*2.2. Method*

The collected data included both a time-series dimension and a cross-sectional dimension, and were thereby transformed into panel-data form. Each firm is observed repeatedly in the vertical dimension with a length of (the number of individuals), I × (the number of periods), T, and the dependent and independent variables K are presented in the horizontal dimension. The overall size of the matrix equals I × T × K observations.

What is typical to panel data and distinguishes them from simple time-series regression is the presence of unobserved heterogeneity that is due to the cross-sectional dimension. Unobserved heterogeneity is the time persistent differences between the individual studied units also called "individual effects" that cannot be estimated with the simple pooled (OLS) regression [38]. When heterogeneity is present in the data, which is typically the case, a model able to take it into account should be used. For this reason, fixed effects and random effects -models that can handle longitudinal and heterogeneous data are used in this research. The fixed effects (FE) or "within"-estimator used has the following form:

$$\mathcal{Y}\_{it} = \beta\_0 + \sum\_{k=1}^{K} \beta\_k \times \mathcal{X}\_{it}^k + e\_{it} + a\_i \tag{4}$$

The within-estimator models the time-invariant heterogeneity in the unknown parameter *a<sup>i</sup>* . The data are transformed by time demeaning all the variables, a.k.a. subtracting the variables' individual means over time from all the variables. The result is a formulation in terms of deviations from the individual means. The *a<sup>i</sup>* term, as well as the constant *β*<sup>0</sup> (see, Equation (4)) that is simply the individual mean, and all the time-invariant independent variables cancel out in this calculation. This eliminates the problem of individual effects, hence it is said to be "fixed" [38].

The coefficients of the FE model can be interpreted as the effect that the unit of change from the individual mean of the respective independent variable has on the same individual's dependent variable from its mean. The main downside of the FE estimators is that one cannot include time-invariant independent variables since they would be canceled out in the model estimation. This simplifies the estimation process but fails to account for the time-invariant variables although they could potentially be significant in determining the values of the dependent variable. To deal with the possible handicaps of the FE in the context of the studied data, a random-effects (RE) model is also applied.

In the random-effects model (RE), the individual differences are allowed and the variation between the individuals is assumed to be random and uncorrelated with the independent variables. The random individual effects are modeled as the error term *u<sup>i</sup>* . The RE-effects model used is defined as follows:

$$Y\_{it} = \beta\_0 + \sum\_{k=1}^{K} \beta\_k \times X\_{it}^k + e\_{it} + u\_i \tag{5}$$

In Equation (5), the intercept corresponds to the mean of the unobserved heterogeneity and the error term *u<sup>i</sup>* is the random time-invariant heterogeneity specific to the individual unit. In the random-effects model, the generalized least squares (GLS) estimator is used. The data are "quasi time-demeaned", which means that a part of the within-individual variation is taken out. For a more comprehensive introduction to the Random Effects model, see [39].

The application of fixed-effects and random-effects models was considered to be sufficient for the purposes of this research in terms of the reliability of the results. We point out that the use of more advanced methods, such as the generalized method of moments (GMM), which control for the violations of the random-effects model and the possible endogeneity problems in the data may reveal deeper and better results from the same data. The use of more advanced methods is left as a topic for future research.
