1 Introduction

The standard interpretation of the law put forward by Gibrat (1931) is that the growth rate of a given firm is independent of its initial size. However, while earlier studies, based on limited samples of well-established and large companies, confirmed Gibrat’s Law, since the contribution by Mansfield (1962) subsequent and recent research has rejected it (see next section). Indeed, the current consensus within the extant empirical literature is that smaller firms show a higher growth rate than their larger counterparts.

One way to approach this puzzle is to take into account the fact that earlier studies focused on companies which were the outcome of a previous (not investigated) market selection and so represented the industrial “core” within which Gibrat’s Law tended to be confirmed. On the other hand, the current literature, based on more comprehensive and large datasets including newly-founded and small firms, tests Gibrat’s Law by investigating a given population of the same firms over time and in doing so magnifies the role of smaller and younger firms (which must grow faster in order to reach a minimum efficient size and survive), and rejects the Law.

The purpose of this study is to test whether a given population of firms tends to converge to Gibrat-like behaviour over time, allowing for market selection and for the correlated exit of the less efficient firms. In this context,—and in contrast with the studies discussed above, the final population is smaller than the initial one and is made up of only the surviving, most efficient companies. In this setting, it may well be the case that Gibrat’s Law is rejected when considering the entire population of firms and the entire period examined (consistently with the current literature), while it is confirmed when considering only the tracked population of surviving firms (consistently with the earlier literature).Footnote 1

In other words, the hypothesis tested in this work is the following: “Gibrat’s Law, although to be rejected in general, might actually be accepted when market selection generates a sort of ‘steady state’, where a much more homogeneous population of surviving firms may behave according to the Law”.

Another important novelty of this study is the method adopted. While authors of earlier and current empirical literature have used econometric techniques, in this work we also include an innovative unsupervised approach generating graphical models able to elicit the intrinsic structure of the data and represent them as a network. More specifically, graphical models represent the data in a fully unsupervised way without testing an a priori assumed functional form. Moreover, they offer an overview of the structural relationship between all variables and not only for each covariate with respect to a dependent variable. However, standard econometric analysis is proposed as a complementary analysis and for the sake of comparison with previous literature.

2 Past and current literature

In 1931 the French engineer and economist Robert Gibrat put forward his Law of Proportionate Effect stating that the proportional rate of growth of a given company is independent of its absolute size at the beginning of the investigated period (now well-known worldwide and since then called Gibrat's Law, or rule of proportionate growth, see Gibrat, 1931).Footnote 2

After the Second World War, Gibrat’s Law of Proportionate Effect was very popular among both economists and statisticians (Santarelli et al. 2006). The main reason was that the Law was fully consistent with a log-normal distribution of firm size (or even considered as the data generation process behind such a distribution). In turn, a log-normal distribution of firm size was (and is nowadays) actually observed in virtually all economic sectors, where a vast majority of small- and medium-sized firms coexist with few larger counterparts. Therefore, as stated by Simon and Bonini (1958), if one “…incorporates the Law of Proportionate Effect in the transition matrix of a stochastic process, […] then the resulting steady-state distribution of the process will be a highly skewed distribution” (1958, p.609). The empirical consistency between Gibrat’s Law and the observed size distribution of firms across different industries was also discussed by Steindl (1965) and treated through examples and simulations by Prais (1976, Chapter 2).

However, although in the long term Gibrat’s Law certainly generates a log-normal firm size distribution, the latter does not necessarily require firms’ proportional rates of growth. Indeed, if we do not limit our attention to the incumbent firms but extend it to the analysis of industrial dynamics, i.e. the entry and exit of companies within a given industry, a log-normal distribution may also emerge, as the consequence of a small group of persisting larger (core) incumbents coexisting with a large fringe of smaller firms, characterised by churning and turbulence (high entry rates, low survival rates, revolving-door firms, see Geroski, 1995). In other words, Gibrat’s Law is a sufficient, but not necessary condition for generating an observable log-normal distribution of firm size. This argument permits the possibility of refuting the Law, without being in contrast with the revealed skewed distribution of firm size (see below).

