1. From pictures to proof: Why the pipeline matters

After the charts, the report does something refreshingly unromantic: it insists on a workflow. That is not academic fussiness; it is a way to stop a small annual dataset from telling tall tales.

With annual data for Slovenia (1990–2024), the temptation is to run a simple regression and declare Okun “confirmed” or “broken.” The report argues that this is precisely how you end up with confident nonsense. Small samples reduce statistical power; structural change makes relationships shift; and non-stationary variables can create correlations that look meaningful but are mostly shared drift. So the method is sequential: first establish whether the series are well-behaved (stationarity), then whether a long-run relationship exists (cointegration), then estimate dynamics with models that match the data (ARDL/NARDL), and only then test directional predictability (causality).

That sequencing is the economic content of the methodology. If you skip the steps, you may still get coefficients, but you won’t know whether they are measuring behaviour or merely mirroring history.

2. First, don’t trip over stationarity: What the unit-root results imply

The unit-root section begins with a plain proposition: before you model “relationships,” you need to know what kind of objects you are relating. A series that drifts with trend is fundamentally different from a series that fluctuates around a stable mean. The report uses two complementary tests, the ERS DF-GLS (where the null is a unit root) and KPSS (where the null is stationarity), precisely because they force you to look at the same question from opposite directions.

The headline result is intuitive and important. In levels, LGDP and the unemployment rate look nonstationary under DF-GLS, with some evidence of trend-stationarity under KPSS, which the report treats as plausible given a transition economy with sustained trend growth and evolving labour-market equilibria. The implication is methodological and economic at once: for levels, you should either difference the variables or treat trend carefully in any long-run model; otherwise you risk conflating growth’s long-run rise with unemployment’s evolving baseline.

In first differences, the picture becomes simpler. DLGDP and DUR are stationary under KPSS, and DUR is also stationary under DF-GLS, supporting the use of differenced series to model short-run dynamics. This supports a “difference Okun” approach if what you want is a clean cyclical sensitivity story, how unemployment changes when GDP growth changes, without claiming the existence of a stable long-run equilibrium.

Then the report makes a crucial pivot to the gap model. Because the output gap and unemployment gap are constructed as cyclical deviations from trend via the HP filter, they behave as they are designed to behave: they are stationary under both DF-GLS and KPSS. That is not just a technical detail. It means the gap variables can be modelled as inherently cyclical objects, more “policy-like” measures of slack, without having to strip out trend again.

Economically, this creates a neat hierarchy: the gap model is naturally aligned with stabilisation talk (above/below potential, slack/tightness), while the difference model is naturally aligned with year-to-year fluctuations (growth surprises and unemployment changes). The unit-root evidence supports both, but it also warns that mixing levels without proper long-run justification is a recipe for spurious certainty.

3. Breaks and nonlinearities: When history rewrites the rules

The report does not stop at “first-generation” tests because Slovenia’s history does not read like a stationary process. The unit-root section explicitly motivates structural-break testing as a way to distinguish genuine nonstationarity from trend changes caused by identifiable episodes (transition dynamics, crises, reforms). In a small annual sample, a single major break can make a stationary relationship look like a random walk, or make a stable mean look like a drifting baseline.

This is why the report layers in structural-break unit-root tests (Zivot–Andrews; Clemente–Montañés–Reyes; Lee–Strazicich), and also uses multiple-break procedures (Bai–Perron and the Ditzen–Karavias–Westerlund approach). The conceptual payoff is straightforward: if the trend or intercept shifts, your model needs to acknowledge it, otherwise the “Okun coefficient” becomes an average of incompatible regimes.

The report’s narrative interpretation, more than any single test statistic, is that Slovenia’s macro-labour relationship has been exposed to episodes large enough to plausibly change time-series properties and slopes. Rather than treat those as “noise,” the workflow treats them as part of the data-generating process. That framing matters later, because it anticipates why models with structural dummies (2009 and 2020) can behave very differently from models without them.

4. The long-run question: Cointegration and what it would mean for policy

Once you know what kind of series you have, the next question is whether output and unemployment share a long-run equilibrium relationship. Cointegration is the report’s way of asking: do these variables move together over decades in a way that makes economic sense, even if they wander in the short run?

