimate of ~2.1% balances stability with responsiveness to cyclical shifts. Potential growth provides the economy's speed limit - how fast it can grow without generating inflation. r* approximates this. The latest GDP report shows the economy is running at roughly its speed limit. Much of this decline in potential growth can be attributed to slower productivity growth.

Taylor Rule vs RBA Cash Rate

The Taylor Rule prescription (~4.55%) currently exceeds the actual cash rate (~3.60%) by about 95 basis points, suggesting policy is somewhat accommodative on this mechanical benchmark. The Taylor Rule was a poor fit for the 15 years following the GFC, when r* was falling and central banks were managing lower-bound risks. With those emergency conditions behind us, it may be more useful as a benchmark again.

Neutral Interest Rate vs RBA Cash Rate

The nominal neutral rate (r* + π^T ≈ 2.2% + 2.5% = 4.68%) compared to actual policy. The current cash rate sits below neutral, implying some policy accommodation. The nominal neutral rate is a theoretical construct: the policy rate that would neither stimulate nor restrain the economy when output is at potential and inflation is at target. It equals the long-run trend r* plus expected inflation (or the inflation target when expectations are anchored).

Inflation Decomposition: Components (Unscaled)

The Phillips Curve decomposed term by term: grey = inflation anchor (expectations/target), orange = demand (unemployment gap), blue = supply (import prices + GSCPI), light blue = residual noise. This is simply the expectations-augmented Phillips Curve visualised.  

The supply and demand contributions are deviations from the inflation anchor, not from zero. The baseline (grey) represents where inflation would settle absent shocks - inflation expectations pre-1993, phased to the target 1993-98, and anchored at 2.5% thereafter. Positive demand bars mean the tight labour market is pushing inflation above this baseline; negative bars mean slack is pulling it below. The same logic applies to supply. So when expectations are anchored at 2.5%, a +1pp demand contribution means demand is adding 1pp to inflation above the 2.5% baseline.

Coefficient Posteriors

All estimated coefficients with 68% and 95% highest density intervals. Key results: all coefficients are statistically different from zero (>99% probability), and all have theoretically expected signs.

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This is a follow-up to Building a Toy Macro Model for Australia. The code is available on GitHub.