Indeed, while earlier studies based on subsamples of large and mature firms had tended to confirm the Law (Hart & Prais, 1956; Hymer & Pashigian, 1962; Simon & Bonini, 1958), further research began to challenge its overall validity. It is important to note that earlier studies were based on limited databases, only comprising large incumbents, namely companies quoted on the London Stock Exchange (Hart and Prais (1956)); the largest 500 Fortune US corporations (Simon and Bonini (1958)); the 1,000 US largest manufacturing firms in the period from 1946–1955 (Hymer and Pashigian (1962)). In other words, earlier consensus regarding the validity of Gibrat’s Law was based on empirical tests limited to the core of larger incumbent companies, so neglecting the role of both incumbent SMEs and newly-founded firms.

The turning point in the literature was the seminal contribution by Mansfield (1962), investigating the U.S. steel, petroleum and tyre sectors in different time periods and finding that Gibrat’s Law was failing in the majority of cases, with smaller firms growing faster than their larger counterparts. Indeed, when studies take SMEs into account it is found that these incorporate their need to reach the minimum efficient size (MES) and so engage in accelerated growth.

Mansfield’s outcome has largely been confirmed by subsequent empirical studies using more comprehensive specifications and also including firms’ age and other controlling regressors. For instance, Hall (1987) studied 1,778 US manufacturing firms which had already reached a certain minimum size (measured in terms of employment) and belonged to two samples spanning the periods 1972–1979 and 1976–1983. Unlike Mansfield (1962), Hall directly regressed growth rates on the logarithm of the initial size and found that the observed negative relationship between size and growth was robust to corrections for both sample attrition and heteroskedasticity.Footnote 3

Evans (1987a) analyzed 100 4-digit manufacturing industries using firm level data drawn from the US Small Business Data Base (42,339 firms). The novel feature of this study was the introduction of age (in addition to size measured in terms of employment) as a possible factor in explaining departure from Gibrat’s Law. A negative relationship between growth and size was found in 89% of the industries examined, while a negative relationship between growth and age was verified in 76% of firms. As in the previous study, the estimation procedure controlled for sample selection bias and heteroskedasticity (see also Evans, 1987b).

The work by Dunne et al. (1989) also supported the rejection of Gibrat’s Law: within each age category, growth rates turned out to decline along employment size classes. Dunne, Roberts and Samuelson obtained these results from data on 219,754 individual plants, rather than firms as in the previous studies, collected in five US censuses of manufacturing (1963-67-72-77-82).

Another important contribution to the investigation of Gibrat’s Law was put forward by Dunne and Hughes (1994), who tested the Law of Proportionate Effect over the periods 1975–80 and 1980–85 using 2,149 quoted and unquoted UK companies belonging to 19 different manufacturing industries. After controlling for sample attrition and heteroskedasticity, Dunne and Hughes found further confirmation that smaller companies tend to grow faster than their larger counterparts; they also found that younger companies, for a given size, tended to grow faster than older ones.

By the same token, Hart and Oulton (1996) used data from 87,109 UK incumbent companies over the period 1989–93 and tested the Chesher-Mansfield specification (see next section), measuring size in terms of employment, sales and net assets. In all cases, they detected an overall estimated coefficient of less than one: on average, small firms grew more quickly than larger ones; however, they also found a not significant relationship between growth and size when considering only the larger firms. In other words, Gibrat’s Law turned out to be rejected in general, but not falsified within the subsample of the core companies (see above).

Audretsch et al. (1999) used an Italian dataset comprising newly-founded manufacturing firms tracked from 1987 to 1993 and found that Gibrat’s Law was indeed rejected in the vast majority of industries, both considering the entire set of firms and the limited set of surviving firms.

On the whole, by the end of the’90 s a new consensus had been reached, partially in contrast with that shared during the previous decades: this was that “Gibrat’s Legacy” (as named by Sutton, 1997; see also Caves, 1998 and Coad, 2009) was defendable not as a general law, but only as a dynamic rule valid for large and mature firms that had already attained the MES level of output but not for their smaller counterparts operating at a sub-optimal scale (Geroski, 1995). In a nutshell, and combining the two consensuses reached by the literature, Gibrat’s Law should be considered rejected when all firms are taken into account, but confirmed when only the core companies within industries are considered. Audretsch et al. (2004), for instance, analysed a sample of firms in the hospitality sector and found mixed results. However, when considering only surviving firms or large firms, Gibrat’s Law was more likely to hold.

