Specification, Heterogeneity and Functional Form in the Determinants of Small-Firm Profitability

1. Introduction

Why one small firm earns more than another is usually answered by reference to how it is financed. The literature on small and medium enterprises is organised around capital structure, and its central result is unusually stable: better capitalised firms are more profitable, as Matias and Serrasqueiro (2017), Dalci (2018), Quoc Trung (2021), Kalash (2023) and Youssef et al. (2023) establish across national panels. This paper asks a question the same data can answer but that has not been posed: when financial structure and the operating model are placed in one equation and measured on the same scale, which carries more of the variation in profitability?

The second term is almost absent from the literature that would have to answer it. Cost composition — the share of value added absorbed by labour, the intensity of purchased services, reliance on leased rather than owned capacity — enters firm-level profitability models as a control when it enters at all. The studies that do examine the operating side approach it through efficiency scores or working-capital ratios rather than through the composition of the cost base: Campisi et al. (2019), Grau and Reig (2021), Chadha et al. (2023) and Tripathi et al. (2024) are typical. The scarcity is measurable. Of 2,827 records retrieved from Scopus, twenty-seven relate labour cost, value added or outsourcing to firm profitability, against 447 relating capital structure to it. This is the first gap the study addresses.

The second is methodological. Endogeneity is treated as a nuisance corrected by internal instruments and reported in a robustness column, as in Canarella and Miller (2018), Vijayakumaran and Vijayakumaran (2019), Dsouza et al. (2025) and Saiz-Sepulveda et al. (2026). Two things are rarely acknowledged. Net worth contains the current year's profit and return on assets is computed on that profit, so part of any contemporaneous association is an accounting identity. And profitability is persistent — Hirsch and Hartmann (2014), Yang et al. (2015), Hirsch et al. (2021) and Choi et al. (2024) establish the regularity across settings — so any lagged financial ratio embedding past profit is correlated with the current error by construction. The persistence literature supplies the premise that invalidates the identification practice of the capital-structure literature, and the two strands rarely meet.

The third gap concerns the two literatures that could test the specification itself. Firm taxonomies built on accounting data, as in Linares-Mustarós et al. (2018), Juntunen et al. (2022) and Kristóf and Virág (2022), select partitions on a single validity index and describe the groups without re-estimating the relationship inside them. Machine learning on accounting data, as in Pap et al. (2022), Shetty et al. (2022), Vajjhala and Strang (2024) and Mahmood et al. (2025), is framed as prediction and evaluated on accuracy, with comparisons against linear models typically made on raw levels rather than on the transformation a fixed-effects estimator uses.

The originality of this study lies in holding one sample and one equation fixed while three methods interrogate different assumptions about it. The setting is the Italian population of certified innovative start-ups and innovative SMEs with a matched control of conventional firms — 91,756 firm-year observations on 14,913 firms — a regime whose evaluations, from Vannoni (2019), Manaresi et al. (2021), Migliaccio and Pavone (2021), Aiello et al. (2024), Anderloni and Harasheh (2025) and Scandurra et al. (2025), estimate average effects while presuming the mechanism. Panel estimation asks whether the association survives the estimator; unsupervised partitioning asks whether one coefficient vector describes every firm; machine-learning regression asks whether the linear form is adequate, on the same firm-demeaned data. The contribution is fourfold: the relative weight of the two blocks is measured within a single specification; the capitalisation coefficient is shown not to be identified by accounting data of this kind, and the reason diagnosed; the structure of profit determination differs across the regulatory populations the scheme defines; and machine learning is used diagnostically rather than competitively.

2. Literature Review

Two hundred studies were retrieved from Scopus and organised into eight themes. The corpus is recent — three quarters published since 2021 — and dominated by a design so standard that it can be described in one sentence: a national panel of firms observed over five to ten years, with a profitability ratio regressed on a capitalisation or leverage ratio and a set of controls, estimated by fixed effects or system GMM. What follows describes each theme, states what the design cannot answer, and locates the present study against it.

2.1. Capital Structure and Profitability in Small Firms

Whether the way a firm is financed shapes what it earns is among the most heavily worked questions in the empirical literature on small and medium enterprises, and its central result is unusually stable across settings. Matias and Serrasqueiro (2017) establish it for Portuguese firms, Dalci (2018) for Chinese manufacturers, Quoc Trung (2021) for Vietnam, Kalash (2023) for an emerging-market panel, Youssef et al. (2023) for non-financial UK SMEs, and Gonçalves et al. (2024), Bhawna and Sahay (2025) and Nkosi et al. (2026) for further national samples. Consistency of the sign conceals a real disagreement about why it holds. Under pecking-order reasoning, profitable firms accumulate retained earnings and borrow less, so causality runs from performance to structure; under trade-off and agency reasoning, leverage disciplines managers and the negative sign becomes a puzzle to be resolved by appeal to distress costs. Singh et al. (2025) and Demiraj et al. (2025) report evidence compatible with more than one mechanism, and the designs employed cannot separate them, because a single contemporaneous correlation is consistent with all of them. Three features of this literature bear on what follows. Samples are almost always listed or bank-financed firms. The equity or debt ratio is almost always the focal variable. And cost-structure variables, where they appear at all, enter as controls without discussion, so the question of whether financial structure matters more than the operating model is never posed.

2.2. Endogeneity, Simultaneity and Identification

The largest theme in the corpus is methodological in substance if not in title. Forty-six of the two hundred studies address endogeneity explicitly, most by instrumenting the focal regressor with its own lags or by moving to system GMM: Canarella and Miller (2018), Vijayakumaran and Vijayakumaran (2019), Samal and Yadav (2025), Dsouza et al. (2025) and Saiz-Sepulveda et al. (2026) are representative. The treatment is remarkably uniform. Endogeneity is acknowledged in a paragraph, an instrumented column is added to the robustness table, and the coefficient it produces is reported as confirming the main result. Instrument validity is tested in a minority of cases, and where a Hansen or Sargan statistic is reported it is almost never allowed to overturn the specification. Two consequences follow. Failures are invisible, because a paper whose instruments reject does not usually report them, so the literature contains no accumulated evidence on whether internal instruments work in this setting. And the specific mechanism that would invalidate them here is unacknowledged: net worth contains the current year's profit and return on assets is computed on that same profit, so part of any contemporaneous association is an accounting identity. Talamas Marcos (2025) and Rastogi and Kumar (2024) are among the few to exploit institutional variation rather than internal lags, and that approach has not become standard practice.

The reporting convention deserves comment in its own right. In this corpus the instrumented column is presented as a robustness check on a preferred fixed-effects estimate, which implies that the two are estimating the same parameter and that agreement between them is the expected outcome. When they disagree, the disagreement is usually attributed to instrument weakness and the fixed-effects estimate retained. That resolution is defensible only if the exclusion restriction is more likely to fail than the exogeneity of the regressor, and no paper in the corpus argues the point explicitly. The alternative reading — that neither estimate identifies the parameter and that the disagreement is the informative result — appears nowhere, though it follows directly from the diagnostics several of these studies already report.

2.3. Profit Persistence and Dynamic Adjustment

A separate literature establishes that profitability reverts towards a firm-specific mean at a measurable rate. Hirsch and Hartmann (2014) and Hirsch et al. (2021) document persistence in European food processing and retailing, Jaisinghani (2016) in Indian pharmaceuticals, Opstad et al. (2022) in restaurants, and Doyran and Santamaria (2019), Killins (2020) and Hoang et al. (2026) in banking and insurance. A parallel strand estimates the speed of adjustment towards a target capital structure — Yang et al. (2015), Sinha and Vodwal (2022), Choi et al. (2024), Abdeljawad and Farhood (2025) and Al Barakat and Ali (2026) — and reaches the same qualitative conclusion from the financing side. Both strands are technically careful and both use dynamic panel estimators. Neither, however, connects persistence to the identification problem of the previous theme, and the omission is consequential. If profitability at t is substantially predicted by profitability at t−2, and if the capitalisation ratio at t−2 mechanically embeds the profit of t−2, then lagged capitalisation is correlated with the current error by construction and cannot serve as an instrument. The persistence literature supplies the premise, the identification literature supplies the practice, and the two are rarely brought into contact.

2.4. Cost Structure, Efficiency and the Operating Model

The operating side of the firm enters the corpus through a different door. Grau and Reig (2021) study operating leverage in European agri-food SMEs, Campisi et al. (2019) and Alhassan and Ohene-Asare (2016) measure efficiency by data envelopment, Chadha et al. (2023) and Tripathi et al. (2024) examine working-capital efficiency in Indian MSMEs, and Bhattu-Babajee and Seetanah (2022) and Prasad and Mondal (2025) relate value added to performance. These are studies of efficiency, not of cost composition: they ask how well inputs are converted into output rather than how the cost base is divided between labour, purchased services, materials and leased capacity. The distinction matters because the second question can be answered from statutory accounts for the whole population of firms, and because the shares are the object a manager can act on. The scarcity is measurable. Of 2,827 records retrieved across eighteen queries, only twenty-seven relate labour cost, value added or outsourcing to profitability at firm level, against 447 relating capital structure to it. Where cost shares do appear in profitability regressions they are controls, reported without comment in a table whose discussion concerns the financial coefficient. No study in the corpus places the two blocks in one equation and asks which carries more of the variation.

