Industry Dynamics and the Distiribution of Firm Sizes: A Non-Parametric Apporoach

Lotti, Francesca; Santarelli, Enrico Working Paper Industry Dynamics and the Distiribution of Firm Sizes: A Non-Parametric Apporoach Quaderni - Working Paper DSE, No. 406 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lotti, Francesca; Santarelli, Enrico (2001) : Industry Dynamics and the Distiribution of Firm Sizes: A Non-Parametric Apporoach, Quaderni - Working Paper DSE, No. 406, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4895 This Version is available at: https://hdl.handle.net/10419/159247 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ 1 INDUSTRY DYNAMICS AND THE DISTRIBUTION OF FIRM SIZES: A NON-PARAMETRIC APPROACH* by Francesca Lotti St’Anna School of Advanced Studies – Pisa, Italy and Harvard University, Department of Economics – Cambridge, MA Enrico Santarelli University of Bologna, Department of Economics Abstract The aim of this paper is to analyze the evolution of the size distribution of young firms within some selected industries, trying to assess the empirical implications of different models of industry dynamics: the model of passive learning (Jovanovic 1982), the model of active learning (Ericson and Pakes, 1995), and the evolutionary model (Audretsch, 1995). We use a non-parametric technique, the Kernel density estimator, applied to a data set from the Italian National Institute for Social Security (INPS), consisting in 12 cohorts of new manufacturing firms followed for 6 years. Since the patterns of convergence to the limit distribution are different between industries, we conclude that the model of passive learning is consistent with some of them, the active exploration model with others, the evolutionary model with all of them. Keywords: Cohorts; Gibrat’s Law; Kernel; Industry Dynamics; Non-parametric; Shakeouts. JEL Classification: L11, L60 This version: June 28, 2001 Corresponding author: Prof. Enrico Santarelli, Università di Bologna, Dipartimento di Scienze Economiche, Strada Maggiore, 45, I-40125, Bologna - ITALY E-mail: [email protected] * A previous version of this paper was presented at the Econometrics Seminars at UC Riverside (18 May 2001). We wish to thank the audience for valuable comments. Discussions with Giovanni Dosi, Samuel Kortum, Luca Lambertini, Markus Mobius, Ariel Pakes, Jack Porter, Aman Ullah and Marco Vivarelli have been very helpful. Financial support from MURST (“Cofinanziamento 2000”, responsible E. Santarelli) is gratefully acknowledged. 2 1 - Introduction Analysis of the size-growth relationship is a commonly used approach to the study of the evolution of market structure. In fact, the firm size distribution (FSD) has received considerable attention - since the seminal works of Herbert Simon and his co-authors between the late 1950s and the 1970s (cf. Simon and Bonini, 1958; and Ijiri and Simon, 1964, 1977) - in most theoretical and empirical studies dealing with the overall process of industry dynamics. The empirical evidence showed a FSD highly skewed to the right, meaning that the size distribution of firms is lognormal, both at the industry level and in the overall economy. This piece of evidence is coherent with the so-called Law of Proportionate Effect (or Gibrat’s (1931) Law): as Simon and Bonini (1958) point out, 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”. Recent evidence based on more complete data sets, suggests that Gibrat’s Law is not confirmed, either for new-born or established firms (for a survey, cf. Geroski, 1995; Lotti et al., 1999), since smaller firms grow more than proportionally with respect to larger ones. This finding should be consistent with a departure of the FSD from the lognormal distribution. In this paper - using quarterly data for 12 cohorts of new manufacturing firms - we account for the evolution of the FSD over time in the case of young firms. Moreover, we try to assess the empirical implications of different models of industry dynamics. The work is organized as follows. Section 2 contains a review of the empirical evidence about Gibrat’s Law and the FSD, as well as an overview of some recent models of industry dynamics. Section 3 describes the data and the methodology used, whereas Section 4 summarizes the main empirical findings. Finally, Section 5 contains some concluding remarks. 