Stationarity, breaks, cointegration, dynamics, causality, the steps that stop you from fooling yourself.

1. From pictures to proof: Why the tests come next

Once the charts have done their job, flagging the obvious co-movements, the dramatic outliers (especially 2009 and 2020), and the suspicion that “one slope” may be too simple, the report moves to the part that feels less intuitive but is more decisive: the formal checklist. The logic is not to worship econometrics; it is to avoid common traps in small annual samples.

With Croatia’s annual data (2000–2024), the report repeatedly warns that power is limited and that results can be sensitive. That is precisely why it insists on a sequential approach. If you don’t know whether a series is drifting or mean-reverting, you can mistake shared trends for relationships. If you ignore breaks, you can treat “structural change” as “random noise” and end up estimating an average that fits no period well. And if you jump straight to causality tests, you can get spurious directionality simply because the underlying variables are not suited to the test environment.

So Part II begins where the report says any credible Okun verification must begin: with stationarity, then breaks, then cointegration, then dynamic models, and only then a cautious look at who predicts whom.

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

The unit-root section is the report’s way of asking: are we looking at stable cyclical objects, or are we looking at series that wander? It uses the classic two-test pairing: DF-GLS (Elliott–Rothenberg–Stock), which treats a unit root as the default under the null, and KPSS, which treats stationarity as the default under the null. In plain English, one test suspects drift; the other suspects stability. If they broadly agree, confidence rises. If they disagree, especially in small samples, interpretation must be cautious.

The report’s baseline unit-root results (summarised in its tables for the 2000–2024 window) suggest a pattern that will feel familiar to anyone who has wrestled with macro series: the level variables, log GDP per capita (LGDP) and the unemployment rate (UR), do not behave like stable, mean-reverting series in a simple way. The report interprets them largely as non-stationary in levels, with clearer stationarity once differenced. That supports the textbook modelling distinction: if you want to work with levels, you need a long-run framework; if you want short-run dynamics, you can often work with differences.

By contrast, the report highlights that the HP-filter “gap” series, output gap and unemployment gap, are designed to be cyclical deviations from trend, and it finds them to behave accordingly. In its discussion of unit-root outcomes, it treats both gaps as effectively stationary and therefore suitable for modelling as cyclical slack measures. Economically, this matters because it clarifies what the “gap specification” is buying you. It is not just a conceptual appeal to “potential output.” It is a statistical and interpretive clean-up: you are working with series that behave like cycles, which fits the stabilisation narrative readers care about.

The report also flags an important limitation: standard unit-root tests assume no breaks. In a country that experienced a global financial crisis and a pandemic shock within a short 25-year annual sample, that assumption is not a technical inconvenience, it is a likely violation.

3. Breaks and structural shifts: when a macro shock is not “just another observation”

Having established that non-stationarity is plausible in levels and stationarity is plausible in gaps, the report then asks whether standard tests are being fooled by structural change. This is where break-adjusted unit-root tests come in: Zivot–Andrews for one endogenous break, Clemente–Montañés–Reyes for two breaks, and Lee–Strazicich as another break-robust approach. The report’s motivation is straightforward: a series can look non-stationary if it is actually trend-stationary with one or two level/trend shifts. In other words, breaks can masquerade as unit roots.

The break tests in the report, again presented as tables and then interpreted in discussion, support the idea that Croatia’s macro series contain structural breaks aligned with major disruptions, notably around the global financial crisis period and the COVID period. The report emphasises that small samples exacerbate this problem: when T is limited, a few years of disruption can dominate statistical inference. So it treats break-aware results as essential context for everything that follows.

The economic meaning here is not abstract. If GDP and unemployment series have breaks, it likely reflects changes in the economy’s structure or in the labour market’s response mechanism, policy shifts, crisis dynamics, recovery regimes, rather than random data noise. That, in turn, implies that Okun’s coefficient might not be stable over the full 2000–2024 window. It may differ between recessionary regimes and recovery regimes, and it may behave differently when shocks are sharp and temporary versus prolonged and persistent.

The report explicitly links these considerations to its broader modelling strategy: break evidence affects whether you include crisis dummies, whether you rely on long-run equilibrium models, and how you interpret causality results that do not explicitly accommodate breakpoints.