2.5. Unsupervised Classification and Firm Taxonomies

Clustering enters the firm-level literature as description. Linares-Mustarós et al. (2018) classify firms by financial performance and distress profile, Juntunen et al. (2022) identify latent classes of accounting outsourcing firms, Ljungkvist and Andersén (2021) and Anton et al. (2021) build taxonomies of small manufacturers and of energy start-ups, Kristóf and Virág (2022) and Salina et al. (2020) partition sectors and banks on financial indicators, and Williams et al. (2025) and Bayaraa et al. (2019) combine clustering with efficiency analysis. Two limitations recur. The first is selection on a single index: silhouette is the default, and because it falls monotonically in the number of clusters for every partitional method, the procedure is biased towards coarse solutions before any substantive judgement is made. Papers reporting silhouette alone report two clusters; papers reporting partition R² alone report seven; the choice is presented as technical when it is in fact undetermined by the criterion. The second is that the partition is where the analysis stops. Groups are profiled and named, and their differences are described in means, but the estimating equation of the accompanying regression analysis is seldom re-estimated inside them, so the reader learns that firms differ without learning whether the pooled coefficients mean anything.

2.6. Machine Learning for Distress, Default and Fraud Prediction

The most established use of machine learning on accounting data is binary classification. Shetty et al. (2022), Gavurova et al. (2022), Papíková and Papík (2024), Cheraghali and Molnár (2026) and Bijoy et al. (2026) predict bankruptcy or distress; Zhang et al. (2025) and Sodnomdavaa and Lkhagvadorj (2026) predict fraud; Ariza-Garzón et al. (2024) and Antar and Tayachi (2025) add explainability tools to the same task. The methodological standard is high and the accuracy gains over logistic regression are consistent. The comparison with a parametric benchmark, however, is almost always conducted on raw levels, and this is where it becomes uninformative for an econometrician. A learner fitted to levels has between-firm variation available to it; a fixed-effects estimator does not. Setting one against the other attributes to functional form a difference that mostly reflects unmodelled firm heterogeneity, and the resulting statement that the learner is twice as accurate cannot be read as a verdict on the linear specification. The point applies with equal force to the continuous-outcome studies of the next theme, and correcting it changes the size of the gap by a factor of nearly three.

2.7. Machine Learning Applied to Firm Performance

A newer strand applies the same algorithms to continuous measures of performance. Pap et al. (2022), Vajjhala and Strang (2024), Garg et al. (2025), Mahmood et al. (2025) and Sultana et al. (2026) model profitability or organisational performance directly, Giudici et al. (2023) compare classification models, and Kristóf and Virág (2022), Vašaničová et al. (2025) and Coronell et al. (2026) combine supervised and unsupervised methods on financial ratios. These studies report accuracy rankings and, increasingly, variable-importance measures. What they do not do is turn the result back on the econometric specification. The gap between a linear model and an ensemble is reported as a reason to prefer the ensemble, not as a measurement of what the linear form omits, and it is not decomposed into components an econometrician could act on: curvature, interaction, or local structure that no polynomial reproduces. Nor is the gap located in the distribution, so it remains unknown whether a linear specification is inadequate everywhere or only in the tails. The tools for these questions — parametric enrichment, interaction statistics, permutation importance computed out of sample, decile calibration — are standard, and the corpus contains almost no example of their being used to validate rather than to replace a regression.

2.8. Innovative Start-Ups and Certified SMEs

The Italian certification regime has generated its own literature. Manaresi et al. (2021) evaluate the Start-up Act, Migliaccio and Pavone (2021) and Vannoni (2019) examine the financial structure of certified firms, Angilella et al. (2023), Aiello et al. (2024), Anderloni and Harasheh (2025) and Scandurra et al. (2025) compare certified firms with conventional peers, and Domma and Errico (2023), Schifilliti and La Rocca (2024) and Di Berardino and Antenozio (2026) study governance and digital orientation within the population. The dominant design is a treatment-effect comparison: certified firms against matched non-certified ones, with growth, employment, survival or access to finance as the outcome. The financing constraint the regime is meant to relieve is treated as given, and the question of whether profit is determined differently across the two populations is not usually asked. It is a different question from whether certified firms perform better on average, and fixed effects absorb the average difference entirely, leaving the structure of determination as the only thing that can still be compared. The regime also offers a natural test of any data-driven partition, since the register provides a classification the analyst did not construct, and no study in the corpus uses it that way.

2.9. Synthesis

Table 1 summarises the eight themes, the studies representing each, the gap each leaves, and what the present study contributes against it. Read down the third column, the gaps share a structure: each literature answers its own question competently and stops at the boundary where it would have to interrogate its own specification. The present study is organised around those boundaries rather than around a new dependent variable.

Three observations cut across the eight themes. The first is that sample composition is remarkably narrow: listed firms and bank borrowers dominate, and the unlisted micro and small firms that constitute the overwhelming majority of European enterprises appear mainly in survey-based work with short panels. The second is that method and question have drifted apart, with the econometric literature refining identification on a specification it does not test and the machine-learning literature testing specifications it does not interpret. The third is that the dependent variable is almost always taken as given, so the mechanical relationship between an accounting ratio and the regressors constructed from the same statements goes unexamined. The present study addresses the three together by holding one sample, one equation and one set of variables fixed across three methods that interrogate different assumptions.

2.10. Methodological Antecedents

The corpus assembled above is deliberately recent, and it inherits its questions from a body of work that predates it. Whether financial structure affects performance at all is the question Modigliani and Miller (1958) answered in the negative under conditions no small firm satisfies, and the departures from those conditions organise everything since: taxes and distress costs in the trade-off tradition, information asymmetry in Myers (1984) and Myers and Majluf (1984), agency costs in Jensen and Meckling (1976). The empirical regularities the recent panels reproduce were established by Titman and Wessels (1988), Rajan and Zingales (1995) and, for the small-firm case specifically, Berger and Udell (1998). Profit persistence has an equally settled pedigree in Mueller (1977), Geroski and Jacquemin (1988) and Fama and French (2000), whose mean-reversion estimates anticipate the autoregressive coefficient reported in Section 4.

The estimators are canonical and their diagnostics older than the applications that use them. The choice between fixed and random effects rests on Mundlak (1978) and Hausman (1978), with the variance test of Breusch and Pagan (1980); the bias that motivates a dynamic specification is Nickell (1981), and the instrumented first-difference estimator is Anderson and Hsiao (1981), generalised by Arellano and Bond (1991) and Blundell and Bond (1998). The overidentification statistic is Hansen (1982), and the weak-instrument literature that governs how a first-stage F should be read — and, more importantly, how it should not — is Bound et al. (1995), Staiger and Stock (1997) and Stock and Yogo (2005), whose warning that a large F reflects sample size as much as instrument strength is directly relevant to Appendix B. Inference clustered by firm follows Petersen (2009), and the multiway alternative discussed in Section 10 is Cameron et al. (2011).

The unsupervised and supervised tools are likewise inherited. The algorithms compared in Section 5 are MacQueen (1967), Ward (1963), Dempster et al. (1977), Bezdek (1981) and Ester et al. (1996); the validity criteria on which they are scored are Caliński and Harabasz (1974), Dunn (1974), Davies and Bouldin (1979) and Rousseeuw (1987), and the stability measure is the corrected index of Hubert and Arabie (1985). On the supervised side the learners are Breiman (2001) and Friedman (2001), and the interaction statistic used in Section 6 is Friedman and Popescu (2008). Naming these sources is not a formality: the silhouette coefficient favours coarse partitions by construction, and the first-stage F is not a measure of instrument strength in the sense that controls bias amplification. Both properties are stated in the original papers and lost in the applied literature that cites neither

3. Data and Methodology

The panel is assembled from thirty-one AIDA extractions covering the universe of Italian certified innovative start-ups and innovative SMEs together with a matched set of conventional small and medium enterprises. Each extraction reports 137 harmonised accounting variables replicated over ten financial years; the reference year of each column is reconstructed from the balance-sheet closing date, which is serialised differently across extractions. Regulatory status and sector codes are merged from the business register maintained by the Ministry of Enterprise, matched on fiscal code and added only after the accounting variables are built.

The estimating sample comprises 91,756 firm-year observations on 14,913 firms between 2014 and 2025, and it is identical across all three methods and all four appendices, so that differences between results are attributable to method rather than to sample.

Table 2. Composition of the estimating sample.

Population	Observations	Firms	Share of observations
Innovative start-ups	12,864	5,173	14.0%
Innovative SMEs	19,565	2,756	21.3%
Conventional SMEs	59,327	6,995	64.7%
Total	91,756	14,913	100.0%

Note. Firm counts by population sum to slightly more than the total because eleven fiscal codes appear in two registers, having graduated from innovative start-up to innovative SME within the window; each is assigned to its earliest population.

Table 2. Composition of the estimating sample.

Population	Observations	Firms	Share of observations
Innovative start-ups	12,864	5,173	14.0%
Innovative SMEs	19,565	2,756	21.3%
Conventional SMEs	59,327	6,995	64.7%
Total	91,756	14,913	100.0%

Note. Firm counts by population sum to slightly more than the total because eleven fiscal codes appear in two registers, having graduated from innovative start-up to innovative SME within the window; each is assigned to its earliest population.

The panel is unbalanced in a way correlated with regulatory status: start-ups file meaningful accounts in roughly one year in three against more than nine in ten for conventional SMEs, so conventional SMEs contribute close to two thirds of the observations while accounting for under half the firms. Appendix A establishes how much the weighting scheme matters.

The dependent variable is return on assets. The regressors form two blocks. The financial block comprises the equity ratio, the liquidity ratio, capital turnover and the logarithm of total assets; the operating block comprises the labour cost share of value added, purchased services and materials as proportions of the value of production, and payments for leased assets on the same basis. All continuous variables are winsorised at the first and ninety-ninth percentiles within year.

Two measurement decisions determine the sample. Cost ratios expressed against the value of production become uninformative where production approaches zero, so the sample is confined to observations with positive value of production, positive value added, and value of production of at least one per cent of total assets. The restriction removes 8.8 per cent of observations, but not neutrally: 29.3 per cent of start-up observations against 1.0 per cent of conventional ones, because pre-operating firms are concentrated among the certified. Rescaling the cost ratios on total assets is not an alternative, since return on assets carries assets in its denominator and the rescaled ratios would stand in an accounting identity with it — regressing ROA on the asset-scaled versions yields an R² of 0.488 against 0.088 for the output-scaled ones.