2 - Theory or Stylized Facts? Gibrat’s Law, applied to the analysis of market structure, represents the first attempt to explain in stochastic terms the systematically skewed pattern of the size distribution of firms within an industry (Sutton, 1997). In effect, the Law cannot be rejected if a) firm growth follows a random process and is independent from initial size, and b) the resulting distributions of firms’ size are lognormal1. Although, from a theoretical viewpoint, labeled as “unrealistic” since Kalecki’s (1945) study on the size distribution of factories in US manufacturing, this result was initially consistent with some empirical studies dealing with incumbent, large firms (Hart and Prais, 1956; Simon and Bonini, 1958; Hymer and Pashigian, 1962). In recent years, most studies have instead shown that these exhibit a different behavior, identifying an overall negative relationship between initial size and post-entry rate of growth (cf., among others, Mata, 1994; Hart and Oulton, 1999). However, Lotti et al. (2001) found that, in the case of new-born firms, the growth rates are negatively correlated with their initial size only during their infancy: Gibrat’s Law fails to hold in the years immediately following start-up, when smaller firms have to rush in 1 Of course, a FSD skewed to the right implies only that Gibrat’s Law cannot be rejected. However, one cannot a priori exclude that the skewness is the result of turbulence, namely of the presence of new-born small firm in the right tail of the distribution. 3 order to reach a size large enough to enhance their likelihood of survival; but in the subsequent years, the patterns of growth of entrants do not differ significantly from the landscape of the industry as a whole. One possible way to explain this phenomenon of self-selection, is to consider the firms’ learning and evolution processes put forward by Jovanovic (1982), Ericson and Pakes (1995), and Audretsch (1995). By following such perspectives, entrants are uncertain about their relative level of efficiency, and only once into the market they learn about their possibilities of survival and growth. The main advantage of these models is that they allow for a) heterogeneity among firms, b) idiosyncratic sources of uncertainty and discrete possible events, c) entry and exit. Boyan Jovanovic’s model of passive learning hypothesizes that firms are initially endowed with uncertain, time-invariant characteristics (i.e. efficiency parameters), of which the firm does not know the distribution. But, once into the market, the firm learns passively about the true efficiency parameter. As a consequence, in every period the firm has to decide its strategy: whether to exit, continue with the same size, grow in size, or reduce its productive capacity. One of the consequences of this model is that, due to a particular kind of selection process, the most efficient firms survive and grow, while the others are bound to shrink or to exit from the market. Like in the passive learning model, Richard Ericson and Ariel Pakes’s model of active learning (1995) assumes that all the decisions taken by the firms are meant to maximize the expected discounted value of the future net cash, conditional on the current information set. Unlike in Jovanovic’s model, a firm knows its own characteristics and its competitors’ ones, along with the future distribution of industry structure, conditional on the current structure. Accordingly, this model can be usefully employed in explanation of ‘entry mistakes’ (as defined by Cabral, 1997), namely the fact that in every period and every industry more firms enter than the market can sustain. Within an active learning perspective, such mistakes occur due to lags in observation of rivals’ entry decision or just because entry investments take time (Cabral, 1997). In a subsequent work, Pakes and Ericson (1998), using two cohorts of firms from Wisconsin, belonging to the retail and the manufacturing industries, found that the structure of the former industry is compatible with the passive learning model, while that of the latter with their model of active exploration (learning). The retail cohort, after eight years seems to have reached the size distribution of the industry as a whole, while the manufacturing one, even if showing higher growth rates, after that period is still far from the limit distribution. David Audretsch (1995) expanded the passive learning approach put forward by Jovanovic (1982) into an evolutionary perspective, allowing for inter-industry