Mark the Graph: Regime-Switching Phillips Curves and Productivity Puzzles: A Day of Model Building

One of the enduring puzzles in macroeconomics is the flattening of the Phillips curve - the relationship between unemployment and inflation that once seemed so reliable has become frustratingly weak in recent decades. Today's work on our Bayesian state-space model for the Australian economy tackled this head-on, with some satisfying results.

The Productivity Puzzle in Hourly Compensation

Previously we had one wage-based Phillips curve using Unit Labour Costs (ULC). In today's work this was extended to two wage equations: one for ULC and one for Hourly Compensation of Employees (HCOE). These measure different things:

ULC = Compensation / Output: What employers pay per unit produced
HCOE = Compensation / Hours: What workers earn per hour worked

The relationship between them is labour productivity: ULC = HCOE / Productivity.

This distinction matters. When we first estimated the HCOE equation with just unemployment gaps and price pass-through, the Phillips curve slopes were weakly identified - the credible intervals were wide and uncomfortably close to zero.

The fix was theoretically motivated: HCOE should include a productivity term. If workers are more productive, they should earn more per hour, even holding labour market conditions constant. ULC already "controls" for productivity by construction (it's in the denominator), but HCOE doesn't.

Adding MFP (multi-factor productivity) growth to the HCOE equation solved the identification problem. The productivity coefficient (ψ) came in around 0.82, meaning workers capture roughly 82% of productivity gains in their hourly pay. The remaining 18% accrues to capital - a plausible result consistent with labour's share of income.

With productivity properly accounted for, the HCOE Phillips curve slopes became well-identified, all showing the expected negative sign with credible intervals excluding zero.

The Case for Regime-Switching

The traditional Phillips curve assumes a stable relationship: when unemployment falls below its natural rate (NAIRU), inflation rises. Simple enough. But decades of evidence suggest this relationship isn't constant. The curve was steeper in the 1980s and early 1990s, flattened dramatically post-GFC, and may have steepened again during the post-COVID inflation surge.

Rather than forcing a single slope across 40 years of data, we implemented regime-switching with three distinct periods:

Pre-GFC (to 2008Q3): The "normal" Phillips curve with moderate responsiveness
Post-GFC (2008Q4–2020Q4): The flattened curve that puzzled central bankers for a decade
Post-COVID (2021Q1+): A potentially steeper relationship as inflation returned

The results were striking. All three price Phillips curve slopes are clearly negative (as theory predicts), but the magnitudes differ meaningfully. The post-GFC period shows the flattest slope—unemployment gaps had little impact on inflation during this era of anchored expectations and globalised supply chains.

We applied the same regime structure to wage equations, finding similar patterns. The Phillips curve isn't dead; it's just context-dependent.

Note on interpretation: The Phillips curve slopes (γ) shown above are quarterly rates. A slope of -1.0 means a 1 percentage point unemployment gap reduces inflation/wage growth by 1 percentage point per quarter, or approximately 4 percentage points annualized. The relative patterns across regimes are the key finding - the curves flattened post-GFC and steepened post-COVID - but multiply by 4 for rough annual magnitudes.

When Regime-Switching Doesn't Help

Good empirical work isn't just about finding effects - it's about knowing when they're not there. We tested regime-switching on Okun's Law (the relationship between output gaps and unemployment changes) and found ... nothing particularly interesting.

All three regime-specific Okun coefficients had posterior medians that fell within each other's 90% credible intervals. The data simply doesn't support different Okun relationships across these periods. We reverted to a single time-invariant coefficient, documenting the test for posterity.

This kind of negative result is valuable. It tells us the output-unemployment relationship has been more stable than the inflation-unemployment relationship - a useful insight for understanding which structural relationships we can rely on.

Flooring MFP: When Negative Productivity Isn't Real

A technical issue arose with our productivity measure. MFP is derived as a Solow residual - the growth in output not explained by capital or labour inputs. During downturns, this measure can turn sharply negative. The GFC and COVID-19 both produced negative MFP readings.

But does productivity actually decline during downturns? Probably not. What we're observing is capacity underutilisation - firms holding onto workers and capital even as output falls, making measured productivity look terrible. True technological progress doesn't reverse because of a demand shock.

We implemented a simple fix: floor MFP at zero before using it as an input to potential output growth. Negative readings get treated as zero - capacity underutilisation, not technological regress. The unfloored series remains available for diagnostics, but the input to the production function respects the asymmetry between genuine productivity growth (can be positive) and cyclical mismeasurement (shouldn't drag down potential).

The result: potential GDP growth no longer shows implausible collapses during recessions. The output gap does the work of capturing cyclical variation, as it should.

Note: The un-floored charts are as follows:

Infrastructure Improvements

Beyond the economic modelling, we split the estimation pipeline into two stages. Stage 1 runs the expensive MCMC sampling (~5 minutes) and saves results. Stage 2 loads saved results and generates diagnostics and charts (~1 minute). This means we can iterate on visualisations and decompositions without re-running the sampler - a significant quality-of-life improvement for development.

What We Learned

Today's work reinforced a few principles:

Let the data speak, but guide it with theory. Regime-switching helped where economic narratives supported it (Phillips curves) but not where they didn't (Okun's Law).
Identification problems often have economic solutions. The weak HCOE slopes weren't a statistical problem - they reflected a missing variable that theory told us should be there.
Respect asymmetries. Productivity growth has a natural floor; recessions don't actually destroy technology.

The model now estimates well-identified Phillips curves across three regimes, with all coefficients showing correct signs and meaningful magnitudes. Tomorrow's work: actually looking at what the model tells us about the current state of the Australian economy.