The equation is then estimated three times by methods that interrogate different assumptions: panel estimation, which tests whether the average association survives the choice of estimator and whether the equity ratio can be instrumented; unsupervised partitioning, which tests whether one coefficient vector describes every firm; and machine-learning regression, which tests the adequacy of the linear form on the same firm-demeaned data the fixed-effects estimator uses. Figure 1 sets out the design.

4. Panel Estimation

The equation relates return on assets to the four financial and the four operating regressors defined in Section 3. It is estimated by pooled OLS, by the between estimator on firm means, by fixed effects with firm dummies and with firm and year dummies, and by random effects; it is then put through four robustness treatments — weighted least squares under two weighting schemes, fixed effects with every regressor lagged one year, and a dynamic specification in first differences following Anderson and Hsiao (1981). Standard errors are clustered by firm in every specification. The five static estimators use the full 91,756 observations on 14,913 firms; the lagged and the dynamic specifications lose the opening years of each firm's series and fall to 76,261 and 64,031 observations respectively. Sample, variable definitions and winsorisation are held fixed across every column, so that what differs between columns is the estimator and nothing else.

Which estimator serves as the reference is settled by test rather than by preference, and the three tests that settle it interrogate different nulls.

Table 4. Panel structure tests.

Test	Statistic	p	Conclusion
F(uᵢ = 0), two-way FE	5.60	< 0.0001	Pooled OLS rejected
Breusch–Pagan LM	25,011.2	< 0.0001	Individual variance present
Hausman (FE vs. RE)	117.01	< 0.0001	Random effects inconsistent
First-stage F (Anderson–Hsiao)	18,229.8	—	Instrument strong

Note. Hausman on eight degrees of freedom. The first-stage F refers to ROA(t−2) as instrument for the lagged difference in the dynamic specification ofTable A2.

Table 4. Panel structure tests.

Test	Statistic	p	Conclusion
F(uᵢ = 0), two-way FE	5.60	< 0.0001	Pooled OLS rejected
Breusch–Pagan LM	25,011.2	< 0.0001	Individual variance present
Hausman (FE vs. RE)	117.01	< 0.0001	Random effects inconsistent
First-stage F (Anderson–Hsiao)	18,229.8	—	Instrument strong

Note. Hausman on eight degrees of freedom. The first-stage F refers to ROA(t−2) as instrument for the lagged difference in the dynamic specification ofTable A2.

The F statistic asks whether firm effects exist at all; the Breusch–Pagan multiplier asks whether their variance is non-zero; the Hausman statistic asks whether they are correlated with the regressors. Rejecting the first two without the third would license random effects, which are the more efficient estimator and are widely used in this literature on precisely that ground. Rejecting all three closes the route and fixes two-way fixed effects as the reference specification. Table 3 reports it, with the between and the firm-only fixed-effects columns set alongside so that the two comparisons that matter most can be made without turning to the appendix.

All eight regressors are significant at the one per cent level, and the within R² of 0.4232 indicates that the specification accounts for a substantial share of the variation in profitability that occurs inside firms over time. The coefficients cannot, however, be compared with one another as they stand, because the regressors are measured on incompatible scales: a unit of capital turnover and a percentage point of the labour share are not comparable quantities, and reading the column downwards would make turnover appear to dominate by a factor of thirty. Figure 2 therefore rescales every coefficient to the effect of a one-standard-deviation increase, which places all eight on the single axis of percentage points of ROA.

Capital turnover dominates under every estimator and the labour share follows immediately. A one-standard-deviation increase in turnover is worth approximately four points of ROA and a comparable increase in the labour share costs close to four; capitalisation is worth about three; purchased services, materials and leased assets between one and two points each; liquidity is negligible throughout. The financial and the operating block therefore contribute on the same order of magnitude, which the unrescaled coefficients of Table 3 conceal entirely, and the ordering is unchanged across the four estimators plotted. What the Figure Aannot show is the difference between the two sources of variation those estimators exploit, which Figure 3 isolates by plotting the between and the within estimate as a pair for each regressor.

The comparison of between with within is more informative than any single coefficient, and it is the diagnostic least often reported in this literature. On the labour share and on purchased services the two estimates almost coincide, and the natural reading — that a relationship holding both across firms and inside them over time is structural rather than compositional — is the wrong one here. An accounting relation holds in both dimensions indifferentially, since it is a property of how the statements are constructed and not of how firms behave, so coincidence of the two estimates is the first trace of the identity examined in Section 7 rather than evidence against it. The diagnostic retains its value, but with the sign of its interpretation reversed: what would be informative on the labour share is divergence, not agreement. The other two comparisons are unaffected, because neither variable enters the identity. On size the two estimates diverge sharply and change sign, so within-firm asset growth operates differently from the advantage of being structurally large; liquidity changes sign in the same way and is small in both dimensions. Capitalisation is higher within than between, which is what the mechanical component of that coefficient would predict and is not evidence of a stronger structural effect.

That objection is elementary and the literature reporting a positive capitalisation coefficient rarely confronts it. Net worth contains the current year's profit, and return on assets is computed on that same profit, so part of the association is an accounting identity rather than an economic relationship. The objection is raised here because it bears on the coefficient this section traces, but it is not confined to it: four of the eight regressors are ratios of an item subtracted in the income statement to the aggregate it is subtracted from, and the operating block is exposed to it more severely than the financial one. Section 7 develops the point in full and revises the reading of the operating coefficients accordingly; what follows should be read in that light.

Three treatments separate identity from relationship, and they do not agree with one another. Lagging every regressor by one year reverses the sign, from +0.177 to −0.068, with significance unchanged. Modelling persistence instead — first differences with ROA(t−2) instrumenting the lagged difference, in the manner of Anderson and Hsiao — returns the coefficient to positive and larger than the static estimate at +0.228, with an autoregressive parameter of 0.271 and a first-stage F of 18,230. Instrumenting the equity ratio directly, across five instrument sets, produces a negative estimate in every case. Figure 4 places all fourteen specifications on a single axis.

The figure is the finding. The five static estimators cluster between +0.12 and +0.18 and the two weighted variants stay within that band; lagging moves the coefficient below zero; the dynamic specification moves it back above every static estimate; each instrumental-variable specification places it between −0.12 and −0.28. These are not overlapping confidence intervals around a common parameter but qualitatively different answers to the same question, and no ranking of the estimators by their own diagnostics resolves the disagreement, because the specifications that reverse the sign are precisely the ones designed to address the identity problem. The sign of the capitalisation coefficient is not identified by these data, and that instability is the result rather than an obstacle to it.

The operating coefficients behave in the opposite way — across seven of the eight estimators the labour share ranges only between −0.115 and −0.141, and the ranking of the four operating variables never changes — but the contrast is not the one it first appears to be, and Section 7 takes it up in full. Return on assets and the labour share are consecutive lines of the same income statement, so an estimator-invariant coefficient is what that construction produces and cannot be read, without further test, as evidence of a relationship more robust than the financial one. What the comparison does establish is an asymmetry worth stating plainly here: the equity ratio is not mechanically determined by the cost base and its sign is nonetheless unidentified, while the labour share is mechanically related to the outcome and stable for that reason. Of the two, the literature has organised itself around the variable whose association with performance is the weaker and the less reliably signed.

The instrumental-variable estimates are reported in full in Appendix B rather than here, because they fail on their own terms and a failed identification strategy does not belong in the body of the argument. All five instrument sets are strong: first-stage F statistics run from 24 for the province-year peer mean to 2,327 for the second lag of the equity ratio. Strength is not validity, and the overidentified specification using the second and third lags rejects the Hansen test with a statistic of 77.18. The mechanism of failure is identifiable rather than merely suspected: with ROA persistent at 0.271, the equity ratio at t−2 mechanically embeds profit at t−2, which predicts profit at t through persistence alone, so the exclusion restriction fails by construction and no lag length escapes it while profitability remains persistent. The one instrument with a defensible exclusion restriction, net external equity injections, is too weakly related to the regressor to be usable: its within-firm correlation with the equity ratio is 0.103, which amplifies any violation of exclusion roughly tenfold. Reporting the attempt with its diagnostics forecloses an obvious referee request and is more informative than omitting it.

Two further features of the panel bear on how the coefficients should be read. The panel is unbalanced in a way correlated with regulatory status, so the weighting scheme determines whose experience the estimates describe. Weighting each firm equally rather than each observation raises capitalisation from 0.177 to 0.199 and size from 0.646 to 0.855, so the relationship is somewhat stronger among firms observed briefly. Weighting by total assets instead, which recovers the coefficient relevant to the economic aggregate rather than to the median firm, halves capitalisation to 0.099 while raising capital turnover from 5.29 to 6.89 and materials from −0.114 to −0.149. In the firms that carry the aggregate the operating model matters more and financial structure less. Neither weighting is correct in the abstract; the two answer different questions and both are reported in Table A2.

The second feature is that a single coefficient vector has been imposed on three populations that the regulation itself treats as distinct. Estimating the equation separately by regime, on the same observations and the same specification, tests that restriction directly, and Figure 5 plots the three vectors on the standardised scale of Figure 2.

