Reconciling quantile autoregressions of firm size and variance–size scaling

Abstract

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.

Appendix

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:

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

and

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

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. $\left| {\frac{{a^{\prime}}}{a}s_{t - 1,j} } \right| > \left| {\frac{{\kappa^{\prime}}}{\kappa }s_{t - 1,j} } \right|$