Reconciling quantile autoregressions of firm size and variance–size scaling


The aim of this paper is to understand what economic mechanisms may cause the Law of Proportionate Effect to break down for fast-growing and shrinking firms. Recent evidence has highlighted that the first-order coefficients of quantile auto-regression of firm size decline across quantiles. Our theoretical results show that negative variance–size scaling is sufficient to yield a decline in quantile auto-regression coefficients if firm log-size is Laplace-distributed, conditional on size one period ahead. However, it is sufficient only for declining auto-regression coefficients for fast-growing firms under Asymmetric Laplace conditional log-size if skewness is decreasing with size. In other words, if the growth of large firms is less dispersed and more left-skewed, size is a disadvantage for the growth of fast-growers, but not necessarily an advantage for fast-decliners. Thus, size-related determinants of negative growth skewness, such as diseconomies of growth, market power, and managerial attention issues, impact on how the LPE is violated. Using data on Dutch manufacturing companies from the Business Register of Enterprises observed between 1994 and 2004, our empirical estimates of quantile regression models confirm the evidence of declining quantile regression coefficients for small–medium firms (20–199 employees) mainly in the right-most quantiles, and for the same subsample, we find that growth rates variance and skewness are decreasing with size. The theoretical propositions of the paper are thus corroborated.

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  1. 1.

    The test is a two-tailed test, which we also use in our empirical analysis in order to be consistent with the literature on testing for unit roots.

  2. 2.

    These works control for sample selection bias, which can be due to the higher hazard rates of smaller companies.

  3. 3.

    See Lotti et al. (2009) for more on the theoretical underpinnings of Gibrat’s law as a long-run regularity.

  4. 4.

    The relevant right-most quantiles in fact correspond to the whole distribution when κ  0, i.e. when skewness is large and positive.

  5. 5.

    The case κ' < 0, which is harder to justify in economic terms, would imply that a decreasing variance–size relationship is a necessary condition for declining quantile auto-regression coefficients for \( p < \kappa^{2} /\left( {1 + \kappa^{2} } \right) .\)

  6. 6.

    The above theoretical analysis is based on probability distributions defined on the real line. The discrete skew Laplace distribution introduced by Kozubowski and Inusah (2006) shares many properties of the continuous Asymmetric Laplace, suggesting that our results are not affected by the assumption that size is continuous.

  7. 7.

    Figure 1 depicts the log-size distribution for 2004 as an example. The kernel density estimates have been obtained by using the kdens module for the Stata software package, with options epan2 for the kernel, over-smoothed for the bandwidth, and reflection with zero lower boundary. An over-smoothing bandwidth was chosen to reduce the effect of discreteness for low size values, while the reflection option was chosen to take into account the zero lower boundary of log size.

  8. 8.

    When using a QQ-plot, not reported here, for comparing our empirical distribution of firm log-size (in the cross-sectional wave of year 2004) with a Gaussian theoretical distribution, the right-skewness of the empirical distribution is even more evident.

  9. 9.

    The discrete nature of a variable such as the number of employees is partly the reason behind the remarkable stability of the conditional median.

  10. 10.

    Let Q p be the p-quantile. The scale is defined as \( \tfrac{{Q_{0.75} - Q_{0.25} }}{{Q_{0.75} + Q_{0.25} }} \), the skewness as \( \tfrac{{Q_{0.75} + Q_{0.25} - 2Q_{0.50} }}{{Q_{0.75} - Q_{0.25} }} \), and the kurtosis as \( \tfrac{{Q_{0.90} - Q_{0.10} }}{{Q_{0.75} - Q_{0.25} }} \). See also Machado and Mata (2000).

  11. 11.

    The kernel density estimates have been obtained by using the kdens module for the Stata software package, with options epan2 for the kernel, and over-smoothed for the bandwidth. An over-smoothing bandwidth was chosen to reduce the effect of zero frequencies for very low and very high log sizes (troughs of the distribution). Given the log scale of the vertical axis, the cases in which the density is equal to zero have been substituted by interpolation (the effect is perceivable only for the highest size class where the number of firms is lower).

  12. 12.

    Again, this is partly due to the “natural threshold” of zero employees.

  13. 13.

    It should be remembered that the size distribution of micro-firms (size class 1) is left-censored because the number of employees is bounded from below by a “natural threshold”, i.e. zero employees. This might be distorting the results for micro-firms (Capasso and Cefis 2012). Also, the similarities in the estimates for size class 1 and the whole sample estimates arise because micro-firms are the most numerous in the sample.

  14. 14.

    Indeed, to prove that firm growth is unrelated to current size and depends only on the sum of idiosyncratic shocks, it is not enough that the first-order regression coefficient in Eq. 7 be equal to 1. If the error term is serially correlated, firm growth rates in one period depend on firm growth rates in the previous periods. Hence, the LPE might be violated even if the first-order regression coefficient is equal to 1.

  15. 15.