Wald tests reject equality of the coefficient vectors in all three pairwise comparisons: 87.18 between start-ups and innovative SMEs, 181.94 between start-ups and ordinary SMEs, and 138.08 between the two SME categories, each on eight degrees of freedom with p below 10⁻⁴. What differs is not the level of profitability, which the fixed effects absorb, but the structure of its determination. Capital turnover carries a coefficient of 7.02 among start-ups and 7.35 among innovative SMEs against 4.17 among ordinary SMEs, while the labour share weighs most heavily on ordinary SMEs at −0.163; capitalisation is highest among start-ups at 0.232. Certification schemes rest on the premise that innovative firms operate under constraints different from those facing conventional ones, and evaluations customarily measure average effects on growth or employment while treating the constraint as given. The evidence here is that the difference is measurable in the structure of the profit equation itself and survives every specification attempted. It also motivates Section 5, which asks whether the partition that matters is the regulatory one or one the data would choose for themselves.

5. Unsupervised Structure

The panel estimates of Section 4 impose a restriction the data have not been asked to confirm: that a single coefficient vector describes every firm. The restriction can be tested without leaving the equation, by partitioning firms on the equation's own variables and re-estimating within each partition. What the exercise can establish is whether the pooled coefficients mean what they appear to mean or whether they average structures that differ; what it cannot establish is any causal effect, and no claim of that kind is made here. The partition is a description of coefficient heterogeneity, and it is built from accounting variables alone.

Each firm is summarised by the decade median of the nine variables entering the equation, standardised to zero mean and unit variance, on the same 14,913 firms and the same 91,756 observations used in Section 4. Regulatory status is withheld from the procedure entirely and used afterwards only as an external check, so that any correspondence between the partition and the register is a result rather than an assumption. Thirty-two configurations were evaluated across six algorithm families — k-means, Ward agglomerative, Gaussian mixture and fuzzy c-means for k between two and seven, density-based clustering over five radii, and random-forest proximity clustering for k between two and four — of which twenty-nine returned a partition with at least two groups; the three discarded are density-based solutions at the widest radii, where every firm falls into a single component. Each retained configuration was scored on eleven validity criteria: silhouette, Calinski–Harabasz, Davies–Bouldin, Dunn, maximum diameter, minimum separation, the Pearson gamma between distances and co-membership, entropy, the Herfindahl–Hirschman index of cluster balance, partition R², and, where defined, the Bayesian information criterion.

Selecting on any single index misleads, and the direction of the error is systematic rather than random. Silhouette falls monotonicatablely in k for every partitional method, so on that criterion alone the two-cluster solution always wins; Calinski–Harabasz behaves identically. Partition R² and the Dunn index move in the opposite direction, rewarding finer partitions. A paper reporting silhouette alone would therefore report two clusters, and a paper reporting R² alone would report seven, from the same data and the same algorithm. The two density-based configurations attain the highest silhouette values in the entire battery while classifying between four and thirty-three per cent of firms as noise, which is what a high silhouette means when a procedure is free to discard the observations that fit badly. Averaging ranks across the nine directional indices selects k-means at k = 4: it places first on partition R² among the balanced solutions and keeps its smallest group at 14.9 per cent of firms. Fifty bootstrap resamples at eighty per cent give a mean adjusted Rand index of 0.9375 with a minimum of 0.6167, so the partition is not an artefact of the particular sample. The neighbouring three-cluster solution ranks immediately behind and was estimated in full as a robustness check; the coefficient pattern reported below survives it.

The four groups are best read on the standardised scale, where each cell states how far a group's typical firm lies from the sample mean in standard deviations.

Figure 6. Cluster profiles, standardised means.

Figure 6. Cluster profiles, standardised means.

The colours identify the archetypes and the medians in Table 5 give them their magnitudes in the units of the accounts. Cluster 0, a third of all firms, is labour-intensive: a labour share of 89.6 per cent of value added, the lowest capitalisation in the sample at 17.3 per cent, and the lowest median return at 3.03. Cluster 1, twenty-seven per cent, is materials-intensive and the largest in scale, with materials at 47.4 per cent of output and median log assets of 9.75. Cluster 2, twenty-four per cent, is composed of micro service firms: purchased services at 57.4 per cent of output, a labour share of 0.63 per cent, the smallest scale in the sample and a median return of 7.36. Cluster 3, fifteen per cent, is the only group defined financially rather than operationally, with capitalisation of 70.1 per cent, a liquidity ratio of 3.70, the lowest capital turnover and the highest median return at 10.23.

Two features of the separation matter more than the labels. The first is that the financial variables behave differently from the operating ones in how they organise the partition. Capitalisation and liquidity are extreme in Cluster 3 and virtually identical across the other three groups, at 0.01, 0.01 and −0.60 standard deviations for the equity ratio: they identify one archetype sharply and say almost nothing about the remaining eighty-five per cent of firms. The labour share and purchased services, by contrast, take a distinct value in each of the four groups and order them. Financial structure characterises an archetype; it does not organise the partition. The second is that the partition was formed without reference to profitability being a target, yet median ROA rises monotonically across the four groups from 3.03 to 10.23, which is the first indication that these are not merely descriptive categories.

Regulatory status was withheld from the procedure, so the correspondence between the partition and the register is an external test of both.

Figure 7. Cluster composition by regulatory regime.

Figure 7. Cluster composition by regulatory regime.

Partition and register correspond strongly without coinciding: χ²(6) = 6,821.9 with Cramér's V of 0.4780. Ordinary SMEs concentrate almost entirely in the two established operating archetypes, 46.0 and 46.9 per cent, with only 7.1 per cent elsewhere. Innovative start-ups distribute almost inversely, 53.1 per cent in the micro-service group and 24.2 per cent in the capitalised one, which is what the certification regime would predict for firms that are small, service-oriented and recently equity-funded. Innovative SMEs spread across all four groups, consistent with a category reached either by graduation from start-up status or by direct entry from conventional activity. The regulatory categories therefore carry real information about the operating model, but they are not the same partition: a Cramér's V of 0.478 leaves a great deal of the variation unexplained, and the analysis of Section 4 by regime and the analysis here by archetype are answering different questions.

The point of the partition is what happens when the equation is re-estimated inside it. The same specification, the same estimator and the same observations are used; only the grouping changes.

Table 6. Equation re-estimated within each cluster (selected coefficients).

Regressor	C0 labour	C1 materials	C2 micro service	C3 capitalised
Capital turnover	3.0860***	5.9623***	7.8726***	13.1552***
	(0.1742)	(0.2216)	(0.3498)	(0.6774)
Labour cost / Value added (%)	-0.1403***	-0.1317***	-0.1347***	-0.1383***
	(0.0038)	(0.0047)	(0.0061)	(0.0084)
Purchased services / Output (%)	-0.0980***	-0.1510***	-0.2475***	-0.2852***
	(0.0075)	(0.0139)	(0.0155)	(0.0176)
Leased assets / Output (%)	-0.1909***	-0.2800***	-0.3051***	-0.4937***
	(0.0234)	(0.0505)	(0.0390)	(0.0545)
Equity ratio (%)	0.2314***	0.1015***	0.2002***	0.1280***
	(0.0091)	(0.0069)	(0.0104)	(0.0136)
Observations	34,377	34,632	12,804	10,062
Within R²	0.4293	0.4955	0.4420	0.4759

Note. Two-way fixed effects, firm-clustered standard errors in parentheses. *** p<0.01. The full eight-regressor estimates are Table A5.

Table 6. Equation re-estimated within each cluster (selected coefficients).

Regressor	C0 labour	C1 materials	C2 micro service	C3 capitalised
Capital turnover	3.0860***	5.9623***	7.8726***	13.1552***
	(0.1742)	(0.2216)	(0.3498)	(0.6774)
Labour cost / Value added (%)	-0.1403***	-0.1317***	-0.1347***	-0.1383***
	(0.0038)	(0.0047)	(0.0061)	(0.0084)
Purchased services / Output (%)	-0.0980***	-0.1510***	-0.2475***	-0.2852***
	(0.0075)	(0.0139)	(0.0155)	(0.0176)
Leased assets / Output (%)	-0.1909***	-0.2800***	-0.3051***	-0.4937***
	(0.0234)	(0.0505)	(0.0390)	(0.0545)
Equity ratio (%)	0.2314***	0.1015***	0.2002***	0.1280***
	(0.0091)	(0.0069)	(0.0104)	(0.0136)
Observations	34,377	34,632	12,804	10,062
Within R²	0.4293	0.4955	0.4420	0.4759

Note. Two-way fixed effects, firm-clustered standard errors in parentheses. *** p<0.01. The full eight-regressor estimates are Table A5.

Pairwise Wald tests reject equality of the coefficient vectors for all six pairs, with statistics between 84.45 and 374.34 on eight degrees of freedom. Rejection at this sample size is not by itself informative, since almost any restriction is rejected on ninety thousand observations; what matters is the magnitude and the pattern of the differences, which Figure 8 shows by expressing each within-cluster coefficient as a multiple of the pooled estimate from Table 3.

The two remaining dimensions are stable for opposite reasons. Liquidity is insignificant in three groups of four, so the pooled estimate is right that it does not matter anywhere. The labour-share coefficient varies by a factor of only 1.1, lying between −0.132 and −0.140 with overlapping confidence intervals throughout, and invariance of that kind invites an obvious objection: return on assets and the labour share are consecutive lines of the same income statement, so a coefficient that does not move across groups may be recording a construction rather than a relationship. The objection is taken up in Section 7 and it carries a testable implication, because the identity fixes the slope at −VA / A: a coefficient this close to constant across archetypes requires value added over total assets to be close to constant across them as well.

It is not. Value added over assets ranges from 0.217 in the micro-service cluster to 0.602 in the labour-intensive one, a factor of 2.8, against a factor of 1.07 in the coefficient. The invariance therefore survives the test that would have exposed it as arithmetic, and it survives it on a partition constructed from the very variables that generate the arithmetic. Of the eight dimensions of the pooled vector, one is faithful because the underlying relationship really is common to all four groups, one is faithful because the variable is inert everywhere, and on the remaining six the pooled estimate averages quantities that differ by factors of two to four.