The most recent literature has generally supported this overall conclusion. For instance, Calvo (2006), analysing 1,272 Spanish manufacturing firms over the period 1990–2000, found smaller firms growing faster than larger ones. By the same token, Oliveira and Fortunato (2006), using an unbalanced panel of Portuguese manufacturing firms over the period 1990–2001, found that large and mature firms do have smaller growth rates than small and young firms. Daunfeldt and Elert (2013) studied Swedish firms within five-digit NACE-industries during the period 1998–2004 and confirmed the rejection of Gibrat’s Law when considering the entire population of investigated companies; however, Gibrat’s Law was more likely to be rejected for industries characterised by a higher MES, while it was more likely to hold in mature industries, in industries with a high degree of group ownership, and in industries with a high market concentration. Tang (2015) studied the Swedish energy market, using an unbalanced longitudinal dataset covering 2,185 firms during the 1997–2011 period, and found an interesting twofold result: on the one hand, Gibrat’s Law was rejected, with smaller firms found to grow faster than their larger counterparts; on the other hand, when examining each firm individually, it was found that many Swedish energy firms do behave in accordance with Gibrat’s Law, namely those in a steady state, i.e. the larger and more mature ones. Distante et al. (2018) ran quantile regression models using annual data covering US manufacturing firms over six decades (1950–2010) and found that, conditional on survival, small establishments grow faster than their larger counterparts. Arouri et al. (2020) studied the pattern of growth of Tunisian firms over the period 1996–2010: their key finding was that consistently with the extant literature, Gibrat’s Law was rejected overall, with smaller firms growing faster than their larger counterparts; however, the negative impact of the initial size was found to be larger and more significant for young firms rather than for mature, larger incumbent firms. Elston and Weidinger (2023) investigated MENA companies listed on the United Arab Emirates (UAE) stock exchanges and found that in most industries smaller firms grow faster than larger firms, with three notable exceptions: energy, telecommunications and industrial manufacturing.Footnote 4

All in all, the extant literature seems to support the general idea that Gibrat’s Law should be rejected when all firms are taken into account, but can be revived when the core of larger and older incumbents is singled out. As mentioned in the previous section, the purpose (and the novelty) of this paper is to test whether a given population of firms tends to converge to Gibrat-like behaviour over time, allowing for market selection and for the correlated exit of the less efficient firms. In particular, and differently from the previous studies discussed above, we will start from a brand-new population (1,720 newly-founded Italian manufacturing firms) and will track them over 11 years to test whether convergence to Gibrat-like behaviour emerges over time. Instead of separating different groups of firms (i.e. core vs fringe), we will deal with the same population of companies over time, allowing for market selection and so for the exit of the less efficient firms, the purpose being to investigate whether earlier literature can be reconciled with more recent research, i.e. to test whether the rejection of Gibrat’s Law ex ante can be coupled with the defence of the Law ex post (see the hypotheses proposed in the next section).

To our knowledge, only two previous studies have attempted this kind of experiment. In a first work, Lotti et al. (2003) ran quantile regressions using data for 855 Italian manufacturing firms founded in January 1987 and tracked for six years; their main result was that in five industries out of six, Gibrat’s Law fails to hold in the years immediately following start-up, whereas it holds, or fails less severely, when firms approach maturity and market selection has done its job. In a later study, the same authors (Lotti et al., 2009) focused on the Italian radio, TV, and communication equipment industry over the period 1987–1994, studying the growth patterns of all the incumbent firms which were active in the sector at the beginning of the examined period (3,285 companies). Consistently with the former study, their results were twofold: on the one hand, Gibrat’s Law is rejected over the entire period, with smaller firms growing faster than larger ones; on the other hand, a convergence toward the validity of the Law occurs over time, once the annual regressions are run over the sub-population of surviving firms.

In what follows, the econometric regressions are similar in nature to the work described in Lotti et al. (2003, 2009), while the graphical model approach in testing Gibrat’s Law is applied for the first time, at least to our knowledge.

Table 1 summarizes the empirical literature discussed so far and highlights samples and main results regarding the testing of Gibrat’s Law from each cited paper.