differences in the likelihood of survival of newborn firms. Accordingly, industry-specific characteristics, such as scale economies and the endowment of innovative capabilities, exert a significant impact on entry, exit, and the likelihood of survival of newborn firms. For example, in industries characterized by higher minimum efficient scale (MES) levels of output, smaller firms face higher costs that are likely to push them out of the market within a short period after start-up. Thus, only the most efficient among newborn firms will survive and grow, whereas the other are pushed out of the market (cf. Audretsch et al., 1999). In this case, the presence of more potential entrants than firms with a significant likelihood to survive in the long run can bring about a shakeout (cf. Klepper and Miller, 1995). In turn, a shakeout occurring at a certain point in the 4 industry’s history is likely to affect the long-run size distribution of firms within the same industry, depending on “how the opportunities vacated by exited firms are reallocated among surviving firms” (Sutton, 1998, p. 260; cf. also Klepper and Graddy, 1990). Conversely, in industries with a lower MES level of output the likelihood of survival will be independent of the firms’ ability to grow (cf. Amaral et al., 1977; Brock, 1999). With this theoretical and empirical background in mind, we look at the evolution of 12 cohorts of newborn firms in selected industries, in order to analyze the process of convergence of the firm size distribution, in terms of number of employees, with respect to the overall industry landscape. The aim of this analysis is to show i) whether the findings by Herbert Simon and his co-authors concerning the Skewness to the right of the FSD are confirmed also in the case of newborn, small firms and ii) whether the FSD resulting from application of the Kernel density estimator is consistent with models of industry dynamics - such as those surveyed above - which identify in the learning processes occurring at the firm level, and in the level of sunk costs that characterizes each industry, some possible theoretical explanations for these facts. 3 - Data and Methodology We use a data set from the Italian National Institute for Social Security (INPS), dealing with 12 cohorts of new manufacturing firms (with at least one paid employee) born in each month of 1987, and their follow up until December 1992. Since all private Italian firms are compelled to pay national security contributions for their employees to INPS, the registration of a new firm as “active” signals an entry into the market, while the cancellation of a firm denotes an exit (this happens when a firm finally stops paying national security contributions). For administrative reasons - delays in payment, for instance, or uncertainty about the current status of the firm - some firms are classified as “suspended”. In the present work we consider these suspended firms as exiting from the market at the moment of their transition from the status of “active” to that of “suspended”, while firms which have stopped their activity only temporarily were included again in the sample once they turned back active. We carry out also an accurate cleaning procedure, aimed at identifying internal inconsistencies and entry or exit due to firm transfers and acquisitions. As regards acquisitions, these are denoted as “extraordinary variations” in the INPS database, and firms involved in such activities can therefore be easily identified and cancelled from the database itself. A correct identification of firms disappeared via acquisitions permitted to avoid acquiring firms to be drawn disproportionately from the low end of the size distribution. As pointed out by Sutton (1998; cf. also Hart and Prais, 1956; Hymer and Pashigian, 1962) this would have caused a violation in the proposed bound and altered the significance of the overall analysis. We focus our analysis on four industries - Electrical & Electronic Engineering, Instruments, Food, and Footwear & Clothing - mainly for two main reasons: the first one concerns their very different market structure in terms of cost of entry (sunk costs), and the second the fact that the latter two industries are less technologically progressive than the former two ones2. 2 And this would allow to draw some conclusions on whether the FSD is or is not sensitive to technological factors. 