The ordering of the equity-ratio coefficient is worth isolating, because it is the coefficient Section 4 could not sign. Within clusters it is positive everywhere, but it is weakest in the materials-intensive group of larger firms at 0.102 and strongest in the labour-intensive group of smaller, thinly capitalised ones at 0.231 — the pattern a financing-constraint reading predicts, with capitalisation mattering most where it is scarcest. That is a pattern, not an identification, and it does not resolve the instability documented in Figure 4; the accounting identity operates inside each cluster exactly as it does in the pooled sample. What the partition adds is that the pooled coefficient of 0.177 is not merely uncertain in sign but heterogeneous in magnitude, so that the specification search of Section 4 was conducted on a parameter that does not have a single value.

Two implications follow for how the rest of the paper reads. The first is that the regulatory partition tested in Section 4 and the operating partition tested here are both real and neither is the other: certification tracks the operating model strongly enough to produce a Cramér's V of 0.478 but not so strongly that policy categories can substitute for cost structure in an empirical model. The second is that the linear specification has now been questioned in one direction only. Section 5 has asked whether one coefficient vector fits every firm and answered that it does not; it has not asked whether the coefficients are constant within a group, or whether the relationship between each regressor and profitability is linear at all. That is the question Section 6 addresses, on the same observations, by fitting learners that impose no functional form and measuring what they recover that the linear equation does not.

6. Machine-Learning Regression as Validation

The panel specification rests on two assumptions beyond those already tested. Section 5 examined the first, that a single coefficient vector describes every firm, and rejected it. The second is that each regressor enters linearly and additively. Examining that one requires estimators that impose no functional form and can therefore be asked how much of the variation in profitability a linear equation leaves unexplained. The purpose here is validation rather than prediction: nothing in this section forecasts profitability, no learner replaces the equation, and no coefficient is revised on the strength of a fit. The questions are how large the gap between the linear form and an unrestricted one is, what kind of structure accounts for it, and whether the ordering of variables implied by the panel coefficients survives an independent measure of predictive content.

Thirteen algorithms were fitted to the same 91,756 observations on the same eight regressors, with five-fold cross-validation grouped by firm so that no firm contributes to both the training and the test partition. They were compared on out-of-sample R², the cross-fold standard deviation of that R², root mean squared error, mean absolute error and fitting time, reported in full as Table A1. Selection follows the same principle as the clustering of Section 5: a battery rather than a single index. Histogram gradient boosting attains the highest accuracy at 0.7978, the lowest errors at 5.932 and 3.476, and a cross-fold standard deviation of 0.0070 among the lowest in the battery, in 3.3 seconds against 120.5 for the random forest that comes third on accuracy. It is used for every result that follows. The comparison also has a finding of its own: the thirteen algorithms fall into three bands rather than a continuum, with tree ensembles and the neural network above 0.72, k-nearest neighbours and the single decision tree between 0.64 and 0.70, and the entire linear family, including the linear support-vector machine, below 0.48.

The comparison that matters is nevertheless not that one, because it is the comparison most often made incorrectly. The fixed-effects estimator explains variation within firms, after firm means have been removed; a learner fitted to raw levels also has the between-firm variation available to it. Setting a within R² of 0.42 against a levels R² of 0.80 therefore attributes to functional form what mostly reflects a different decomposition of the variance. The comparison is made three times instead, under three treatments of the between-firm information.

Table 7. Like-for-like comparison of linear and non-linear fits.

Data transformation	Linear R²	SD	Gradient boosting R²	SD	Gap	Ratio
Levels (pooled)	0.4730	0.0112	0.7978	0.0070	0.3248	1.69x
Firm-demeaned (within)	0.4153	0.0126	0.5360	0.0107	0.1206	1.29x
Levels + firm means (Mundlak)	0.4795	0.0116	0.7986	0.0054	0.3191	1.67x

Note. Five-fold cross-validation grouped by firm. SD is the standard deviation of R² across folds. The second row applies to both models exactly the transformation the fixed-effects estimator applies.

Table 7. Like-for-like comparison of linear and non-linear fits.

Data transformation	Linear R²	SD	Gradient boosting R²	SD	Gap	Ratio
Levels (pooled)	0.4730	0.0112	0.7978	0.0070	0.3248	1.69x
Firm-demeaned (within)	0.4153	0.0126	0.5360	0.0107	0.1206	1.29x
Levels + firm means (Mundlak)	0.4795	0.0116	0.7986	0.0054	0.3191	1.67x

Note. Five-fold cross-validation grouped by firm. SD is the standard deviation of R² across folds. The second row applies to both models exactly the transformation the fixed-effects estimator applies.

The three rows answer the same question under three different treatments of between-firm variation, and reading them together is what makes the comparison interpretable. The first gives the learner access to differences between firms as well as within them. The second removes firm means, replicating exactly what the fixed-effects estimator does. The third restores the between-firm information to the linear model in the form of firm means, following Mundlak. Comparing the first with the third isolates how much of the linear model's apparent disadvantage on levels is simply unmodelled firm heterogeneity; comparing the second with either isolates what remains attributable to functional form.

Figure 9. Out-of-sample accuracy under three transformations.

Figure 9. Out-of-sample accuracy under three transformations.

On raw levels the non-linear model is 1.69 times more accurate. On firm-demeaned data, which is what the fixed-effects estimator uses, the ratio falls to 1.29 and the absolute gap from 0.325 to 0.121: the linear specification recovers 0.415 of within-firm variance against 0.536 attainable. Supplying the linear model with firm means in the manner of Mundlak raises it only from 0.473 to 0.480, which locates the source of the remaining difference: on levels the learner's advantage is mostly firm heterogeneity that fixed effects already absorb, and what survives the demeaning is functional form. The honest statement of the gap is therefore 0.121 rather than 0.325, and everything that follows concerns those twelve points of R².

The next question is what kind of structure they represent, and the first candidate is one the linear model can be given directly. If the missing structure were polynomial, adding squares and interactions to the equation would recover it, at a cost in parameters that can be counted.

Table 8. How much of the gap parametric terms recover.

Specification	Terms	R²	SD	Gap closed (%)
Linear, 8 regressors	8	0.4153	0.0126	0.0
+ squared terms	16	0.4319	0.0116	13.7
+ pairwise interactions	36	0.4355	0.0125	16.7
+ squares and interactions	44	0.4401	0.0114	20.6
Full cubic expansion	164	0.3672	0.1890	-39.9
Histogram gradient boosting	—	0.5360	0.0107	100.0

Note. Firm-demeaned data, five-fold cross-validation grouped by firm. Gap closed is the share of the 0.1206 difference between the linear specification and gradient boosting that each enrichment recovers.

Table 8. How much of the gap parametric terms recover.

Specification	Terms	R²	SD	Gap closed (%)
Linear, 8 regressors	8	0.4153	0.0126	0.0
+ squared terms	16	0.4319	0.0116	13.7
+ pairwise interactions	36	0.4355	0.0125	16.7
+ squares and interactions	44	0.4401	0.0114	20.6
Full cubic expansion	164	0.3672	0.1890	-39.9
Histogram gradient boosting	—	0.5360	0.0107	100.0

Note. Firm-demeaned data, five-fold cross-validation grouped by firm. Gap closed is the share of the 0.1206 difference between the linear specification and gradient boosting that each enrichment recovers.

The table turns each specification into a price, and the figure shows what is bought.

Figure 10. Parametric enrichment against gradient boosting, firm-demeaned data.

Figure 10. Parametric enrichment against gradient boosting, firm-demeaned data.

Sixteen terms buy 13.7 per cent of the gap, forty-four buy 20.6, and one hundred and sixty-four buy nothing at all: the full cubic expansion performs worse than the original eight-regressor model, with a cross-fold standard deviation of 0.189 against 0.013, because it overfits the tails of a heavy-tailed dependent variable. The cross-fold standard deviation is the column that explains the failure rather than merely recording it, and it is why the cubic row is reported at all. Roughly four fifths of what the learner captures is therefore not polynomial. Nor is it interaction: Friedman H-statistics are small throughout, the strongest pair — the equity ratio with capital turnover — reaching 0.042, so the residual structure is univariate curvature rather than joint dependence between regressors. What the linear form omits is local structure: thresholds, saturation, and regions of a variable's range where it matters against regions where it does not.

Partial dependence makes those shapes explicit and explains why polynomials approximate them badly. The labour share declines steeply over its lower range and then flattens, so its marginal effect is concentrated among firms whose wage bill is a moderate share of value added and close to nil among those where it is already dominant. Capital turnover rises and saturates. Purchased services are flat over most of their range before bending at the upper end. The equity ratio is close to monotone and mild, which is consistent with the modest positive fixed-effects coefficient rather than with the negative instrumental-variable estimates of Section 4. A quadratic term must curve everywhere in order to curve anywhere, and a cubic must reverse; the surface the learner recovers is instead flat in some regions and steep in others, which is exactly the structure a low-order polynomial cannot represent and a tree ensemble represents naturally.

A second question is independent of fit. Permutation importance measures how much out-of-sample accuracy is lost when a predictor is randomly reshuffled, so it ranks variables by contribution to genuine prediction rather than by coefficient magnitude or by significance, and it is computed on held-out data.

Figure 11. Permutation importance, firm-demeaned data.

Figure 11. Permutation importance, firm-demeaned data.

The labour share alone accounts for 63.5 per cent of predictive content, capital turnover for 13.3 and purchased services for 9.6. The operating block carries roughly ninety per cent of the information, against 6.5 per cent for the equity ratio and 1.1 for liquidity, and the error bars are small relative to the distance between the labour share and everything else, so the ordering is not a sampling artefact. This is a sharper hierarchy than the panel coefficients suggest — Figure 2 put capitalisation and the labour share within a point of one another on the standardised scale — and it is reached by a route that uses no standard errors and no functional form.

