Abstract
Firms grow and decline by relatively lumpy jumps which cannot be accounted by the cumulation of small, “atom-less”, independent shocks. Rather “big” episodes of expansion and contraction are relatively frequent. More technically, this is revealed by the fat-tailed distributions of growth rates. This applies across different levels of sectoral disaggregation, across countries, over different historical periods for which there are available data. What determines such property? In Dosi et al. (The footprint of evolutionary processes of learning and selection upon the statistical properties of industrial dynamics. Industrial and corporate change. Oxford University Press, Oxford, 2016) we implemented a simple multi-firm evolutionary simulation model, built upon the coupling of a replicator dynamic and an idiosyncratic learning process, which turns out to be able to robustly reproduce such a stylized fact. Here, we investigate, by means of a Kriging meta-model, how robust such “ubiquitousness” feature is with regard to a global exploration of the parameters space. The exercise confirms the high level of generality of the results in a statistically robust global sensitivity analysis framework.
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Notes
In the present case it may be more appropriate to call the choice of the sampling points in the parameters space as quasi-experiment, as the conditions imposed for selecting the observations for the sample are specified by the NOLH.
Here, we closely follow, whenever possible, the analytical framework employed by those authors and refer the readers to their paper for additional details and references.
The rescaled Beta distribution was preferred because of its superior flexibility in terms of parametrization and the bounded support. Other than Beta, Laplace and Gaussian, Log-normal and Poisson distributions were also tested in Dosi et al. (2016). Different distributions did not qualitatively affect the results.
The simulation model is coded in C++ and it is run inside the LSD simulation platform (Valente 2014) which is also employed for the NOLH sampling procedure, as explained below.
The parameter \(\mu \) of the distributions (Beta, Laplace, Gaussian) was chosen in order to produce an average innovation shock of 0.05 (or 5% increase in the productivity of adopted innovations). This value is loosely connected to the order of magnitude of advancements in process innovation for many industries. Similarly, \(\mu _{max}\) (0.20) represents an upper limit to the innovation process for distributions with infinite support (Laplace, Gaussian), also loosely based on empirical evidence. The remaining distributions’ parameters were set to keep at least 80% of the mass of the distributions below \(\mu _{max}\). The number of firms was chosen to be in line with empirical datasets when the analysis is done at 2–3 digits. The parameter A and \(\gamma \) were set to 1 arbitrarily. The initial size was set to 1 / N (all firms equal) with no loss of generality, as the model is ergodic (see Dosi et al. 2016) and initial conditions are not relevant for a sufficiently long time frame. Initial productivity is set to 1 as a reference.
Subbotin parameters estimation is performed by the maximum-likelihood method using the Subbotools package (Bottazzi 2014).
Second order polynomials with full interactions were evaluated but systematically produced meta-models with worse fitting than the original model, even when more samples are added to the DoE, as the interactions and nonlinearities are usually better modelled by the correlation function. The Kriging trend function coefficients are estimated using generalized least squares.
Definitions for other correlation function alternatives can be found in Roustant et al. (2012).
The Kriging correlation function (kernel) coefficients are estimated by means of numerical maximum likelihood. For the details on the technical implementation applied, see Roustant et al. (2012).
The technical feasibility criterion adopted was the minimally “normal” operation of the market, measured by the survival of at least two firms during the majority of simulation time steps. Also, some of the parameters’ test ranges limit, in practice, the possible ranges of variation for other parameters (e.g., the distribution average \(\mu \) must be lower than the upper support of distributions \(\mu _{max}\)).
Noise is used in the entire estimation process to evaluate observations. Samples under too much noise (sampling variance over 10 times the average) are discarded in the estimation process. Table 4 presents the effective number of observations used.
However, the \(Q^2\) statistic is not lower-bounded to zero, like the \(R^2\), being possibly negative in the case the model performs worse than the “no-model” estimate (the mean of the sample). To avoid confusion, we lower bounded the values of \(Q^2\) to zero.
The meta-model estimation (using GLS for the trend and numerical ML for the correlation function coefficients) and the following sensitivity analysis (using Sobol decomposition) was performed using the DiceKriging, DiceOptim and DiceEval packages (Roustant et al. 2012; Dupuy et al. 2015) in R (R Core Team 2016).
The Matèrn correlation function—the Fourier transform of the Student distribution density function—in its 5/2 formulation can be specified as (Rasmussen and Williams 2006):
$$\begin{aligned} {\text {corr}}(\delta (\mathbf {x}_i), \delta (\mathbf {x}_j)) = \left( 1 + \sqrt{5} h + \frac{5}{3} h^2 \right) \exp \left( - \sqrt{5} h \right) , \quad h = \sum _{g=1}^{k} \psi _g |x_{g,i}-x_{g,j}|. \end{aligned}$$(11)One may question how such non-linear interactions can be captured if the employed Kriging trend function is a polynomial of order zero or one. The answer is to be found on the Kriging correlation function as the role of interactions was excluded only in the global trend. Note, instead, that the correlation function does capture the interactions among the parameters, which are indeed stochastic (spatial correlation) and not deterministic.
Kriging predictions becomes more precise as the interpolated point gets closer to one of the DoE points, where the error of the model is always zero by construction—and vice versa—so \(\epsilon \) is not constant.
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Acknowledgements
We thank Francesca Chiaromonte for helpful comments and discussions. We gratefully acknowledge the support by the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 649186 - ISIGrowth and by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), process No. 2015/09760-3.
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Dosi, G., Pereira, M.C. & Virgillito, M.E. On the robustness of the fat-tailed distribution of firm growth rates: a global sensitivity analysis. J Econ Interact Coord 13, 173–193 (2018). https://doi.org/10.1007/s11403-017-0193-4
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DOI: https://doi.org/10.1007/s11403-017-0193-4
Keywords
- Fat-tailed distributions
- Kriging meta-modeling
- Near-orthogonal latin hypercubes
- Variance-based sensitivity analysis
- ABMs validation