Table 1 Previous literature: synoptic view
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While the above discussion is specifically devoted to Gibrat’s Law, it is also worth briefly referring to the related vast amount of literature focusing on the so called High-Growth-Firms (HGFs, often referred to as "gazelles"). This literature is somewhat tangential but relevant to our current topic, since the role of size is often taken into account, albeit not being its main topic of analysis. As summarised by Coad et al. (2014), researchers in this field have explored whether factors such as innovation and industry affiliation affect a company's likelihood of becoming a HGF, coming to the conclusion that innovation (Audretsch et al., 2014; Cefis & Marsili, 2006; Colombelli et al., 2013; Goel & Nelson, 2022; Zhang & Mohnen, 2022) and sector-specificity (Acs, 2011) can enhance both a firm's survival prospects and its rate of growth.Footnote 5 However, even within this strand of literature, there remains a lack of consensus on how a firm's size influences growth dynamics: while most studies tend to confirm a negative relationship between size and growth, some are more nuanced. For instance, Coad et al. (2014) highlight the stylised fact that HGFs tend to be young, but not necessarily small. Additionally, Delmar et al. (2003) show that when growth is measured in absolute terms, large firms tend to grow more, and Acs (2011) demonstrates that there is a subset of high-growth firms which are large.Footnote 6

As we will discuss later in the empirical section, we engage with this literature by incorporating controls for innovativeness, focusing on a single cohort of firms to account for the impact of firm age, and introducing sectoral dummies.

3 Data, hypotheses and preliminary econometric specification

The analysis is based on AIDA-BvD data, which contains comprehensive information on all the Italian firms required to file accounts. Specifically, we acquired a dataset comprising the entire populationFootnote 7 of 1,720 new (founded in 2009) Italian manufacturing firms with at least one employee, tracked for 11 years, i.e. until 2020. We selected the following variables of interest: Employees (E); Regional belonging (dummies corresponding to the NUTS-2 classification, R); Sectoral belonging (dummies corresponding to the 2 digits NACE classification, S); Innovativeness (a dummy I that indicates whether a firm is registered as “innovative” according to the Italian decree “innovative firms act 221/2012”, see Guerzoni et al., 2021)Footnote 8; Profitability, computed as the ratio between “earnings before taxes” and “revenues from sales and services” (P).

By utilising data on Italian manufacturing firms newly established in 2009, the validity of Gibrat's Law was examined over the entire period of 2010–2020, as well as year-by-year. Through this set of analyses, we aimed to test the following hypotheses jointly, with the objective of reconciling if possible the diverging evidence discussed in Section 2:

  • (H1) Gibrat's Law is rejected over the entire period (a priori hypothesis);

  • (H2) A convergence towards a Gibrat-like steady state emerges among the population of surviving firms (a posteriori hypothesis).

If both these hypotheses were to be confirmed, this would mitigate the apparently controversial debate surrounding the validity of Gibrat's Law (see Section 2). While the Law may be rejected when examining the overall evolution of a given ex-ante population of companies (a priori hypothesis), it may still accurately describe the patterns of growth for well-established firms within the ex-post sub-population that results from market selection and learning processes (a posteriori hypothesis).

The specification used to test Gibrat’s Law econometrically is the following (the same adopted by Evans, 1987a, b, Lotti et al., 2009):

$${G}_{i,t} = {\beta }_{0}+ {\beta }_{1}log\left({E}_{i,t-1}\right)+{{\beta }_{1}R}_{i,t}+{\beta }_{1}{S}_{i,t}+{\beta }_{1}log\left({P}_{i,t-1}\right)+{\beta }_{1}{I}_{i}+{\varepsilon }_{i,t},$$
(1)

where \({G}_{i,t}=({{E}_{i,t}-E}_{i,t-1})/{E}_{i,t-1}\) is the employment growth rate of firm i at time t; \({\beta }_{0}\) is the intercept of the model and serves as a reference point, indicating the expected outcome when no explanatory factors are at play; \({E}_{i,t-1}\) is the number of employees the firm has i at time t-1; regional dummies, sectoral dummies, profitability (P) and innovativeness (I) act as controls.Footnote 9

Chesher (1979) pointed to the coefficient \({\beta }_{1}\) in order to test the validity of Gibrat’s Law through the significance of the relevant parameter. In particular, if \({\beta }_{1}\)= 0, Gibrat’s Law holds; if \({\beta }_{1}\)< 0, smaller firms grow at a higher rate than their larger counterparts, while the opposite is the case if \({\beta }_{1}\)> 0.