The interpretation of the leading entry requires the qualification developed in Section 7, and stating it here rather than there prevents the figure from being read as a third independent confirmation. A learner recovers an accounting relation almost perfectly, and return on assets and the labour share are consecutive lines of one income statement, so a substantial part of the 63.5 per cent measures the predictive content of a construction rather than of a behaviour. The permutation exercise agrees with the panel estimates and with the clustering because all three inherit the same arithmetic, not because three independent routes converge. What the Figure Aoes establish independently is the position of capital turnover, which enters no identity and which the panel coefficients understate: second on this measure at 13.3 per cent, and the variable whose coefficient varies most across archetypes. And the entry that is informative by its smallness is the equity ratio, at 6.5 per cent, since capitalisation is not mechanically tied to the outcome and its low share is therefore a statement about explanatory content rather than about construction.

The last diagnostic asks where in the distribution the two models differ, which a single R² cannot show.

Figure 12. Calibration by decile of realised profitability.

Figure 12. Calibration by decile of realised profitability.

In the bottom decile of demeaned ROA, where the realised mean is −16.5 points, the linear model predicts −7.9; in the top decile, realised at +16.6, it predicts +6.6. The linear model compresses the distribution towards its centre by roughly a factor of two at the extremes, and gradient boosting by a factor of 1.8, so the non-linear model is better calibrated in the tails without being well calibrated there. In the middle six deciles both are accurate, with biases below 1.7 points and below 0.5 for four of the six. Essentially the whole difference between the two specifications arises in the two deciles at each end, which is worth stating explicitly because it identifies exactly which observations drive the result: a study that trimmed the tails would find the linear and the non-linear specification nearly equivalent, and would be right about the firms it retained and silent about the rest.

Three conclusions follow for the panel estimates, and they are conclusions about interpretation rather than about validity. The coefficients of Section 4 are local linear approximations to a mildly curved surface and should be described as average slopes rather than as constant marginal effects, with the approximation weakest for the most and least profitable tenths of the sample. The ranking of variables they imply is corroborated rather than overturned by an independent measure of predictive content, and corroborated by a wider margin than the coefficients themselves suggest. And the residual gap is not a matter of sample size: the learning curve of Appendix D shows the linear model flat from the first ten thousand observations, its test score rising from 0.4005 to 0.4153 over a twenty-fold increase in training data, while the boosting test score converges towards 0.536 from above. What the linear specification lacks is form, not information, and the twelve points of R² it cannot reach set a realistic ceiling for any equation of this kind estimated on these variables.

7. The Accounting Identity in the Operating Block

The argument of Section 4 against the capitalisation coefficient applies with greater force to the coefficient this paper reports as its main finding, and the objection is best raised here rather than answered defensively later. Net worth contains the current year's profit, which is why the equity ratio stands in a partly mechanical relationship with return on assets. The labour share of value added stands in a relationship that is mechanical by construction, because the two quantities are consecutive lines of the same income statement.

Figure 13. Where the regressors sit in the income statement that generates the dependent variable.

Figure 13. Where the regressors sit in the income statement that generates the dependent variable.

Under the Italian civil-code scheme, value added is the value of production less materials, purchased services, leased assets and other operating costs; EBITDA is value added less labour cost; and the operating result is EBITDA less depreciation and provisions. Writing s for the labour share and A for total assets, ROA = (VA / A)(1 − s) − D / A. The derivative of the dependent variable with respect to the regressor is therefore −VA / A exactly, before any estimation. Four of the eight regressors are ratios of an item subtracted in that cascade to the aggregate it is subtracted from, so the operating block as a whole is exposed to the objection, and the labour share most of all.

Two readings advanced earlier in the paper have to be withdrawn as a consequence. The stability of the labour-share coefficient across estimators, reported in Section 4 as evidence that the relationship is structural, is what an accounting relation produces, since such a relation does not depend on the estimator. And its 63.5 per cent share of permutation importance measures in part the predictive content of a construction rather than of a behaviour, since a learner recovers an identity almost perfectly. A third reading looks like the same point in another form — the near-invariance of the coefficient across the four clusters, at −0.132 to −0.140 — and Section 7.1 shows that it is not. The reason the three can be told apart is that the identity fixes not only the sign of the coefficient but its magnitude, and magnitude is checkable.

7.1. How Much of the Coefficient Is Mechanical

The identity predicts a slope of −VA / A, so the test requires one quantity the estimating equation does not report: the distribution of value added over total assets in the sample itself.

Table 10. Value added over total assets in the estimating sample.

Population	p5	p10	p25	Median	p75	p90	Mean	Obs.
All firms	0.055	0.105	0.211	0.353	0.604	1.040	0.498	96,052
Innovative start-ups	0.015	0.031	0.088	0.213	0.435	0.727	0.321	14,412
Innovative SMEs	0.034	0.066	0.156	0.304	0.515	0.753	0.370	20,907
Ordinary SMEs	0.130	0.174	0.257	0.394	0.697	1.256	0.584	60,733

Table 10. Value added over total assets in the estimating sample.

Population	p5	p10	p25	Median	p75	p90	Mean	Obs.
All firms	0.055	0.105	0.211	0.353	0.604	1.040	0.498	96,052
Innovative start-ups	0.015	0.031	0.088	0.213	0.435	0.727	0.321	14,412
Innovative SMEs	0.034	0.066	0.156	0.304	0.515	0.753	0.370	20,907
Ordinary SMEs	0.130	0.174	0.257	0.394	0.697	1.256	0.584	60,733

Note. Value added and total assets as reported in the statutory accounts, winsorised at the first and ninety-ninth percentiles. The ratio is not among the nine variables of the estimating equation and was therefore computed separately, by reapplying the sample construction of Section 3 to the source extractions: positive value of production, positive value added, and value of production of at least one per cent of total assets. The resulting 96,052 firm-year observations on 15,501 firms exceed the estimating sample of Table 2 by 4.7 per cent, because the ratio requires only two of the nine variables and is defined for firm-years that the full specification drops for missingness elsewhere. The difference is not selective on any dimension relevant to the comparison: the median of the ratio changes by less than 0.01 when the sample is restricted to firm-years with all nine variables present.

The median is 0.353 and the middle half of the sample lies between 0.211 and 0.604. The identity therefore predicts a coefficient of about −0.355 for the median firm and between −0.21 and −0.60 for the interquartile range. The estimated coefficient is −0.1365.

Figure 14. The distribution of value added over total assets, and the slope the identity implies.

Figure 14. The distribution of value added over total assets, and the slope the identity implies.

The estimate is not the identity. It is roughly two-fifths of the magnitude predicted at the median and falls below the first quartile of the predicted range. For the coefficient to be mechanical in full, value added would have to be 13.7 per cent of total assets, and just under fourteen per cent of firm-years are at or below that level — concentrated among start-ups, whose median is 0.213 against 0.394 for ordinary SMEs. The identity is unmistakably present in these data, and it over-predicts what is estimated by a factor of between two and three.

Two mechanisms account for the gap and both can be quantified. The first is conditioning. The equation includes capital turnover, revenue over assets, which absorbs much of the variation in value added over assets, so the reported coefficient is a partial effect holding scale intensity fixed rather than the unconditional slope the identity describes. The second is the covariance between the labour share and value-added intensity within firms: regressing VA / A on the labour share with firm effects returns a slope of −0.00068, so a firm whose labour share rises by ten points sees value added over assets fall by 0.007, and the total derivative implied by the identity moves from −0.355 to −0.375 at the median, and to −0.283 when firms are weighted by the within-firm variance of the labour share that the fixed-effects estimator actually exploits. Neither adjustment brings the prediction near −0.1365.

A second and independent comparison is already available in the estimates of Section 4. Lagging every regressor by one year breaks the identity, because the labour share of t−1 and the profit of t are computed from different statements. What survives the break bounds from below whatever part of the coefficient is not contemporaneous arithmetic.

Figure 15. What survives when the identity is broken by lagging.

Figure 15. What survives when the identity is broken by lagging.

The labour share retains a quarter of its magnitude at −0.0346 and remains significant at one per cent; leased assets retain 29 per cent, purchased services 8 per cent, and materials become insignificant. The retained quarter would require a value added of 3.5 per cent of total assets to be mechanical in its turn, a level reached by 3.2 per cent of firm-years, so the lagged coefficient is unlikely to be an artefact of construction. It is not, however, a clean estimate of behaviour either, since lagging also changes the question a cost structure is being asked to answer: it predicts next year's profit rather than this year's, and some of the loss is dynamics rather than arithmetic.

A third comparison settles the matter, and it uses the partition of Section 5 rather than the time dimension. The identity fixes the slope at −VA / A, so it predicts not one magnitude for the sample but a different magnitude for every group of firms whose value-added intensity differs. The four archetypes differ in that respect considerably: median value added over total assets runs from 0.217 in the micro-service cluster to 0.602 in the labour-intensive one, a factor of 2.8. If the coefficient were mechanical it would have to vary by the same factor, and be largest in absolute value precisely where value added per unit of assets is largest.

They do not. Across the four archetypes the predicted slope ranges by a factor of 2.8 and the estimated coefficient by a factor of 1.07, from −0.132 to −0.140, with confidence intervals that overlap throughout. The ordering is partially preserved — the labour-intensive cluster does carry the largest coefficient in absolute value, as the identity requires — but the amplitude is not, and amplitude is what the identity fixes. The ratio of estimate to prediction falls from 0.62 in the micro-service cluster to 0.23 in the labour-intensive one, so the attenuation is itself systematic, and largest exactly where the mechanical component should be strongest.