    Without additional assumptions, the data alone are not able to identify which of the two solutions of the system given by (17) and (18) pertains to β p and which one pertains to ρ p . In principle, the left hand side of (21) could be β p and the left hand side of (20) could be ρ p . Here, we follow the additional assumption of Chesher (1979), also used by Fotopoulos and Giotopoulos (2010), that the log size autoregressive coefficient corresponds to the one of the two solutions that is closer to 1. As can be seen in our results, this occurs with the current form of (20) and (21).

  16. 16.

    Quantile regressions have been run by means of the Stata software package. The standard function qreg has been used for the whole sample, while the module bsqreg, which adopts bootstrapping techniques to estimate the standard errors, has been used for the three size classes. In particular, 20 bootstrap replications have been used for size class 1, and 50 bootstrap replications have been used for size classes 2 and 3. We were obliged to choose different estimation functions across different subsamples according to the subsample size and the related computational burden.

  17. 17.

    The discreteness of the number of employees as a size proxy tends to create subsamples of firms that have exactly the same growth rate, i.e. exactly the same proportion between log size at time t and log size at time t-1. This is particularly evident for the conditional quantiles that are close to the median, because many firms keep the same number of employees in two consecutive years. For this reason, for the quantiles close to the median the parameter is often estimated equal to one with standard error equal or close to zero. For the micro-firms, the effect of proxy discreteness can be so strong that even unstable firms tend to move in the same proportion, in particular when they shrink: e.g., firms having one employee can only shrink to zero employees, which can create the case of a size autoregression parameter equal to zero at low quantiles with standard error equal to zero.

  18. 18.

    Standard errors for \( \hat{\beta}_{p}\) are retrieved from the standard errors of \( \hat{\gamma }_{1,p} \) and \( \hat{\gamma }_{2,p} \) using the delta method, as in Fotopoulos and Giotopoulos (2010). For computing the confidence interval, the naïve method has been used of multiplying the standard error by 1.96 before summing it to (subtracting it from) the parameter estimate.

  19. 19.

    Quantiles 0.005, 0.01, 0.05, 0.95, 0.99, and 0,995 have been omitted, since the regression coefficient estimates on those quantiles are based on few observations.

  20. 20.

    Such results are not reported here and are available upon request.


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The authors would like to thank Koen Frenken, Michael Fritsch (the Associate Editor), Federico Tamagni, two anonymous referees, and the participants at 7th European Meeting on Applied Evolutionary Economics (EMAEE), Pisa, 2011, for helpful comments and suggestions. The empirical analysis in this research has been carried out at the Centre for Research of Economic Microdata at Statistics Netherlands (CBS). The views expressed in this paper are those of the authors and do not necessarily reflect the policies of Statistics Netherlands. The authors thank the on-site and the remote-access staff of CBS for their collaboration. This work was supported by Utrecht University [High Potential Grant (HIPO) to E. Cefis and K. Frenken]; and the University of Bergamo (Grant ex. 60 %, no. 60CEFI10, Department of Economics, to E. Cefis).

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Correspondence to Elena Cefis.



Proof of Proposition 1

In the first-order Taylor approximation of the quantile function (Eq. 10), \( \beta_{p} = \frac{{\partial \;Q_{p} \left( {s_{t,j} |\bar{s}} \right)}}{{\partial \;s_{t - 1,j} }} \). For β p to decline across quantiles (i.e. to be decreasing in p), the second derivative of the quantile function with respect to s t−1,j and p must be negative. In the Laplace case (Eq. 11) the following conditions must hold:

$$ \frac{{\partial^2 Q_{p} \left( {s_{t - 1,j} } \right)}}{{\partial s_{t - 1,j} \partial p}} = \frac{{a^{\prime}}}{p} < 0\quad {\text{if}}\;0 < p \le 1/2,\quad {\text{and}}\;\frac{{\partial^{2} Q_{p} \left( {s_{t - 1,j} } \right)}}{{\partial s_{t - 1,j} \partial p}} = \frac{{a^{\prime } }}{1 - p} < 0\quad {\text{if}}\;1/2 \le p < 1. $$

Both are true if a′ < 0. Because s t,j is conditionally Laplace by assumption,

\( a\left( {s_{t - 1,j} } \right){ = }\sqrt {\frac{{V\left[ {s_{t,j} |s_{t - 1,j} } \right]}}{2}} \), hence a′ < 0 in turn requires \( V\left[ {s_{t,j} |s_{t - 1,j} } \right] \) to be decreasing with respect to s t−1,j .

Proof of Proposition 2

The second derivative of the Asymmetric Laplace quantile function with respect to s t−1,j and p reads:

$$ \frac{{\partial^{2} Q_{p} \left( {s_{t - 1,j} } \right)}}{{\partial s_{t - 1,j} \partial p}} = \frac{{a^{\prime}\kappa + a\kappa^{\prime}}}{p}\quad {\text{if}}\;0 < p \le \frac{{\kappa^{2} }}{{1 + \kappa^{2} }}, $$


$$ \frac{{\partial^{2} Q_{p} \left( {s_{t - 1,j} } \right)}}{{\partial s_{t - 1,j} \partial p}} = \frac{{a^{\prime}\kappa - a\kappa^{\prime}}}{1 - p}\quad {\text{if}}\;\frac{{\kappa^{2} }}{{1 + \kappa^{2} }} \le p < 1. $$

The former is negative if \( a^{\prime}\kappa + a\kappa^{\prime} < 0 \), the latter if \( a^{\prime}\kappa - a\kappa^{\prime} < 0 \).