We estimated Eq. (1) for each period t on the subsamples of firms still existing at t; moreover, we estimated overall employment growth over the entire investigated period, with growth \({G}_{i,t}=-100\%\) for firms that have exited the market (as in Evans, 1987a, b).

Table 2 presents the summary statistics for each period, showing the progressive reduction in the population as well as the corresponding growth rate and other variables included in the model for each year. It is noteworthy that only 688 companies out of 1,720 survived until the end of the investigated period, pointing to a ten-year survival rate in Italian manufacturing, equal to 40% (this is not surprising and in line with the stylised facts pointed out by Geroski, 1995). In order to survive, newly-founded firms must grow, with an average size increasing from 14.10 employees at the beginning of the period to 21.40 at the end of the period. Profitability is hard to achieve for these young firms, although more likely in the later years, and innovativeness (as defined in this study) is a really exceptional attribute.

Table 2 Descriptive statistics (year by year and overall)
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4 A novel graphical model approach

One of the main contributions of this study is that we made use of unsupervised graphical models (GM) to gain a different and more comprehensive perspective on whether a given population of firms tends to converge to Gibrat-like behaviour over time. In fact, GM allow us to describe jointly the overall structure of dependency between variables and in this way we can capture multiple relationships, including non-linear and conditional dependencies, and both direct and indirect influences. In particular, in our context GM can depict not only whether the proxy for size directly affects growth, but also any other mediating effects.

In more detail, graphical models are a framework combining network representation and probability theory to specify conditional independence relationships between random variables in a given dataset. These relationships are represented through a graphical representation, specifically a graph \(\mathcal{G}\)(V,L), where V is a finite set of nodes corresponding to the variables of interest and L is the set of links in the network, representing the conditional dependence between any pair of variables (Lauritzen, 1996).Footnote 10

One of the central problems in GM representation is the estimation of the underlying probability distributions of the variables from a finite sample. Chow and Liu (1968) proposed an approach for discrete variables that approximates their probability functions via probability distribution of the second-order tree dependence. The connection between nodes of the tree represents the unknown joint probability of the nodes (or associated variables), providing information on their mutual dependence or mutual information. Specifically, Chow and Liu (1968) found that a probability distribution of a tree dependence approximates the true value probability of a set of discrete random variables composing the tree, if and only if the latter has maximum mutual information. Under the assumption that the cell probabilities of discrete random variables factorise according to an unknown tree τ written as \({\mathcal{G}}_{D}=(\Delta ,{L}_{\Delta })\), they can be written as:

$$p(d|\tau )=\frac{\prod_{u,v \in {E}_{\Delta }}p({d}_{u},{d}_{v})}{{\prod }_{v\in \Delta }p({d}_{v}{)}^{{d}_{v} - 1}}$$
(2)

where \({d}_{v}\) is the number of links incident to node v, namely the degree of v. According to Eq. (2), the maximized log-likelihood, up to a constant, turns out to be \(\sum_{(u,v) \in {L}_{\Delta }}{I}_{u,v}\), where \({I}_{u,v}\) is the mutual information between nodes u and v. It is worth noting that the mutual information between two variables is defined as a measure of their closeness (Lewis, 1959), and therefore is a dimensionless, non-negative, and symmetric quantity which measures the reduction of uncertainty about a random variable, given the knowledge of another. Put differently, Chow and Liu’s (1968) algorithm employs the concept of mutual information to quantify the strength of the connections which link the variables depicted in the graphical representation (Riso et al., 2023).