The report runs multiple cointegration approaches because each has different sensitivities, especially in small samples and in the presence of breaks.

Engle–Granger: Weak in levels, stronger in gaps, fragile in differences

Engle–Granger provides limited support for cointegration in levels (LGDP and UR): across variants with/without trend and with/without augmentation, the tests fail to reject no cointegration, which the report interprets as consistent with breaks or parameter instability over the full period.

In the first-difference pairing (DLGDP and DUR), the report notes that some specifications show strong evidence of cointegration (even very strong under certain variants), but that this evidence weakens materially once lag augmentation is included, making the conclusion sensitive to lag length. That sensitivity is a recurring theme: in short annual samples, “robustness” is not a luxury; it is the entire point.

The gap model is where Engle–Granger becomes more consistent. The report finds cointegration evidence between output gap and unemployment gap that is stronger and more stable across specifications than in levels, and still reasonably supportive even when the direction is reversed (though generally weaker). The economic interpretation is appealing: slack measures may share a stable equilibrium relationship even when levels are distorted by long-run transitions.

Gregory–Hansen: Cointegration strengthens once you allow breaks

The Gregory–Hansen test is introduced as a realism upgrade: it allows the cointegration relationship itself to shift with a structural break. In levels, the report largely finds no cointegration across specifications, with only an occasional significant-looking outcome that is not representative of the full set.

But in first differences and in the gap model, the Gregory–Hansen results are described as strong and robust, with break dates clustering around economically meaningful periods (the report points to mid-2000s for first differences and mid-1990s to late-1990s for gaps). The message is not that Slovenia “has breaks”; it is that the long-run relationship becomes visible once you stop forcing it to be constant across regimes.

The report’s methodological interpretation is explicit: for Slovenia, break-adjusted cointegration testing is not an optional flourish; it is a prerequisite for credible long-run claims in a transition-era dataset.

Johansen: System-based evidence, improved with crisis dummies

The Johansen approach is used as a system method, and the report pays attention to what can make or break it: lag length selection, deterministic components, and residual diagnostics. It then highlights that including structural dummies for 2009 and 2020 improves model fit and helps isolate exogenous shocks that could otherwise distort long-run inference.

The economic implication is subtle: it suggests that Slovenia’s long-run output–unemployment linkage is not best understood as a single smooth equilibrium, but as an equilibrium path repeatedly jolted by large shocks. A system method that can incorporate controls for those jolts will naturally tell a clearer story than a method that assumes stability by default.

5. Estimating Okun for Slovenia: ARDL results and what the coefficients say in human terms

Having gathered evidence that cointegration is more plausible, especially in gap terms, and especially when breaks are respected, the report moves to ARDL models as a practical estimation framework suited to small samples and mixed integration properties.

It estimates four annual ARDL-type setups with dummy variables for 2009 and 2020 and uses SIC to guide lags. The report then focuses on two linear ARDL models and one NARDL (nonlinear ARDL) robustness check.

Difference-style ARDL with UR as dependent variable: A conventional Okun story, cautiously earned

In the first model (UR as dependent; GDP and lags; plus dummies), the report describes the relationship as negative and statistically significant in short and long run, consistent with Okun’s Law. It highlights an error-correction term around −0.223, implying gradual adjustment toward equilibrium (roughly a fifth of disequilibrium corrected per year). Bounds testing supports cointegration, and diagnostic tests (serial correlation, heteroskedasticity, normality, RESET) are reported as passed, with high explanatory power.

Economically, this says: when output improves, unemployment tends to adjust downward, and there is a long-run anchor in this specification. The report also notes that the crisis dummies are not significant here, which it interprets as possibly reflecting that the effect of those episodes is already channelled through GDP dynamics or that the small sample makes dummy effects hard to detect cleanly.

Gap-model ARDL with unemployment gap as dependent variable: The clearest “policy” version

The second model is the one the report treats as most compelling: unemployment gap modelled on output gap. Here the long-run coefficient is reported as negative and larger in magnitude (around −0.227), and the error-correction coefficient is very strong (about −0.796), suggesting rapid adjustment, with most disequilibrium corrected within a year. Bounds tests strongly support cointegration; diagnostics again support model adequacy; and the model is framed as particularly suitable for policy simulation because it maps cyclical output slack to cyclical labour slack.