4. Cointegration: Does Croatia have a long-run “Okun equilibrium,” or only short-run co-movement?

Once integration orders are understood, the report turns to the long-run question. This is where Okun’s Law becomes more than a recession story. Cointegration tests ask whether output and unemployment share a stable equilibrium relationship that anchors short-run fluctuations, even if the levels themselves drift. If cointegration exists, it justifies modelling in levels with an error-correction mechanism. If it does not, then levels-based long-run claims become fragile, and differenced or gap-focused models become more defensible.

The report does not rely on a single cointegration test. It runs multiple approaches: Engle–Granger as a baseline, Gregory–Hansen to allow for regime shifts, Johansen as a system method, and the Bayer–Hanck meta test to pool evidence.

The headline that emerges, stated explicitly in the report’s own summary of cointegration evidence, is more nuanced than a simple yes/no. It concludes “with a high degree of certainty” that there is no long-run equilibrium relationship between the GDP growth rate and the unemployment rate over the research period. That is a sharp statement, and it matters because it narrows what you should try to “verify.” Croatia’s Okun relationship, in this reading, is not a long-run equilibrium mapping from growth rates to unemployment rates. It is more plausibly a relationship in other formulations, particularly those that use levels with cointegration support, or slack variables, or models explicitly designed for mixed integration and small samples.

At the same time, the report also presents evidence from system and meta approaches that suggests long-run binding between LGDP and UR in levels can be present under certain specifications. In particular, it later notes that Johansen and Bayer–Hanck provide supporting evidence for at least one cointegrating relationship between LGDP and UR, which it then uses to interpret causality results in levels more credibly. The key is to read this in the report’s own framing: the “no cointegration” conclusion is aimed at the growth-rate formulation; the levels formulation can show a long-run link under the right structure and assumptions.

Economically, this is an important distinction. If the growth-rate version of Okun does not have a stable long-run equilibrium in Croatia, it does not mean output and unemployment are unrelated. It means that the relationship may work through levels dynamics, through cyclical slack (gaps), or through regimes that shift across crises and recoveries. Croatia’s labour market may respond to sustained growth and accumulated output changes rather than to year-to-year growth fluctuations in a mechanically stable way.

5. Dynamic estimation: ARDL and NARDL as “small-sample realism”

After cointegration comes estimation, done in a way that respects small samples and mixed integration. The report chooses ARDL models precisely because they can model both short-run dynamics and long-run relationships in a single framework and are typically considered suitable when variables are a mix of I(0) and I(1). It also extends to NARDL to allow for asymmetry: the idea that negative growth shocks may raise unemployment more than positive growth shocks reduce it.

The report estimates ARDL-type models for both the standard (difference/level) formulations and the gap formulations, and it pays close attention to the error-correction term. That coefficient is the model’s “speed of adjustment”: it tells you how quickly deviations from long-run equilibrium are corrected.

Two interpretive points stand out in the report’s narrative discussion of these models.

First, the models can produce statistically significant short-run and long-run relationships consistent with Okun’s Law in certain specifications. The report’s language is careful but positive where warranted: it describes significant coefficients and well-behaved diagnostics in some ARDL setups, particularly in gap-based versions. It also notes that the error-correction term in those models implies relatively quick adjustment toward equilibrium, meaning that labour-market slack responds meaningfully to output slack within a plausible time horizon.

Second, and equally importantly, it warns against overconfidence. It returns to sample-size limitations and the risk of overfitting. When you have 22 or 23 usable observations after lags, the model can fit the past neatly without guaranteeing stable inference. The report also flags that the absence of explicit trend or break structures in some model variants could limit reliability given Croatia’s macro disruptions. In other words, even when ARDL produces “nice results,” the report insists on reading them in the light of breaks and small-sample fragility.

Economically, the ARDL/NARDL step is where the Okun story becomes policy-usable: it translates “co-movement” into “adjustment.” If output improves, how quickly does unemployment respond? If slack closes, how quickly does labour slack close? But the report keeps the reader honest: the answer depends on specification, and the Croatian sample is short enough that any numeric estimate should be treated as a range of plausibility rather than a timeless coefficient.

6. Bayer–Hanck: Pooling evidence, with a caveat Croatia can’t ignore

Because individual cointegration tests can disagree, especially in small samples, the report uses the Bayer–Hanck meta cointegration test to strengthen inference by combining evidence across tests. It reports that under both a constant model and a constant-plus-trend model, the hypothesis of no cointegration is rejected, supporting the existence of a long-run relationship between the series it tests.