Prior to the development of the Chow-Liu algorithm, various other algorithms were created to determine the probabilistic structure and corresponding maximum-likelihood estimator. Specifically, Kruskal (1956) proposed a simple and efficient solution to this problem by starting with a null graph and adding the edge with the highest weight at each step, as long as it does not form a cycle with previously chosen edges. Edwards et al. (2010) extended the Chow-Liu algorithm to be applied to mixed data sets X, using mutual information between discrete and continuous variables. This algorithm relies on the use of mutual information between a discrete variable, \({D}_{u}\), and a continuous variable, \({C}_{v}\). It is characterised by the marginal model, which turns out to be an ANOVA model (Edwards, 1995). It is worth noting that when dealing with mixed variables, the evaluation of the mutual information \(I\left({d}_{u},{c}_{v}\right)\) between each pair of nodes requires distinguishing between the case when the variance of \({C}_{v}\) is distributed homogeneously across the level of the discrete variable \({D}_{u}\) from the case when it is heterogeneously distributed (Edwards, 1995). As pointed out by Edwards et al. (2010), one of the disadvantages of selecting a tree on the basis of maximum likelihood is that it always includes the maximum number of edges, even if the latter are not supported by data. They therefore suggested the use of one of the following measures to avoid this drawback:

$${I}^{BIC }= I({x}_{i},{x}_{j}) - log(n){k}_{{x}_{i},{x}_{j}} ; {I}^{AIC }= I({x}_{i},{x}_{j}) - 2{k}_{{x}_{i},{x}_{j}}$$
(3)

The degrees of freedom associated with the pair of variables \({x}_{i}\) and \({x}_{j}\) are represented by\({k}_{{x}_{i},{x}_{j}}\), and are determined based on the nature of the variables involved, continuous or discreteFootnote 11. These measures are used in an algorithm proposed by Edwards et al. (2010) to determine the best spanning tree. The algorithm stops when the graph has reached its maximum number of edges. The algorithm can generate either a tree or a forest, where a forest is a group of trees.

To test the validity of Gibrat's Law, we employed the extension of the Chow-Liu algorithm (Chow & Liu, 1968) proposed by Edwards et al. (2010) for mixed datasets. This methodology allowed us to map the conditional dependence relationships of the variables involved in this analysis onto a graph \(\mathcal{G}\) (V,L), where V is a finite set of nodes with direct correspondence to the variables of interest and L is the set of links in the network (Lauritzen, 1996). The links represent the conditional dependence between any pair of variables.Footnote 12 Specifically, the GM employed in this paper belong to the class of multivariate distributions, whose conditional independence properties are encoded in a tree/forest in the following way: the absence of a link between two nodes represents conditional independence between the corresponding variables (Jordan, 2004).

In the context of this study, if there is a direct connection between node G (employment growth rate at time t) and node E (number of employees at time t-1), or the connection is mediated by another node, Gibrat's Law does not hold. Conversely, if node G is not connected with node E, Gibrat's Law holds. In other words, we are examining the dependence between variable G (employment growth rate at time t) and variable E (number of employees at time t-1) using the GM to validate Gibrat's Law over time. In more detail, the GM methodology allowed us to understand how the relationships between the variables involved in the model change over time; in particular, we built both a GM for each year and one to test the overall relationships over the entire period; therefore, we exactly mimic the econometric setting put forward in the previous section.Footnote 13

On the whole, the unsupervised learning approach proposed in this study does not supplant the econometric approach but rather enhances it, enabling us to test Gibrat’s Law without being constrained by the assumptions inherent in econometric modelling.

5 Empirical findings

Table 3 presents the output of the regressions, which can be considered as our preliminary baseline. In the last column, which displays the results over the entire investigated period, the key coefficient log (E) is negative and significant, rejecting Gibrat’s Law and supporting Hypothesis 1.Footnote 14 However, the regressions on single periods tell a different story: the initial size displays a significant and negative impact only in the first seven years, whereas—allowing for market selection—this significance disappears as of 2017 (supporting Hypothesis 2). As far as the controls are concerned, while profitability boosts growth, innovation does not seem to play a significant role.Footnote 15

Table 3 Regression analyses (year by year and overall); dependent variable: employment growth rate
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Figure 1 illustrates the GMs generated using the algorithm introduced by Edwards (1995) and depicts the investigated nodes, with the key variables growth (G) and employment (E) highlighted in orange and blue respectively.

