Model Specification: Defining the Three-Level Multilevel Model

Given the specific data structure of this analysis, a three-level model will be applied. In this context, cross-sectional data consists of social enterprises (Level1) nested within regions (Level2) in countries (Level3).

In the following, multilevel analysis is carried out in three-step regressions: Firstly, the unconditional models are provided, which give information on the variability of the dependent variable at each level of specification (Table 6). Secondly, the fixed effects are estimated (Table 7) after having added covariates at each level of specification and thirdly, the random effects treatment models are assessed (Tables 8-11) which, in addition to the fixed effects estimations, include randomly varying slopes across regions.

Unconditional Models

The first step in a multilevel analysis usually develops an unconditional model to partition the variance of the outcome variable into its within- and between-groups components (Heck et al., 2010). This is helpful to determine how much of the variance in the outcome variable lies between the regions in the sample. Moreover, the first unconditional model does not specify

independent variables at any level. The unconditional model for social enterprise in region in country is described by (notation follows the one used by Raudenbush & Bryk, 2002):

, (i)

The unconditional model examines social enterprise growth as a function of the regional mean , e.g. the mean growth rate in region in country , plus a random error , that is the deviation of social enterprise ’s growth rate from the regional mean. The random effects are supposed to be normally distributed with a mean equal to and a variance . Between regions, variation in intercepts can be viewed as an outcome varying randomly around some country mean . It can be represented as:

, (ii)

where the deviation of a region ’s mean from the country mean is represented by . The random effects are assumed to be normally distributed with a mean equal to and a variance . Within each of the countries, the variability among regions is supposed to be the same (Raudenbush and Bryk, 2002). The country mean , on the other hand, is modelled as randomly varying around a grand mean :

. (iii)

where is the random country effect, i.e. the deviation of country ’s mean from the sample’s grand mean. These effects are also assumed to be normally distributed with a mean of 0 and a variance of .

For the unconditional models, it is possible to examine the decomposition of variance of the outcome into its three components: Among social enterprises within regions (Level1),

, among regions within countries (Level2), , and among countries (Level3), .

This model provides a measure of dependence within each Level2 and Level3 unit by way of the interclass correlation (ICC). The ICC describes the proportion of variance that is common to each unit, as opposed to variation that is associated with social enterprises within their units (Heck et al., 2010). According to Hox (2002), it can be thought of as the population estimate of the amount of variance explained by the grouping structure. The ICC is represented as:

, (iv)

where represents the variance and the subscripts B and W stand for between regions and within regions. The higher the ICC, the more homogeneous are the units, i.e. there exists substantial variability between regions and countries. The level of ICC is a guideline on whether the choice of multilevel modelling with regional- and country-effects is justified on the present dataset (Heck et al., 2010). If ICC is too small, researchers often use 0.05 as a rough cut-off point, the higher level grouping does not affect the estimates in any meaningful way. In these cases, a single-level analysis would suffice.

For a three-level model the proportion of variability (ICC) in outcomes at Level3 is defined as:

, (v)

For Level2 the ICC is defined as:

, (vi)

And for Level1 the ICC is equal to:

. (vii)

Conditional Models

For three-level models, coefficients at Level1 are captured by coefficients. Level2 and Level3 coefficients are captured by and respectively. For a social enterprise in region in country , the general Level1 model is described by (notation follows the one used by Raudenbush & Bryk, 2002):

∑ , (viii)

where dependent variable is represented by either: 1. Employment growth, 2. Revenue growth or 3. Social impact development. Further, is an intercept for region in country , represents Level1 predictors , such as social networks (informal and

formal), operational strategies (operational models, diversity, complexity), geography, enterprise maturity (age), employment in 2008, revenues in 2008, assets in 2008 and industrial sector (nace). For social enterprise in Level2 (unit ) and Level3 (unit ), the term

represents the corresponding Level1 coefficients. Level1 variance is assumed to be normally distributed with a mean equal to and a variance .

At Level2, the general regional model is defined as:

∑ , (ix)

where is the intercept for country in modelling the regional effect . In addition,

are Level2 characteristics (q , such as entrepreneurial culture, informal capital, social trust, population density, GDP per capita, expenditure on public health and risk of poverty. Moreover, are corresponding Level2 coefficients which represent the direction and strength of association between regional characteristics and . The random effects on Level2 are represented by .

There are equations in the Level2 model, depending on the number of Level1 coefficients. The random effects in these equations are assumed to be correlated (Raudenbush

& Bryk, 2002). Formally, it can be assumed that the set of are multivariate normally distributed each within a mean of 0, variance and covariance between elements and

of (Raudenbush et al., 2004). All these variances and covariances are collected in a matrix labelled whose dimensionality depends on the number of coefficients specified as random in the Level2 model.

A similar modelling process is replicated at the country level. Level3 random effects depend on the number of randomly varying effects in the model. Between countries, a general model can be defined as:

∑ , (x)

where is the intercept, are Level3 predictors (s , represents the corresponding Level3 coefficients and it further provides information on the direction and strength of association between country characteristic and . Level3 random effects are captured by and are comprised in the corresponding variance-covariance matrix. For

each country there are ∑ equations in the Level3 model. Here too, the variances and covariances are collected in matrix . The dimensionality of depends on the number of Level2 coefficients that are formulated as random (Raudenbush & Bryk, 2002;

Raudenbush et al., 2004). According to Raudenbush & Bryk (2002), there are many alternative modelling possibilities. Based on the theoretical framework and the hypotheses of this thesis, the present analysis will include randomly varying Level2 coefficients. The fixed effects and random treatment estimation outcomes will be discussed in detail in the homogeneous are the units which indicated that substantial variability between regions and countries exists. Moreover, the analysis of ICC is important to ensure that the application of multilevel modelling is warranted (Heck et al., 2010).

Table 6 Unconditional models and intra class correlation (ICC).

Unconditional models Model 1a:

In order to avoid instability in the estimates of regression coefficients, e.g. triggered by a high Type I error rate, the dependent variables employment growth and revenue growth are