Now, recall that for an AL conditional log-size, \( V\left[ {s_{t,j} |s_{t - 1,j} } \right] = a^{2} \left( {{\kappa}^{2} + {\kappa}^{ - 2} } \right) \). Take natural logarithms on both sides to yield:

$$ {\text{ln}}\;V\left[ {s_{t,j} |s_{t - 1,j} } \right] = 2\;{\text{ln}}\;a{\text{ + ln}}\left( {1 + \kappa^{4} } \right) - 2\;{\text{ln}}\;\kappa $$

Taylor-expand \( {\text{ln}}\left( {1 + {\text{e}}^{{4\:{\text{ln}}\:\kappa }} } \right) \) as a linear function of ln κ around some \( {\text{ln}}\:\kappa_{0} \), plug into the above, and take first derivatives with respect to s t−1,j :

$$ \frac{{d\;{\text{ln}}\;V\left[ {s_{t,j} |s_{t - 1,j} } \right]}}{{d\;s_{t - 1,j} }} \approx 2\frac{{d\;{\text{ln}}\;a}}{{d\;s_{t - 1,j} }} + 2\frac{{\kappa_{0}^{4} - 1}}{{\kappa_{0}^{4} - 1}}\frac{{d\;{\text{ln}}\;\kappa }}{{d\;s_{t - 1,j} }} $$

The above goes to \( 2\left( {\frac{{d\;{\text{ln}}\;a}}{{d\;s_{t - 1,j} }} - \frac{{d\;{\text{ln}}\;{\kappa}}}{{d\;s_{t - 1,j} }}} \right) \) as \( {\kappa}_{0} \to 0 \), and to \( 2\left( {\frac{{d\;{\text{ln}}\;a}}{{d\;s_{t - 1,j} }} + \frac{{d\;{\text{ln}}\;{\kappa}}}{{d\;s_{t - 1,j} }}} \right) \) as \( {\kappa}_{0} \to \infty \). Hence, given κ′ > 0 (and thus \( \frac{{d\;{\text{ln}}\;\kappa }}{{d\;s_{t - 1,j} }} > 0 \)),

$$ \left( {\frac{{d{\text{ln}}a}}{{ds_{t - 1,j} }} - \frac{{d{\text{ln}}\kappa }}{{ds_{t - 1,j} }}} \right) \le \frac{1}{2}\frac{{d{\text{ln}}V\left[ {s_{t,j} |s_{t - 1,j} } \right]}}{{ds_{t - 1,j} }} < \frac{{d{\text{ln}}a}}{{ds_{t - 1,j} }} + \frac{{d{\text{ln}}\kappa }}{{ds_{t - 1,j} }} $$

or, putting it another way,

$$ \left(\frac{{a^{\prime}}}{a} - \frac{{\kappa^{\prime}}}{\kappa}\right)s_{t - 1,j} \le \frac{1}{2}\frac{{d{\text{ln}}V\left[{s_{t,j} |s_{t - 1,j} } \right]}}{{ds_{t - 1,j} }} <\left(\frac{{a^{\prime}}}{a} + \frac{{\kappa^{\prime}}}{\kappa}\right)s_{t - 1,j} $$

As an implication, a negative variance–size relationship implies that \( a^{\prime}\kappa - a^{\prime}\kappa < 0 \), which is sufficient for \( \frac{{\partial^2\;Q_{p} \left( {s_{t - 1,j} } \right)}}{{\partial \;s_{t - 1,j} \partial \;p}} < 0\) (i.e. β p decreasing in p) in an AL distribution only when \( p \ge \frac{{\kappa \left( {s_{t - 1,j} } \right)^{2} }}{{1 + \kappa \left( {s_{t - 1,j} } \right)^{2} }} \), namely for the right-most quantiles—proving Proposition 2.

Notice also that a negative variance–size relationship is implied by \( a^{\prime}\kappa + a\kappa^{\prime} < 0 \), meaning that the width parameter a is more elastic with respect to firm size than the skewness parameter κ, i.e. \( \left| {\frac{{a^{\prime}}}{a}s_{t - 1,j} } \right| > \left| {\frac{{\kappa^{\prime}}}{\kappa }s_{t - 1,j} } \right| \)

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Capasso, M., Cefis, E. & Sapio, A. Reconciling quantile autoregressions of firm size and variance–size scaling. Small Bus Econ 41, 609–632 (2013).

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  • Firm growth
  • Law of Proportionate Effect
  • Quantile regression
  • Variance–size scaling
  • Skewness

JEL Classifications

  • L11
  • L25
  • L26
  • L60