*3.2. Econometric Methods*

In early empirical testing of Gibrat's Law using econometrics, the ordinary least squares (OLS) method of estimation was used. Due to the presence of the lagged dependent variable, this induces endogeneity issues through the feedback, or looping, mechanism, as shown by Chesher (1979). Consequently, as Chesher (1979) and Jang and Park (2011) have pointed out, this means that OLS will be inconsistent unless the number of variables representing firm size is equal to the number of time periods. When there are more than a few time periods, this becomes, at a minimum, inefficient, and infeasible at most. Even so, many researchers have used OLS to test Gibrat's Law (Daunfeldt and Halvarsson 2015). This is true for the previous studies that have analysed the validity of Gibrat's Law in camping sites (Italian and Dutch).

An alternative approach is to use the generalized method of moments (GMM) and, specifically, those methods that are specifically created for dynamic panel data scenarios. Arellano and Bond (1991) proposed such a method for panel data to ensure a consistent evaluation of the parameters. They exploited the moment conditions of the first differenced error terms, which allowed for the use of the lagged level of two periods prior as instruments for the first differenced equation. The estimator has been called the AB or FD GMM method. Some researchers have used it to test Gibrat's Law (Ivandi´c 2015), but when the autoregressive parameter (β) approaches unity, the instruments used become weaker. In the case that Gibrat's Law holds, β = 1, the instruments are entirely invalid, as they are not correlated with the first differenced equation. This leads to inconsistent and downwardly biased estimators of β, as has been shown in several studies using Monte Carlo simulations (Blundell and Bond 1998; Jang and Park 2011; Moral-Benito et al. 2019). As a result, using the Arellano–Bond estimator will tend to lead to a rejection of Gibrat's

Law too often. Due to the difficulties of the first difference GMM estimator (Arellano and Bond 1991), Arellano and Bover (1995) and Blundell and Bond (1998) developed an improved version of the dynamic panel data GMM estimator, which combines the lagged level instruments for the differenced equation with differenced instruments for the level equation. The method has been called the system GMM (SYS-GMM) estimator and has proven to be a very powerful dynamic panel data estimator, even when the autoregressive parameter β approaches unity. This is what the testing of Gibrat's Law requires, and it has been used in many articles (Donati 2016; Giotopoulos 2014; Jang and Park 2011; Oliveira and Fortunato 2008). In addition, due to the information contained in the level equation, one can estimate time-invariant variables, such as location and sector, which is not possible with the FD-GMM estimator. The System GMM estimator is the estimator of choice in a dynamic model with short panel data.

There are two main arguments against the SYS-GMM estimator, however. First is the issue of using instrumental variables, as they will always run the risk of becoming weak (as with the FD-GMM estimator). This risk is circumvented to a degree by the SYS-GMM estimator by using sets of equations with two sets of instruments, in addition to the fact that the instruments are the 'same' variable as those being instrumented. The instruments do require the sacrifice of a time period, which can be crucial in short panels. The second contention is more serious, as, contrary to the FD-GMM estimator, the SYS-GMM estimator requires the assumption of mean stationarity of the cross-sectional observations (firms). That is, by including the first difference instrument for the level equation, it assumes that all cross-section observations have reached a steady state (Allison et al. 2017; Moral-Benito et al. 2019). This translates into the assumption that each of the firms has reached their natural size and are in a steady state at the beginning of the sample period, which seems to be an unrealistic assumption for the camping sector (and many other sectors for that matter). Additionally, the moment conditions that both GMM estimators rely on require that there is no second order autocorrelation for validity of the instruments. If this does not hold, trice-lagged instruments need to be used, which would induce weaker instruments, and the sacrifice of another time period.

Consequently, Allison et al. (2017) and Williams et al. (2018) developed a maximum likelihood estimator based on simultaneous equations as an alternative to the GMM estimators; this is called the ML-SEM (maximum likelihood structural equation modelling) estimator. This is a computationally intensive method, which avoids both the instrumental variables issue of GMM, the limitations of FD-GMM, and the unrealistic assumptions of SYS-GMM. The ML-SEM estimator is slightly more precise and unbiased than the SYS-GMM estimator under a variety of conditions while relying on the same weak regulatory assumptions as the FD-GMM estimator, without using instrumental variables. For our study, the ML-SEM estimator's largest drawback is its novelty, as it is not fully optimized with our software, and we are not aware of any published article testing Gibrat's Law by applying ML-SEM.