5 To examine the effect of firms’ age on the distribution of their sizes, we study each cohort at each quarter after start-up, and this for their first six years in the market. In Tables 1A-1D and in Table 2 some descriptive statistics are reported. In general, all industries experience a shakeout period during which the number of survivors, among new entrants, declines by 40 per cent or more. From Tables 1A, B, C, and D it turns out that, on average, the survival rate at the end of the period (i.e., after 21 quarters) is much higher within the cohorts belonging to the Electrical & Electronic Engineering and the Instruments industries, than it is the case with the Food and the Footwear & Clothing industries. Thus, consistently with Audretsch’s (1995) hypothesis, industry specific characteristics, such as the commitment to innovative activities, seem to set in motion a pre-entry selection mechanism that selects only those start-ups that find in their endowment of innovative capabilities a possible competitive advantage. Looking at Table 2 Figure 1, one immediately observes that - with the sole exception of the Food industry - the standard deviation of firm sizes is much higher at the end of the relevant period than in the first quarter. Dispersion of firm sizes tends therefore to widen as surviving firms reach the MES level of output and specialize in one of the many clusters of products which - according to John Sutton's (1998, pp. 597-605) "independent submarkets" hypothesis - characterize each industry. In turn, firm size increases along with its age for the Electrical & Electronic Engineering and the Instruments industries, but only for the first 13 and 12 quarters respectively, corresponding with a period comprised approximately between December 1989 and January 19913. Afterwards, a decline in average firm size emerges, which is consistent with views of recessions (the period between 1991 and 1993 has been characterized in Italy by a significant slowdown in the GDP growth rates) as times of “cleansing” (cf. Boeri and Bellmann, 1995). In fact, the sectoral data reported in Table 3 show for both industries a significant decrease in the growth rates of value added since 1989, with a trough in 1993. This pro-cyclical pattern of the average firm size is even more marked in the Footwear & Clothing industry, in which the average size starts to decline after the eight quarter in the market (as early as 1989, that is the initial year of the recession). The data on the Footwear & Clothing industry show a substantial stability of the average firm size over time. This result is to some extent consistent with the dynamics of value added in the same industry: Table 3 points out alternate peaks and troughs in the Footwear & Clothing industry growth rates that are unlikely to affect firm size, since this needs time to adjust its patterns to variations in value added. In a recent paper by Machado and Mata (2000) the Box-Cox quantile regression method is used to estimate the distribution of firm sizes and, accordingly, to analyze industry dynamics in Portugal. This approach consists in modeling each quantile as a function of a number of industry characteristics that are expected to affect firm size. Since our database doesn’t provide any details about industry characteristics, in the present study we use instead a non-parametric approach. The basic idea is to look if, with the passing of time, the empirical distribution of firm sizes converges towards a lognormal distribution, under the hypothesis that this represents the limit distribution. To characterize the distribution, we used the Kernel density estimator (Pagan and Ullah, 1999), which can be summarized as follows. 3 In effect, since the 12 cohorts include firms born in each month of 1987, each column in Table 2 deals with all firms and all cohorts. 