Fig. 1
figure 1

Graphical Models: direct and conditional dependences between variables (year by year and overall). Notes: Each subfigure represents the graphical model for a distinct year, while 'ALL' pertains to the pooled dataset. Nodes represent variables, and connections indicate the absence of statistical independence. G denotes the employment growth rate, E stands for employment, I is innovativeness, S represents sectoral belonging, P denotes profitability, and R stands for the NUTS-2 region

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In Fig. 1, each node represents a variable and the presence of a link indicates a relation of conditional dependences. If two nodes are connected via a third variable, the two variables are in a dependence relation, conditional to the distribution of the third one. For instance, in 2010 growth (G) depends directly on employment (E), while it is independent of the dummy variable for innovative firms (I).

Over the entire period (Fig. 1, last panel: All) growth is associated with the initial size, through the mediation of sectoral belonging. This implies that for each given sector, growth and size are not independent. Consequently, Gibrat's Law is rejected, since size does have an effect on growth, albeit varying across sectorsFootnote 16 (this evidence supports Hypothesis 1).

However, shifting to the annual analyses, our results present a different picture. In the first three periods, growth is directly linked to employment: even independently from sectoral belonging, size does influence the growth rate of firms. In the subsequent four periods, this relationship persists, but is mediated by profitability (and sectoral belonging in 2015). This evidence suggests that Gibrat’s Law is initially rejected but the dependency becomes weaker over time. Finally, as was the case for the regressions, as of 2017 growth and employment were no longer connected, corroborating the interpretation of Gibrat’s Law as a steady state convergence (thus supporting Hypothesis 2).

It is noteworthy that the results from the graphical models not only align with the regression results, but also provide a more intuitive and interpretable representation of the data. Specifically, we were able to observe an initial direct correlation between growth and size, then a temporary mediation effect of profitability, while eventually any type of either direct or indirect dependency fades away as market selection proceeds.

6 Conclusions

As discussed in detail in Section 2, in the extant (and partially controversial) literature the conclusion has been reached that Gibrat’s Law can be rejected when all firms are taken into account, but can be confirmed when the core of larger and older incumbents is isolated.

Differently from most previous studies, in this paper we did not single out different groups of firms (i.e. core vs fringe), instead tracking a brand new population of companies over time, allowing for market selection and so discovering whether a given population of firms tends to converge to Gibrat-like behaviour through time. In so doing, we tested whether Gibrat’s Law, albeit being general refutable, can actually be accepted when market selection generates a sort of “steady state”, where a much more homogeneous population of surviving firms may behave according to the Law (see Section 1).

Using both standard econometrics and a novel unsupervised approach generating graphical models, in this paper it has been shown that the early and current literature testing Gibrat’s Law can be indeed reconciled; in particular, the rejection of Gibrat’s Law ex ante can be coupled with a defence of the Law ex post. More specifically, while we have confirmed rejection of the Law when firms were considered over the entire investigated period, we have shown the opposite when we allowed for market selection and we tracked only the surviving companies. Indeed, the growth behaviour of the re-shaped (smaller) population of surviving most efficient firms was in line with Gibrat’s intuition.

This twofold evidence may have interesting implications in terms of both applied and theoretical research on the one hand, and policy options on the other. In particular, policy makers should take into account the fact that employment growth crucially depends on the combination of different factors characterising industrial dynamics, such as new firm formation, firm size, survival rates and market selection. Indeed, the evidence supporting rejection of Gibrat’s Law has laid the groundwork for policies aimed at bolstering young and small firms; if young and smaller firms exhibit more substantial growth than mature and larger incumbents, thereby contributing to increased employment and added value in the economy, this provides support for interventions in favour of entrepreneurial activities.

However, if this deviation from Gibrat’s Law is primarily driven by the survival dynamics of underperforming firms in their initial years, the policy implications might be different. Indeed, if after this initial period surviving firms stabilise and conform to Gibrat’s Law (as shown in this study), interventions should be more cautious and selective. In particular, the policy focus should not be solely on assisting small firms in general but rather on ensuring that market mechanisms operate efficiently during a company's early stages, facilitating the survival of the most promising ventures. In this context, policies guaranteeing liquidity and survival chances to struggling firms may not be the optimal choice. Conversely, greater emphasis should be placed on institutions and organisations capable of channelling funds to the most promising enterprises, such as venture capital firms or sector-specific industrial policies targeting high-potential sectors.