The economic reading is exactly the kind of Okun story policymakers want, but with an important twist: it is not “growth reduces unemployment” in the abstract. It is “closing the output gap closes the unemployment gap,” and it does so relatively quickly in this estimated framework.

NARDL: Asymmetry is allowed, but the data refuse to dramatise it

The NARDL model is introduced to test a realistic hypothesis: labour markets may react differently to expansions than to contractions. For Slovenia, however, the report finds nearly identical long-run coefficients for positive and negative output-gap components and no meaningful evidence of long-run asymmetry. The cointegration evidence is described as statistically inconclusive in this nonlinear setup (just below a key threshold), and explanatory power is lower than in the linear ARDL gap model.

The report’s interpretation is pragmatic: the NARDL is a useful robustness check, but in this dataset the linear gap ARDL appears sufficient. It also notes a sample-imbalance issue (few negative growth observations in one context), which can make nonlinear decomposition unstable in small samples.

Taken together, the estimation step yields a ranking the report is comfortable defending: the gap ARDL is the best-fitting and most interpretable model, with strong adjustment and robust diagnostics; the difference-style ARDL is valid and supportive; the nonlinear extension is informative but not decisive.

6. Bayer–Hanck: A meta-test that largely sides with the gap story

Because individual cointegration tests can disagree, especially in short samples, the report uses the Bayer–Hanck meta cointegration test to combine evidence from several methods.

The pattern matches what earlier sections suggested: levels are weak and sensitive, while the gap model is stronger and more consistent, particularly under a constant specification (and weaker once trends are included). The report treats this as corroboration: long-run equilibrium claims are more defensible in cyclical slack terms than in raw levels, where trend, breaks, and deterministic choices can dominate.

Methodologically, this step is the report’s way of saying: “don’t let one test crown itself king.” Economically, it reinforces the view that Slovenia’s most reliable Okun relationship is the one expressed in terms of gaps, how far output is from potential and how far unemployment is from its natural rate.

7. Causality: Who leads whom, and why dummies change the answer

Finally, the report asks the question politicians love and econometricians fear: does output “cause” unemployment, or can unemployment be a leading indicator of output? It uses the Toda–Yamamoto procedure precisely to reduce the risk that pre-testing decisions drive the conclusion.

Without structural dummies, the report finds no significant Granger causality in either direction across levels, first differences, and gap models. The message: if you treat the sample as one homogeneous regime, you do not get clear directional predictability.

Then the report adds dummy variables for 2009 and 2020, and the narrative shifts. In levels, it finds borderline bidirectional causality at the 10% level. In first differences, it still finds no meaningful causality. But in the gap model with dummies, the report finds something cleaner: a statistically significant, one-way link from output gap to unemployment gap, while the reverse direction remains unsupported.

That is a big interpretive payoff, and the report draws it explicitly: cyclical output deviations appear to be a predictor of cyclical unemployment deviations once major discontinuities are accounted for. In plainer English: when you control for the years where the world fell off its axis, the remaining cyclical relationship looks more like the textbook story, output slack leads labour-market slack, not the other way around.

Methodologically, the section’s moral is that breaks can mask causality; economically, the moral is that Slovenia’s Okun mechanism is most visible as a slack-to-slack transmission channel, rather than as a simple year-to-year growth-to-unemployment-change machine.

8. Bridge to Part III: What the workflow adds up to

Part II’s pipeline produces a coherent conclusion, but not a simplistic one. The report’s weight of evidence points to Okun’s Law being most credible and policy-relevant in Slovenia when framed in gap terms, supported by cointegration evidence strengthened by break-aware methods, by ARDL estimation with strong adjustment, and by causality that becomes clear once structural dummies are included. Part III will do what the report itself sets up but does not fully exhaust in the results narrative: integrate these findings into a policy reading, what they imply for stabilisation strategy, labour-market responsiveness, and the kinds of shocks that still have the power to loosen even Slovenia’s relatively disciplined Okun whisper.