6 Table 1A - Number of firms active at the end of each quarter – Electrical & Electronic Engineering Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Cohort 1 128 125 121 120 117 113 112 109 108 107 106 105 104 103 102 102 97 95 93 92 90 Cohort 2 64 61 59 56 53 51 52 51 50 50 50 50 49 47 44 43 40 38 36 37 38 Cohort 3 72 68 65 62 60 61 61 61 57 55 53 53 53 51 51 48 48 48 48 47 43 Cohort 4 49 46 47 47 47 47 45 43 43 43 42 41 40 41 41 39 38 34 33 33 33 Cohort 5 59 53 53 52 53 50 50 47 46 48 46 44 44 43 41 40 37 37 35 34 34 Cohort 6 71 68 65 64 62 62 63 59 58 55 49 49 49 48 47 45 44 42 41 37 36 Cohort 7 41 41 41 41 39 38 38 37 37 36 34 30 30 29 28 27 27 27 25 24 23 Cohort 8 18 18 18 17 17 17 17 17 16 15 15 15 15 15 14 14 14 14 14 12 12 Cohort 9 72 67 63 63 64 62 60 58 58 57 57 57 55 56 52 52 53 52 50 50 49 Cohort 10 60 58 54 50 49 50 52 49 47 47 44 44 44 41 42 42 42 42 40 39 38 Cohort 11 57 53 55 53 53 51 51 51 50 48 46 46 43 42 40 41 39 38 39 39 39 Cohort 12 29 28 26 25 25 25 25 26 25 25 24 23 23 23 23 22 22 22 21 20 19 Total 720 686 667 650 639 627 626 608 595 586 566 557 549 539 525 515 501 489 475 464 454 Table 1B - Number of firms active at the end of each quarter – Instruments Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Cohort 1 62 61 60 60 59 56 56 56 55 53 51 51 50 50 48 46 43 41 40 42 40 Cohort 2 38 37 35 36 35 35 34 34 34 33 32 29 28 27 27 27 26 24 24 25 25 Cohort 3 34 32 33 33 31 31 30 30 28 27 27 26 24 23 22 21 19 20 20 20 20 Cohort 4 26 26 25 24 23 23 20 19 19 18 18 17 17 17 17 16 17 17 17 17 17 Cohort 5 20 20 20 19 19 19 19 19 18 19 18 17 17 15 14 14 14 14 14 13 13 Cohort 6 33 33 32 31 28 28 28 27 27 25 24 23 21 21 21 21 21 21 21 17 19 Cohort 7 35 34 30 30 30 28 27 25 25 25 24 25 25 24 23 23 22 21 21 22 22 Cohort 8 11 11 10 10 10 10 10 10 10 10 10 10 10 10 10 10 87776 Cohort 9 27 27 25 24 24 23 23 23 23 22 22 22 21 20 20 20 19 20 18 18 18 Cohort 10 32 30 28 26 26 27 25 24 23 24 22 21 21 20 19 18 18 18 18 17 17 Cohort 11 26 25 25 24 24 22 22 19 19 19 18 17 17 17 17 17 16 16 15 15 15 Cohort 12 18 18 17 16 15 14 14 14 14 14 14 13 13 12 11 11 11 11 11 11 10 Total 362 354 340 333 324 316 308 300 295 289 280 271 264 256 249 244 234 230 226 224 222 7 Table 1C - Number of firms active at the end of each quarter – Food Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Cohort 1 93 88 88 83 78 76 73 72 70 70 68 67 65 63 61 59 58 56 57 55 54 Cohort 2 47 43 40 37 34 34 33 33 29 28 28 27 24 24 24 24 22 23 23 21 21 Cohort 3 46 43 42 39 40 37 37 34 34 33 30 27 26 27 25 21 21 23 23 19 19 Cohort 4 40 35 30 29 30 29 29 29 28 28 29 27 26 25 23 19 19 20 20 19 19 Cohort 5 41 38 35 33 34 35 34 32 29 28 27 27 25 24 23 22 22 21 21 21 19 Cohort 6 44 42 37 35 32 29 29 29 28 28 25 25 25 25 25 25 24 24 24 24 22 Cohort 7 46 35 35 34 38 35 33 33 35 30 30 27 25 24 24 23 22 21 22 22 21 Cohort 8 20 16 15 15 14 13 12 89888888897777 Cohort 9 30 27 22 19 20 19 18 17 18 19 17 18 16 17 15 15 14 15 14 13 13 Cohort 10 51 40 34 32 32 30 30 26 29 26 23 24 26 21 19 18 23 19 18 16 19 Cohort 11 110 65 53 47 72 49 42 40 67 40 32 31 40 33 31 30 57 38 30 28 43 Cohort 12 80 42 23 23 47 29 21 18 49 19 12 12 22 10 10 937 25 12 11 27 Total 684 514 454 426 471 415 391 371 425 357 329 320 328 301 288 273 328 292 271 256 284 Table 1D - Number of firms active at the end of each quarter – Footwear & Clothing Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Cohort 1 164 159 158 156 145 143 136 132 129 126 121 120 113 112 110 110 103 100 98 95 93 Cohort 2 92 89 84 80 74 69 68 67 61 55 55 55 53 50 46 46 43 42 40 37 35 Cohort 3 85 79 76 73 71 65 62 60 59 56 51 50 48 45 45 41 40 40 38 38 37 Cohort 4 97 91 83 77 72 70 69 64 64 62 58 51 51 45 40 40 37 36 35 34 34 Cohort 5 100 93 86 83 83 79 78 74 74 70 68 66 67 65 59 55 55 48 40 49 45 Cohort 6 89 87 81 77 74 72 72 70 69 64 63 59 58 53 51 50 49 45 44 43 41 Cohort 7 88 80 73 69 69 65 63 60 57 55 54 55 53 52 48 44 43 43 42 41 41 Cohort 8 36 28 24 26 25 23 22 23 22 21 19 18 17 16 15 13 13 13 13 12 12 Cohort 9 97 95 87 84 78 75 70 68 67 63 65 63 60 59 57 56 55 55 52 51 49 Cohort 10 104 99 88 81 78 75 78 71 66 62 61 62 61 56 56 55 54 52 46 46 43 Cohort 11 96 93 86 78 75 68 63 61 61 57 54 51 49 47 43 41 40 40 38 37 34 Cohort 12 51 46 43 41 39 35 34 34 35 31 29 27 28 28 27 26 26 26 26 24 20 Total 1099 1039 969 925 883 839 815 784 764 722 698 677 658 628 597 577 558 540 522 506 484 8 Table 2 – Average Size and Standard Deviation, each quarter, all industries. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Electrical & Electronic Eng. Average Size 4.61 6.33 7.23 7.77 8.24 8.78 9.11 9.23 9.48 9.69 9.98 9.91 10.51 10.42 10.53 10.34 9.81 9.84 9.73 9.67 9.66 Standard Deviation 9.01 10.89 12.45 13.24 14.15 15.7 16.06 16.02 16.25 16.87 17.56 18.47 28.27 31.86 31.53 31.06 28.88 29.92 28.52 30.23 29.03 Instruments Average Size 3.37 4.66 6.02 7.31 7.9 8.2 8.63 9.14 9.36 9.37 9.43 9.72 9.59 7.91 8.01 8.15 8.05 7.97 8.07 9.68 9.85 Standard Deviation 7.77 11.03 15.77 20.97 25.21 25.98 27.29 29.02 29.39 29.83 29.67 30.47 29.97 17.79 17.72 17.85 17.62 17.47 18.05 36.59 37.3 Food Average Size 4.15 4.39 4.44 4.43 4.66 4.65 4.49 4.46 4.87 4.6 4.53 4.43 4.59 4.38 4.31 4.22 4.52 4.28 4.21 4.06 4.16 Standard Deviation 8.28 8.51 10.16 10.15 9.72 10.04 9.4 9.4 9.74 9.63 10.29 10.45 10.83 10.65 11.04 11.18 11.77 11.33 11.47 11.43 11.45 Footwear & Clothing Average Size 6.31 8.67 9.36 9.76 9.78 9.81 9.76 9.88 9.81 9.64 9.39 9.28 9.16 9.14 8.91 8.76 8.68 8.43 8.08 7.74 7.17 Standard Deviation 10.26 13.95 14.85 16.03 16.29 16.98 17.34 17.7 17.7 18.02 17.84 17.83 17.81 18.32 15.57 18.62 18.87 18.92 18.59 18.32 17.5 15 Table B.1 – following d13 Kernel Density Estimate LN_Q13 -.236693 6.63529 .000024 .409879 13th Q: Pr(Sk) = 0.012 Pr(Kur) = 0.155 Pr(χ2) = 0.0180 d14 Kernel Density Estimate LN_Q14 -.237653 6.85906 .000012 .406529 14th Q: Pr(Sk) = 0.026 Pr(Kur) = 0.150 Pr(χ2) = 0.0320 d15 Kernel Density Estimate LN_Q15 -.245488 6.83853 .000017 .389453 15th Q: Pr(Sk) = 0.030 Pr(Kur) = 0.173 Pr(χ2) = 0.0388 d16 Kernel Density Estimate LN_Q16 -.253139 6.83516 .000017 .390652 16th Q: Pr(Sk) = 0.070 Pr(Kur) = 0.243 Pr(χ2) = 0.0980 d17 Kernel Density Estimate LN_Q17 -.27752 6.79075 .000017 .376148 17th Q: Pr(Sk) = 0.085 Pr(Kur) = 0.588 Pr(χ2) = 0.1909 d18 Kernel Density Estimate LN_Q18 -.255681 6.81812 .000014 .377866 18th Q: Pr(Sk) = 0.130 Pr(Kur) = 0.405 Pr(χ2) = 0.2249 d19 Kernel Density Estimate LN_Q19 -.269731 6.76801 .000021 .396547 19th Q: Pr(Sk) = 0.253 Pr(Kur) = 0.500 Pr(χ2) = 0.4131 d20 Kernel Density Estimate LN_Q20 -.282974 6.85386 .000021 .382259 20th Q: Pr(Sk) = 0.502 Pr(Kur) = 0.434 Pr(χ2) = 0.5859 d21 Kernel Density Estimate LN_Q21 -.278261 6.78105 .000032 .400216 21st Q: Pr(Sk) = 0.333 Pr(Kur) = 0.488 Pr(χ2) = 0.4904 16 Table B.2 – Kernel Density Estimation, log(size), quarterly, Instruments. (continuous line is the Normal Distribution fitted into the data). Test of Normality below. d1 Kernel Density Estimate LN_Q1 -.225597 4.62005 9.2e-06 1.03435 1st Q: Pr(Sk) = 0.000 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d2 Kernel Density Estimate LN_Q2 -.271556 5.19154 .00002 .624652 2nd Q: Pr(Sk) = 0.000 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d3 Kernel Density Estimate LN_Q3 -.293507 5.37491 .000074 .469364 3rd Q: Pr(Sk) = 0.000 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d4 Kernel Density Estimate LN_Q4 -.312026 5.90301 .000043 .397631 4th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.003 Pr(χ2) = 0.0000 d5 Kernel Density Estimate LN_Q5 -.263021 6.18191 .000018 .368495 5th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.001 Pr(χ2) = 0.0000 d6 Kernel Density Estimate LN_Q6 -.264339 6.19923 .00003 .35631 6th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.003 Pr(χ2) = 0.0000 d7 Kernel Density Estimate LN_Q7 -.294019 6.29047 .000042 .346034 7th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.016 Pr(χ2) = 0.0000 d8 Kernel Density Estimate LN_Q8 -.320683 6.38447 .000049 .352399 8th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.024 Pr(χ2) = 0.0000 d9 Kernel Density Estimate LN_Q9 -.321763 6.41307 .000062 .342984 9th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.052 Pr(χ2) = 0.0000 d10 Kernel Density Estimate LN_Q10 -.323088 6.43456 .000063 .336379 10th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.053 Pr(χ2) = 0.0000 d11 Kernel Density Estimate LN_Q11 -.325139 6.43216 .000065 .342386 11th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.055 Pr(χ2) = 0.0001 d12 Kernel Density Estimate LN_Q12 -.345932 6.44849 .000088 .350925 12th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.060 Pr(χ2) = 0.0001 17 Table B.2 – following d13 Kernel Density Estimate LN_Q13 -.284193 6.36641 .000127 .34897 13th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.085 Pr(χ2) = 0.0005 d14 Kernel Density Estimate LN_Q14 -.337478 5.28624 .002275 .350612 14th Q: Pr(Sk) = 0.004 Pr(Kur) = 0.883 Pr(χ2) = 0.0219 d15 Kernel Density Estimate LN_Q15 -.287537 5.19281 .003366 .351572 15th Q: Pr(Sk) = 0.011 Pr(Kur) = 0.996 Pr(χ2) = 0.0432 d16 Kernel Density Estimate LN_Q16 -.298374 5.20365 .003813 .352236 16th Q: Pr(Sk) = 0.033 Pr(Kur) = 0.884 Pr(χ2) = 0.1009 d17 Kernel Density Estimate LN_Q17 -.31063 5.20098 .004129 .352415 17th Q: Pr(Sk) = 0.044 Pr(Kur) = 0.747 Pr(χ2) = 0.1218 d18 Kernel Density Estimate LN_Q18 -.329987 5.21279 .004 .354385 18th Q: Pr(Sk) = 0.033 Pr(Kur) = 0.766 Pr(χ2) = 0.0961 d19 Kernel Density Estimate LN_Q19 -.330859 5.2937 .003897 .34733 19th Q: Pr(Sk) = 0.060 Pr(Kur) = 0.628 Pr(χ2) = 0.1486 d20 Kernel Density Estimate LN_Q20 -.331447 6.73502 .000071 .352541 20th Q: Pr(Sk) = 0.003 Pr(Kur) = 0.126 Pr(χ2) = 0.0066 d21 Kernel Density Estimate LN_Q21 -.350095 6.75697 .000091 .358689 21st Q: Pr(Sk) = 0.005 Pr(Kur) = 0.162 Pr(χ2) = 0.0118 18 Table B.3 – Kernel Density Estimation, log(size), quarterly, Food. (continuous line is the Normal Distribution fitted into the data). Test of Normality below. d1 Kernel Density Estimate LN_Q1 -.200798 4.7334 .00007 .951447 1st Q: Pr(Sk) = 0.000 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d2 Kernel Density Estimate LN_Q2 -.266489 4.79909 .000539 .504765 2nd Q: Pr(Sk) = 0.000 Pr(Kur) = 0.792 Pr(χ2) = 0.0000 d3 Kernel Density Estimate LN_Q3 -.28369 5.27412 .000226 .445048 3rd Q: Pr(Sk) = 0.000 Pr(Kur) = 0.788 Pr(χ2) = 0.0000 d4 Kernel Density Estimate LN_Q4 -.290648 5.2033 .000413 .405044 4th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.487 Pr(χ2) = 0.0000 d5 Kernel Density Estimate LN_Q5 -.279174 5.1312 .00038 .412036 5th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.671 Pr(χ2) = 0.0000 d6 Kernel Density Estimate LN_Q6 -.293577 5.19142 .000639 .366087 6th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.063 Pr(χ2) = 0.0001 d7 Kernel Density Estimate LN_Q7 -.296662 4.83996 .002156 .366913 7th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.010 Pr(χ2) = 0.0001 d8 Kernel Density Estimate LN_Q8 -.301685 4.65839 .004127 .364313 8th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.009 Pr(χ2) = 0.0001 d9 Kernel Density Estimate LN_Q9 -.275682 4.71833 .002945 .365732 9th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.029 Pr(χ2) = 0.0000 d10 Kernel Density Estimate LN_Q10 -.303968 4.72281 .004462 .364385 10th Q: Pr(Sk) = 0.002 Pr(Kur) = 0.002 Pr(χ2) = 0.0003 d11 Kernel Density Estimate LN_Q11 -.315561 5.00691 .002776 .356962 11th Q: Pr(Sk) = 0.003 Pr(Kur) = 0.002 Pr(χ2) = 0.0004 d12 Kernel Density Estimate LN_Q12 -.326039 5.09672 .002499 .347349 12th Q: Pr(Sk) = 0.003 Pr(Kur) = 0.001 Pr(χ2) = 0.0002 19 Table B.3 – following d13 Kernel Density Estimate LN_Q13 -.330384 5.10107 .002784 .341229 13th Q: Pr(Sk) = 0.002 Pr(Kur) = 0.049 Pr(χ2) = 0.0001 d14 Kernel Density Estimate LN_Q14 -.330597 5.02194 .003608 .346798 14th Q: Pr(Sk) = 0.008 Pr(Kur) = 0.002 Pr(χ2) = 0.0008 d15 Kernel Density Estimate LN_Q15 -.338105 5.02024 .004093 .342039 15th Q: Pr(Sk) = 0.008 Pr(Kur) = 0.004 Pr(χ2) = 0.0014 d16 Kernel Density Estimate LN_Q16 -.345728 5.02786 .004519 .337975 16th Q: Pr(Sk) = 0.008 Pr(Kur) = 0.004 Pr(χ2) = 0.0013 d17 Kernel Density Estimate LN_Q17 -.302677 5.01221 .002813 .342929 17th Q: Pr(Sk) = 0.000 Pr(Kur) = 0.218 Pr(χ2) = 0.0002 d18 Kernel Density Estimate LN_Q18 -.334414 5.04394 .003336 .345207 18th Q: Pr(Sk) = 0.002 Pr(Kur) = 0.042 Pr(χ2) = 0.0023 d19 Kernel Density Estimate LN_Q19 -.346665 5.11735 .003687 .33799 19th Q: Pr(Sk) = 0.008 Pr(Kur) = 0.011 Pr(χ2) = 0.0026 d20 Kernel Density Estimate LN_Q20 -.358946 5.14644 .003969 .330228 20th Q: Pr(Sk) = 0.006 Pr(Kur) = 0.003 Pr(χ2) = 0.0011 d21 Kernel Density Estimate LN_Q21 -.34377 5.13956 .002716 .337629 21st Q: Pr(Sk) = 0.000 Pr(Kur) = 0.071 Pr(χ2) = 0.0010 20 Table B.4 – Kernel Density Estimation, log(size), quarterly, Footwear & Clothing. (continuous line is the Normal Distribution fitted into the data). Test of Normality below. d1 Kernel Density Estimate LN_Q1 -.247807 4.78041 .001651 .581436 1st Q: Pr(Sk) = 0.000 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d2 Kernel Density Estimate LN_Q2 -.258221 5.75539 .000408 .346372 2nd Q: Pr(Sk) = 0.000 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d3 Kernel Density Estimate LN_Q3 -.265542 5.82237 .000661 .3417 3rd Q: Pr(Sk) = 0.071 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d4 Kernel Density Estimate LN_Q4 -.26554 5.85279 .000752 .344707 4th Q: Pr(Sk) = 0.369 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d5 Kernel Density Estimate LN_Q5 -.267594 5.82828 .000932 .345516 5th Q: Pr(Sk) = 0.565 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d6 Kernel Density Estimate LN_Q6 -.271028 5.91294 .00088 .344462 6th Q: Pr(Sk) = 0.950 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d7 Kernel Density Estimate LN_Q7 -.273136 5.90076 .00101 .343606 7th Q: Pr(Sk) = 0.807 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d8 Kernel Density Estimate LN_Q8 -.274043 5.8462 .001342 .345438 8th Q: Pr(Sk) = 0.641 Pr(Kur) = 0.049 Pr(χ2) = 0.0000 d9 Kernel Density Estimate LN_Q9 -.277699 5.84604 .001456 .342595 9th Q: Pr(Sk) = 0.535 Pr(Kur) = 