*3.1. The Hypotheses Connected to Gibrat's LPE*

If β > 1, the sector has an explosive trend, where larger firms grow proportionally faster than smaller firms. If this explosive trend persists over time, there will be few companies left, and the sector will convergence to oligopoly or monopoly. In young industries, the explosive trend could be relevant as an initial edge can exacerbate itself in succeeding years. This cannot last forever, however, which is one reason why older sectors tend to not reject the LPE. If β < 1, there is a mean reverting trend in the sector, where the mean growth is stronger among small companies as they are in a state of 'catch-up'. Thus, we can assume there exists a 'natural' or 'perfect' firm size at which firms will eventually return if they diverge from it, their long-run growth being equal. The steady state size of each firm need not be the same, but the firms will revert to some mean. In the extreme case, where β = 0, every deviation from this mean will be cancelled out in the next period, meaning that firms do not deviate for more than one period. In this case, current size is no predictor of future size.

Many papers (Novoa 2011; Oliveira and Fortunato 2008; Shehzad et al. 2009) have tested Gibrat's Law by following the procedure of Tschoegl (1983), which is a stronger version of Gibrat's Law that suggests three propositions (P1–P3). From Equation (1), we can write the growth for company *i* as:

$$\mathbf{y}\_{\text{it}} = \boldsymbol{\alpha} + \beta \mathbf{y}\_{\text{i}, \text{t}-1} + \varepsilon\_{\text{it}},\\\text{where } \varepsilon\_{\text{it}} = \rho \varepsilon\_{\text{it}-1} + \mathbf{u}\_{\text{it}} \text{ and } \mathbf{u}\_{\text{it}} \sim \mathcal{N} \text{ (0, } \sigma^2) \tag{2}$$

The sum of the error term's (uit) deviation (σ) is by construction equal to zero. In addition, the variance of the error term is written as:

$$
\sigma^2\_{\text{it}} = \delta \mathbf{y}\_{\text{it}} + \eta\_{\text{it}} \tag{3}
$$

which gives the three propositions (the null hypothesis) as:

$$\begin{array}{c} \text{P1: } \pounds = 1 \\ \text{P2: } \pounds = 0 \\ \text{P3: } \delta = 0 \end{array}$$

First (P1), the relative growth of each firm is independent of the firm's initial size, and firms follow a random walk. This is the firm size's autoregressive process. Second (P2), if a firm deviates from its growth path in one year, this deviation does not carry over to the next year; that is, extraordinary success/failure in one year does not translate into extraordinary success/failure the next year. The second proposition differs from the first in the sense that the first proposition concerns the firm's trend, whereas the second concerns deviations from this trend. This (P2) is equivalent to the firm size's moving average process. The third proposition (P3) states that the relative variance in firm size is independent of the firm's initial size. That is, small firms do not vary relatively more or less in their size than large firms. This is equivalent to the firm size's heteroscedastic property.

As stated, P1 holds if the firms collectively follow a random walk, meaning if the best prediction of future size is the current size and all deviations from this size follow a random process.

If P2 holds, all outside effects on the firm size for a given year will be completely reflected in the firm size for this particular year, and those effects will have no impact for the firm's growth in the coming years. A success one year will give an increase in the size of the firm, but this increase does not necessarily lead to a further increase in the next year. That is, the deviation in growth in one year will not carry over into the next year. This does not mean that the success/failure of one year disappears the next year, only that its effect is absorbed that year. If ρ = 0, there is no spill-over effect, and growth will normalize to the prior regular growth after an initial shock. If ρ < 0, firms with extraordinary success in one year will have considerably worse results the next year, with a growth below the average. A lucky period is followed by an unlucky following period, and vice versa: a failure one year results in good performance the next year, with growth stronger than the

others. If ρ > 0, extraordinary growth one year will persist into the following year. That is, growth over the average level for a given firm will persist into the following period. If a company has extraordinarily strong growth one year, it will also manage to maintain this above-average growth in the following period. If one year turns out badly for a company with a low growth, this will also yield negative consequences the following year. If firm growth is characterized by ρ > 0, we can view this a persistence in firm success/failure, or 'slowness' in firm growth. On the other hand, ρ < 0 can be viewed as success/failure being 'cancelled out'. Particularly for campsites, P2 can go both ways, depending on whether the visitors are (dis)pleased with more/fewer other visitors at the same campsite.

If P3 holds, the proportional variance of revenue for the companies is independent of their size—that is, firm size is not related to growth volatility. A negative δ value means there is a negative relationship between a firm's growth volatility and its size: smaller firms have relatively more volatility in their revenue stream. One interpretation is that smaller companies experience greater uncertainty than large enterprises, perhaps because smaller firms are more sensitive to consumer tastes and market conditions, whereas larger firms have a more stable revenue stream.

A study by Calvino et al. (2018) concluded that the value is negative, and remarkably stable across 21 selected countries. Goddard et al. (2004) investigated whether the previous year's growth has an impact on the actual growth, and found a positive relationship, but with no significant impact.

Many researchers have included independent and control variables to the estimators to see how this affects growth. By extending the model with other variables, one can test and explain how different factors contribute to growth and analyse why Gibrat's Law is rejected (Oliveira and Fortunato 2008). For instance, Donati (2016) showed how liquidity constraints limited the growth of small firms. Debt leverage as a control variable yields a mixed result (Jang and Park 2011; Phillips 1995). Some report a negative relationship (Billett et al. 2007), because higher debts increased the number of poor projects. On the other hand, a higher debt level can increase firm performance through successful ventures—that is, the level of debt can be seen as risk-taking.