But the report immediately adds a crucial warning: the Bayer–Hanck meta test does not account for structural breaks. In Croatia’s context, that caveat is not pedantry. The earlier unit-root and break analysis already suggests that breaks are visible in the key series. So the report treats the meta-test as supportive, but not definitive. It strengthens the case for a long-run relationship, but it does not eliminate the possibility that the relationship changes across crisis regimes.

This is a recurring theme in the report’s methodology: every method adds evidence, and every method has a limitation that matters for Croatia. The trick is not to find a method with no caveats; it is to build confidence by checking whether multiple methods point in compatible directions.

7. Causality: Who leads whom, and how much weight Croatia’s annual data can carry

The report’s causality section uses the Toda–Yamamoto approach and applies it to levels and differences for LGDP and UR, as well as to the output and unemployment gaps. It motivates this “dual approach” as a way to examine directional predictability in both original variables and cyclical components while mitigating risks of spurious results driven by nonstationarity.

The causality results are mixed, and informative precisely because of that.

In levels (LGDP and UR), the report finds a clear pattern of unidirectional causality from LGDP to UR at the 5% level, while UR does not Granger-cause LGDP. Economically, this is the most straightforward Okun interpretation in the entire causality block: past output contains predictive information about future unemployment. The report reads it as consistent with the theoretical Okun channel, expansions reduce unemployment by raising labour demand, and it interprets the absence of reverse causality as evidence that unemployment does not significantly anticipate GDP in this sample horizon.

In first differences (DLGDP and DUR), the report finds no statistically significant causality in either direction, with very high p-values. It offers two plausible interpretations that stay within its own framework: differencing removes long-run information that might carry predictive content, and short-run annual fluctuations may be dominated by noise or by factors (including breaks) that a bivariate model cannot capture. Economically, the implication is that Croatia’s year-to-year wiggles are not a reliable forecasting device in a simple two-variable causality framework, even if long-run relationships exist in levels.

In the gap framework (output gap and unemployment gap), the report again finds no significant causality in either direction at conventional levels. One direction “approaches” significance but does not cross thresholds; further differencing removes even those marginal effects. The report interprets this as indicating a weak or absent cyclical predictive link in the gaps framework, at least in the strict Toda–Yamamoto bivariate setting.

Taken together, the report’s causality conclusion is cautious but clear. The strongest directional evidence is in levels, running from output to unemployment. In differences and gaps, the directionality does not show up in a statistically persuasive way. That does not mean the cyclical mechanism is absent. It means that, given the sample size and the likely presence of structural breaks, the strict predictability criterion is hard to satisfy in those formulations.

The report then does what good applied work should do: it foregrounds the limitations. It notes that Toda–Yamamoto relies on asymptotic distributions that work better in larger samples, so Croatia’s roughly 22 annual observations constrain power and increase error risks. It also notes that the procedure does not account for breaks in the causal relationship itself. If Croatia’s growth–unemployment linkage changed after major disruptions, a single full-sample causality test can produce attenuated or misleading results. In other words, “no causality detected” in some formulations can reflect instability rather than irrelevance.

Finally, the report ties causality back to cointegration. It notes that Johansen and Bayer–Hanck results indicate at least one cointegrating relationship between LGDP and UR, which increases the credibility of causality testing in levels and supports the interpretation of LGDP → UR predictability as a long-run phenomenon. That is an important integration: causality results are not read in isolation; they are read as consistent with the long-run binding between the variables under certain frameworks.

8. Bridge to Part III: What the results add up to, before policy enters

Part II’s full checklist produces a disciplined conclusion: Croatia’s Okun relationship is visible, but it is specification-sensitive and sample-constrained. Unit-root and break-aware testing reinforces the need to respect structural disruptions. Cointegration evidence is strongest when the right variables and frameworks are used (with the report explicitly dismissing a long-run equilibrium between growth rates and unemployment rates), and dynamic ARDL-type models can produce significant Okun-consistent relationships while still requiring caution about overfitting and omitted break structures. Causality appears in levels from output to unemployment, but not robustly in differences or gaps. Part III will do what the report sets up but does not fully execute in the “results-only” framing: interpret these findings in a coherent policy narrative. That includes what they imply for labour-market adjustment, stabilisation strategy, and how Croatia’s evolving institutional context, including EU membership and euro adoption, should shape expectations about whether “growth will deliver jobs” quickly, slowly, or only under certain conditions.