0.000 Pr(χ2) = 0.0000 d10 Kernel Density Estimate LN_Q10 -.275 5.86971 .001359 .349868 10th Q: Pr(Sk) = 0.590 Pr(Kur) = 0.000 Pr(χ2) = 0.0006 d11 Kernel Density Estimate LN_Q11 -.275927 5.86691 .001376 .351274 11th Q: Pr(Sk) = 0.559 Pr(Kur) = 0.000 Pr(χ2) = 0.0007 d12 Kernel Density Estimate LN_Q12 -.282834 5.80829 .001713 .344931 12th Q: Pr(Sk) = 0.331 Pr(Kur) = 0.000 Pr(χ2) = 0.0003 21 Table B.4 – following d13 Kernel Density Estimate LN_Q13 -.284064 5.80553 .001791 .345228 13th Q: Pr(Sk) = 0.197 Pr(Kur) = 0.000 Pr(χ2) = 0.0004 d14 Kernel Density Estimate LN_Q14 -.287061 5.85541 .001731 .344743 14th Q: Pr(Sk) = 0.112 Pr(Kur) = 0.001 Pr(χ2) = 0.0028 d15 Kernel Density Estimate LN_Q15 -.29608 5.86823 .001788 .337686 15th Q: Pr(Sk) = 0.159 Pr(Kur) = 0.000 Pr(χ2) = 0.0010 d16 Kernel Density Estimate LN_Q16 -.295355 5.85604 .001905 .341262 16th Q: Pr(Sk) = 0.274 Pr(Kur) = 0.000 Pr(χ2) = 0.0013 d17 Kernel Density Estimate LN_Q17 -.296711 5.86123 .001942 .342317 17th Q: Pr(Sk) = 0.335 Pr(Kur) = 0.001 Pr(χ2) = 0.0029 d18 Kernel Density Estimate LN_Q18 -.300745 5.84592 .002268 .339717 18th Q: Pr(Sk) = 0.287 Pr(Kur) = 0.001 Pr(χ2) = 0.0028 d19 Kernel Density Estimate LN_Q19 -.307011 5.8404 .002347 .335186 19th Q: Pr(Sk) = 0.316 Pr(Kur) = 0.000 Pr(χ2) = 0.0008 d20 Kernel Density Estimate LN_Q20 -.31487 5.83633 .002457 .32893 20th Q: Pr(Sk) = 0.223 Pr(Kur) = 0.000 Pr(χ2) = 0.0004 d21 Kernel Density Estimate LN_Q21 -.321095 5.76351 .002884 .325365 21st Q: Pr(Sk) = 0.329 Pr(Kur) = 0.000 Pr(χ2) = 0.0001 22 Appendix C We started the analysis of the moments of the FSD in the various industries by standardizing the distributions, in order to obtain a more reliable comparison with the normal standard distribution. Subsequently, for the standardized distributions and for every industry, we computed, quarter by quarter, the Skewness Index and the Kurtosis Index. The Skewness Index, as a measure of asymmetry (or, more precisely, of the lack of symmetry), was computed as: where s is the standard deviation. Since the Skewness for a normal distribution is zero - whereas it gets negative values for a distribution skewed to the left, and positive values for one skewed to the right - we expect our sequence of Skewness indexes to converge to zero. Figure C.1 – Skewness Index, by quarter and by industry. Looking at Figure C.1, one can note that for three out of four industries (the only exception being the Footwear & Clothing one) the FSD tends to become more symmetric over time, with different patterns of convergence. But even after 21 quarters, the FSD in the Electrical & Electronic Engineering, the Instruments and the Food industries is still skewed to the right, while in the ( ) 3 n 1i 3 i s xx Skewness ∑ = − = -1 -0.5 0 0.5 1 1.5 2 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Electrical & Electronic Eng. Instruments Food Footwear & Clothing 23 Footwear & Clothing industry, starting from a distribution skewed to the right, it turns out to be skewed to the left. The other measure we used to characterize the evolution of the FSD is the Kurtosis index, aimed at assessing whether the data are peaked or flat relative to a normal distribution. In other words, a distribution characterized by a high Kurtosis tends to have a distinct peak near the mean, to decline rather rapidly, and to have heavy tails. On the contrary, a distribution with low Kurtosis tends to have a flat top near the mean rather than a sharp peak. We used the specification centered at 3, or Pearson Kurtosis: If the Kurtosis index is greater than 3, the distribution is said to be leptokurtic (with a peak in correspondence of the mean), while if it is less than 3 the distribution is platykurtic (more flat and less concentrated around the mean with respect to the normal distribution). In Figure C.2 the different values of this index, by industry and for each quarter are reported. Figure C.2 – Kurtosis Index, by quarter and by industry. For all industries, the Kurtosis index shows a convergence towards the normal distribution, although in the case of the Electrical & Electronics and the Instruments industries, at the end of the relevant period, it appears to be more concentrated around the mean than in that of the other two industries, for which it tends to be more spread. 0 1 2 3 4 5 6 7 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Electrical & Electronic Eng. Instruments Food Footwear & Clothing ( ) 4 n 1i 4 i s xx Kurtosis ∑ = − =