Figure 16. The identity prediction against the estimate, by archetype.

Figure 16. The identity prediction against the estimate, by archetype.

This is the decisive comparison of the three, because it holds the identity fixed and varies the quantity the identity says the coefficient must track. A relation that is arithmetic cannot be near-constant across groups whose multiplier differs threefold. The near-invariance reported in Section 5 is therefore not the signature of an accounting relation after all: it is a property of the estimated relationship, and it survives a partition built to separate firms on exactly the dimensions that generate the arithmetic.

The three comparisons can be stated together. The contemporaneous coefficient is smaller than the identity alone would generate, so it is the identity net of the variation the specification conditions away; the lagged coefficient survives a break in the identity, so a part of the association is not contemporaneous arithmetic at all; and the coefficient is stable across groups in which the identity's multiplier is not, so its stability is not mechanical either. The identity is present in these data, it accounts for part of the level of the coefficient, and it accounts for neither its stability nor the whole of its magnitude.

7.2. What the Contribution Becomes

The paper's claim changes in wording rather than in value, and the honest statement of it is this. A large share of the observed variation in return on assets is fixed once the composition of the cost base is known, and part of that share is arithmetic rather than behavioural. This is a measurement the literature reviewed in Section 2 has not reported and which policy has proceeded as though it did not exist, and it is sharper for being bounded on both sides: the identity over-predicts the coefficient by a factor of two to three, and a quarter of the association survives the breaking of the identity altogether.

The comparison with financial structure survives the concession intact and is sharpened by it. The equity ratio is not mechanically determined by the cost base, and its coefficient is unstable in sign across fourteen specifications and identified by no instrument these data admit. The labour share is mechanically related to profitability, and part of its stability follows from that relation — but only part, since the stability itself survives a test the identity fails. A literature that has organised itself around the first variable while treating the second as a control has therefore chosen, of the two, the one whose association with performance is both weaker and less reliably signed. That claim requires no causal interpretation of either coefficient.

The clustering and machine-learning results are affected asymmetrically. The dispersion of the capital-turnover coefficient by a factor of 4.3 across archetypes is untouched, since turnover enters no identity, and it remains the strongest evidence in the paper that a single coefficient vector does not describe every firm. The invariance of the labour-share coefficient, which Section 7 opened by treating as suspect, emerges from Table 12 as a finding rather than an artefact. Only the machine-learning ceiling of 0.536 is qualified: a learner approaches an accounting relation closely, so part of the within-firm predictive accuracy it attains is the identity, and the informative residual is what it recovers beyond that.

Two of the five tests that would separate identity from relationship remain open, and both require re-estimation on the present sample rather than new data. They are listed for completeness and because the direction each would have to take is already specified.

Table 11. Five tests that separate identity from relationship.

Test	Identity predicts	Structure predicts	Status and result	What it establishes
Coefficient equals −VA / A	Equality	Departure from −VA / A	Done. Identity implies −0.355 at the median and −0.283 to −0.375 under alternative weightings; estimate is −0.1365	The estimate is the identity attenuated, not the identity itself
Lagged regressors	Coefficient near zero	Coefficient survives	Done. −0.0346***, a quarter retained; would require VA / A of 0.035, reached by 3.2% of firm-years	A lower bound on the non-mechanical component
Slope tracks VA / A across clusters	Coefficient varies with cluster VA / A	Coefficient invariant while VA / A varies	Done. VA / A ranges 2.8× across archetypes, the coefficient 1.07×, from −0.132 to −0.140	The stability of the coefficient is not mechanical
Dependent variable VA / A, labour share as regressor	No association	Association survives	Outstanding. Re-estimation only, same variables and sample	Whether the labour share predicts a quantity it does not enter
ROA at t + 1 on labour share at t	Coefficient near zero	Coefficient survives	Outstanding. Re-estimation only, one year of observations lost	Predictive content free of the shared statement

Note. The first test uses Table 10 and the two-way fixed-effects estimate of Table 3; the second uses the lagged specification of Table A2; the third uses Table 12. The remaining two require re-estimation on the sample of Section 3 and no new variables. Shading distinguishes completed tests from outstandin.

Table 11. Five tests that separate identity from relationship.

Test	Identity predicts	Structure predicts	Status and result	What it establishes
Coefficient equals −VA / A	Equality	Departure from −VA / A	Done. Identity implies −0.355 at the median and −0.283 to −0.375 under alternative weightings; estimate is −0.1365	The estimate is the identity attenuated, not the identity itself
Lagged regressors	Coefficient near zero	Coefficient survives	Done. −0.0346***, a quarter retained; would require VA / A of 0.035, reached by 3.2% of firm-years	A lower bound on the non-mechanical component
Slope tracks VA / A across clusters	Coefficient varies with cluster VA / A	Coefficient invariant while VA / A varies	Done. VA / A ranges 2.8× across archetypes, the coefficient 1.07×, from −0.132 to −0.140	The stability of the coefficient is not mechanical
Dependent variable VA / A, labour share as regressor	No association	Association survives	Outstanding. Re-estimation only, same variables and sample	Whether the labour share predicts a quantity it does not enter
ROA at t + 1 on labour share at t	Coefficient near zero	Coefficient survives	Outstanding. Re-estimation only, one year of observations lost	Predictive content free of the shared statement

Note. The first test uses Table 10 and the two-way fixed-effects estimate of Table 3; the second uses the lagged specification of Table A2; the third uses Table 12. The remaining two require re-estimation on the sample of Section 3 and no new variables. Shading distinguishes completed tests from outstandin.

8. Discussion

Three methods were applied to one sample, and the value of doing so lies in where they agree and where they do not. On the operating block they agree: the labour share carries the largest standardised coefficient after capital turnover, varies by a factor of only 1.1 across the four clusters, and accounts for 63.5 per cent of out-of-sample predictive content. Agreement of that kind invites an obvious objection, since an accounting relation also holds whatever the estimator, the subsample or the functional form, and Section 7 tests it rather than conceding it. The identity turns out to explain part of the level of the coefficient and none of its stability: it predicts −0.355 at the median firm against an estimate of −0.1365, and it predicts a threefold variation across archetypes where the estimates vary by seven per cent. What the three methods converge on is therefore a relationship that is partly arithmetic in magnitude and not arithmetic in its constancy.

On financial structure the three methods disagree, and the disagreement is informative rather than a defect. The static panel estimates reproduce the sign reported by Matias and Serrasqueiro (2017), Dalci (2018), Kalash (2023) and Youssef et al. (2023): better capitalised firms earn more. Lagging the regressors reverses that sign, the dynamic specification restores it, and every instrument set contradicts both. Because net worth contains the current year's profit, part of the contemporaneous association is mechanical here too, and because profitability persists at 0.271 no lag length escapes the problem — which is why the internal instruments used by Vijayakumaran and Vijayakumaran (2019), Dsouza et al. (2025) and Saiz-Sepulveda et al. (2026) fail. The persistence documented by Hirsch and Hartmann (2014) and Hirsch et al. (2021) and the identification practice of the capital-structure literature are, on this evidence, incompatible.

Setting the two blocks side by side is what the paper can claim without any causal interpretation. The equity ratio is not mechanically determined by the cost base, and its coefficient is unstable in sign across fourteen specifications and identified by no instrument these data admit. The labour share is mechanically related to profitability, yet its coefficient is more stable than that relation alone would produce and remains constant where the relation predicts it should move. Of the two variables, a literature organised around the first has chosen the one whose association with performance is both weaker and less reliably signed, while treating the second as a control — which is the position of Campisi et al. (2019), Grau and Reig (2021), Bhattu-Babajee and Seetanah (2022) and Chadha et al. (2023), none of whom place the blocks in competition.

The clustering and machine-learning results are affected asymmetrically. Capital turnover enters no identity, so its dispersion by a factor of 4.3 across archetypes is a failure of the pooled vector and the strongest evidence that one coefficient set does not describe every firm. The partition earns a second role beyond description, since it is what makes the identity testable: groups that differ in value-added intensity are groups in which a mechanical coefficient would have to differ. The 0.536 ceiling on within-firm predictive accuracy, by contrast, is partly an artefact of the same arithmetic, because a learner approaches an accounting relation closely, and the interesting residual is what it recovers beyond that. Both results qualify the panel estimates without overturning them, which is the sense in which the exercise is validation rather than replacement — a use of these algorithms that the prediction-oriented applications of Pap et al. (2022), Vajjhala and Strang (2024) and Mahmood et al. (2025) do not attempt.

Two implications for practice follow. For the empirical literature, the appropriate response to an unsTable Aoefficient is to report the instability rather than to select the treatment that preserves the expected sign; and the appropriate response to a stable one is to test whether the stability is arithmetic before reading it as structure, which requires only that the identity be written down and its prediction compared with the estimate. The corpus reviewed in Section 2 contains no instance of either check, in either direction. For policy, the certification regimes evaluated by Manaresi et al. (2021) and Scandurra et al. (2025) rest on relieving a financing constraint whose measured association with profitability cannot be signed on this population, while the cost margin that does carry the association receives no instrument at all.

Table 9 states, for each method, what was found and whether it aligns with, qualifies, or contradicts the existing evidence.

9. Policy Implications

Italy's productive structure is dominated by firms that do not grow: the size distribution is compressed towards the bottom, and the number of firms making the transition from micro to medium scale in any decade is small. This is habitually treated as a financing problem, and the policy instruments in place are built accordingly. The evidence assembled here bears on that diagnosis directly, and four implications follow.

The first concerns what certification schemes are for. The Italian Start-up Act, like most European equivalents, presumes that innovative firms are constrained by access to capital and responds with tax relief on equity investment, free access to a public credit guarantee, and relief from company-law rules on capital maintenance. Evaluations by Manaresi et al. (2021), Aiello et al. (2024), Anderloni and Harasheh (2025) and Scandurra et al. (2025) establish that the instruments reach their targets and improve financing outcomes. What the present results add is that the constraint they relieve is not the one that binds on profitability. Capitalisation accounts for 6.5 per cent of out-of-sample predictive content and cannot be signed reliably; the share of value added absorbed by labour accounts for 63.5 per cent and is stable across every estimator, every group and every functional form tried. A firm whose cost structure leaves no margin does not become profitable because equity is cheaper.

The second concerns the sixty-month horizon. Certification expires by statute five years after incorporation, which presumes that firms reach a stable operating configuration within that window. The partition recovered in Section 5 suggests otherwise: 53.1 per cent of certified start-ups sit in a micro-service configuration with a labour share close to zero and the lowest capital turnover in the sample, and the coefficient vector governing their profitability differs significantly from that of established operating firms. Migliaccio and Pavone (2021) and Vannoni (2019) document the same internal heterogeneity from the financial side, and Tran and Santarelli (2014) and Amoa-Gyarteng and Dhliwayo (2023) report comparable patterns for young firms elsewhere. A fixed horizon applied to a population this heterogeneous withdraws support from firms at very different stages of development.

The third concerns instruments that act on the price of capital. Italian firms remain overwhelmingly bank-financed, and the policy response has been to lower the cost of equity through fiscal channels. These instruments operate on a price. If profitability is determined predominantly by operating structure, the price of external capital is not the margin on which the growth decision turns: firms do not scale because they lack the operating configuration that scale requires, and cheaper equity does not supply it. Benkraiem (2016), Kumar and Rao (2016), Tong and Serrasqueiro (2020) and Garcia-Martinez et al. (2023) reach compatible conclusions from the financing side, finding access effects on investment that do not translate into performance. Capital-cost instruments should therefore be complemented by instruments acting on operating capability — managerial capacity, workforce composition, the productivity of labour employed — since that is where the variation in performance is concentrated. Campisi et al. (2019), Grau and Reig (2021), Chadha et al. (2023) and Tripathi et al. (2024) identify the same margin from a different direction.

The fourth concerns eligibility. Certification is established through thresholds on research spending, graduate employment or intellectual property, expressed as ratios to output or to total costs. Ratios of this form are satisfied most easily where the denominator is smallest, so firms with negligible operating activity qualify readily, and the restriction imposed in Section 3 removes 29.3 per cent of start-up observations against 1.0 per cent of conventional ones for precisely that reason. The certified population is therefore shaped by the arithmetic of eligibility as much as by innovative intent. Thresholds expressed as ratios should be paired with an absolute floor on operating activity, and registers should collect the cost-structure indicators used here — the labour share of value added, the intensity of purchased services — which statutory accounts already report.

A final implication is evaluative rather than substantive. Wald tests reject a common coefficient vector across the three regulatory populations at every pairwise comparison. Evaluations that estimate an average treatment effect on growth or survival, as most of the cited work does, measure a quantity silent on this: the populations differ in how profit is determined, not only in how much they earn. Evaluation designs that test for differences in coefficient structure, alongside differences in level, would recover information the current standard discards.

10. Limitations

The most consequential limitation is that none of the designs identifies a causal effect. There is no exogenous variation in financial structure in these data: firms choose their capitalisation, and the regulatory populations compared are self-selected, since certification is voluntary and conditional on criteria correlated with unobserved characteristics. Lagging the regressors, modelling persistence and instrumenting all attenuate simultaneity without removing it, and Section 4 shows that instrumentation performs worse than the estimator it is meant to correct. The results are conditional associations that survive or fail specification, not magnitudes with a causal interpretation. Most of the literature reviewed in Section 2 shares this limitation; the appropriate response is the one taken by Talamas Marcos (2025), Rastogi and Kumar (2024) and Camuffo and Poletto (2024), who exploit institutional or experimental variation rather than statistical corrections.

A second limitation concerns sample composition. Innovative start-ups file meaningful accounts in roughly one year in three against more than nine in ten for ordinary SMEs, and the restriction imposed to obtain interpreTable Aost ratios removes a further 29.3 per cent of start-up observations against 1.0 per cent of ordinary ones. Both selections are correlated with regulatory status and both remove firms at the pre-operating end of the distribution. The estimates therefore describe firms with genuine operating activity and are silent about the population the scheme most distinctively contains — which is the sharpest constraint on their reach. Modelling that selection explicitly remains to be done.

Third, sector controls are incomplete. Sector codes are available from the business register for the certified populations but for a small minority of the control group, so specifications with sector-by-year effects rest on a subsample composed almost entirely of certified firms. Coefficients are stable across those specifications, which is reassuring, but the test is not conducted on the full sample, and the possibility that the operating coefficients partly capture sectoral composition in the control group cannot be excluded. Costa et al. (2017) and Yousaf (2025) show how sensitive determinant estimates are to this kind of conditioning.

Fourth, the measurement of cost intensity is imperfect. Ratios to the value of production are fragile where production approaches zero, and rescaling on total assets is not an alternative because it places the ratios in an accounting identity with the dependent variable. Sample restriction addresses the symptom rather than the issue, which is that statutory accounts measure operating structure with error concentrated in exactly the firms of greatest policy interest. Studies working at sectoral level, such as Grau and Reig (2021) and Prakash and Nauriyal (2020), avoid the problem by aggregating, at the cost of the heterogeneity exploited here.

Fifth, inference is clustered by firm alone. The persistence documented by Hirsch and Hartmann (2014) and Hirsch et al. (2021) implies cross-sectional dependence that two-way or spatially corrected inference would price in, and the intervals reported here are correspondingly optimistic — though not by enough to overturn statistics several orders of magnitude above conventional thresholds.

Sixth, the unsupervised results are descriptive. The partition establishes that firms occupy distinguishable regions of the accounting space and that the boundaries are stable to resampling, but not why a firm occupies the region it does: certification may have placed it there, it may have been there already, or both may reflect an unobserved third factor. The selection procedure also involves discretion — the weighting of the eleven indices and the five per cent floor on cluster size are defensible but not unique — and the three-cluster solution ranks closely enough to be reported alongside. The same caution applies in the taxonomies of Linares-Mustarós et al. (2018), Juntunen et al. (2022) and Salles-Filho et al. (2023).

Finally, two boundaries on the machine-learning results. The comparison establishes what the linear form omits on these nine variables and cannot speak to variables absent from the accounts, so the 0.536 ceiling is a property of the data as much as of the method; and both models are poorly calibrated in the top and bottom deciles, which is where the firms of most interest to a growth-oriented policy are found. Balzano and Magrini (2026) and Cheraghali and Molnár (2026) report the same tail behaviour on Italian and Nordic accounting data. The analysis is confined to one country and one certification regime, and whether the dominance of operating over financial structure generalises to economies with different size distributions and financing systems is open.

11. Conclusions

This study asked which of two explanations of firm profitability carries more weight when both are placed in the same specification: the way a firm is financed, or the way it operates. The question was put to a panel of Italian firms spanning three regulatory regimes and answered three times, by panel estimation, unsupervised partitioning and machine-learning regression on the same observations, which interrogate respectively the average association, the assumption of one coefficient vector, and the adequacy of the linear form.

The first conclusion is that the two explanations are not comparable in the way the question presumes. The share of value added absorbed by labour carries a coefficient stable across every estimator, nearly invariant across four heterogeneous groups, and responsible for most out-of-sample predictive content — and stability of that kind is what an accounting relation produces, since return on assets and the labour share are consecutive lines of one income statement. The association is therefore in substantial part arithmetic rather than behavioural, and what survives when the identity is broken by lagging, a quarter of the coefficient, bounds from below whatever part of it is not. The finding is not that operating structure causes profitability, but that a large share of the variation in profitability is fixed once the composition of the cost base is known.

The second conclusion concerns the variable the literature does report. The coefficient on capitalisation changes sign depending on whether regressors enter contemporaneously or with a lag, changes sign again when persistence is modelled, and turns strongly negative under every instrument constructible from these data — instruments that fail because profitability is persistent, so any lagged ratio embedding past profit is correlated with the current error. Setting the two variables side by side is what can be claimed without a causal interpretation of either. Capitalisation is not mechanically determined by the cost base, yet its sign is not identified; the labour share is mechanically related to profitability, and stable for that reason. A literature organised around the first while treating the second as a control has chosen the weaker and less reliably signed of the two.

The third conclusion concerns heterogeneity. The three regulatory populations differ not merely in the level of their performance but in the structure that determines it, and the tests reject a common coefficient vector in every pairwise comparison. A partition built from accounting data alone, with the register withheld, recovers four operating archetypes that correspond to the regulatory classification without coinciding with it, and within them the capital-turnover coefficient — which enters no identity — ranges fourfold.

Methodologically the study makes a narrower point of wider use. Comparisons between econometric and machine-learning models are frequently made on incompatible variance decompositions; correcting the comparison halves the apparent superiority of flexible learners, and part of what remains is the accounting relation a learner recovers almost perfectly. Of the residual, four fifths is irreducible by polynomial terms, which locates the shortfall of the linear model in local rather than global structure. Machine learning becomes a diagnostic instrument for specification rather than a competitor to it.

What the study cannot claim is causality: no exogenous variation in financial structure exists in these data, and the comparison across regimes is between self-selected groups. Two continuations follow. The first needs no new data — the distribution of value added over total assets fixes the magnitude the identity predicts, and specifications whose dependent variable does not contain the regressor would separate arithmetic from behaviour in the cost coefficients. The second is institutional: the statutory expiry of certification, the eligibility thresholds and the fiscal rates changed repeatedly by legislation offer discontinuities that would support the causal reading this design withholds.