The most widely used analytical tool to evaluate productivity change is the Malmquist total factor productivity (TFP) index. In the agri-food sector, this methodology is applied by several authors, e.g ., Galdeano-Gómez (2006) and Guzmán & Arcas (2008). The Malmquist productivity index has some advantages over alternatives (Bassem, 2014). For instance, it does not require information on the input and output prices, and it allows the decomposition of productivity changes into two components: technical efficiency change (EFC) and technological change (TEC). The former evaluates the degree to which a company’s efficiency improves or worsens (catching up), while the latter reflects the change in the efficiency frontier (innovation) between two-time periods (Cooper et al ., 2007). This index is built in terms of distance functions (Eq. [1]) that can be either input orientated or output orientated. The former is defined as the maximum possible reduction of the inputs given an output vector constant, and the latter considers the maximum proportional expansion of the output given a specific input level constant under a reference technology. We measured productivity changes under the constant returns to scale model (CRS) for the technology and output orientation. Grifell-Tatjé & Lovell (1995) showed that Malmquist index may not correctly measure productivity changes when the variable returns to scale (VRS) model is assumed for the technology. In addition, both input orientation and output orientation provide the same technical efficiency scores when CRS technology is applied. Therefore, the choice of orientation was not relevant in our study. We selected an output orientation because, in agriculture, one usually attempts to maximize output from a given set of inputs, rather than the converse (Coelli & Prasada-Rao, 2005).

For a firm at period t , with t = 1,…, T , the output distance function is defined as:

where S ^t is the production possibility set that transforms the inputs into outputs for period t , x ^t is an input vector, y ^t is an output vector , and sca­lar measures the extent to which y ^t can be maximised while maintaining the input level.

The Malmquist TFP index ( M _i ), which measures the productivity changes between two temporal periods, t and t + 1, can be defined as:

where measures the maximum expansion of output possible in t. Equation [2] is the geometric mean of two productivity indexes: the first is calculated with respect to technology in period , and the second is calculated with respect to technology in the period t + 1.

According to Färe et al . (1994) and Coelli et al . (1998), to assume the presence of productivity inefficiencies, the Malmquist TFP index can be divided into two components (Eq. [3]), where the ratio outside brackets measures the change in efficiency (the relative position with respect to the frontier) between two periods ( t , t + 1) and the geometric mean of the two ratios inside the brackets captures the technological change (the shift of the frontier) between t and t + 1.

Moreover, the efficiency shows how much further or closer away a firm grows to the “best practice companies” situated in the frontier. An index that is greater than, equal to or less than 1 indicates that firms’ efficiency improves, stagnates or reduces, respectively. The technological change indicates that the innovation level of the firms where the index is greater than unity indicates improvements and stagnation or deterioration when the indexes are less than unity (Färe et al ., 1994). Finally, a value of M _i > 1 indicates a TFP of firm growth between t and t + 1, while M _i < 1 shows deterioration in productivity.

To analyse a productivity growth model, we based our analysis on a Cobb-Douglas specification [4]. This is one of the most widely used production function forms in economics, and it represents the relationship between production, input and output:

where Y , L and K , respectively, refer to total production output, labour input and capital input. α is the output elasticity of labour; ß is the output elasticity of capital. A productivity growth specification can be deducted by taking logarithmic derivative of [4]. That is:

which is equivalent to:

where G _Y , G _L and G _K , respectively, refer to the growth rate of Y , L and K . Based on [6], we contrasted interaction effects in productive characteristics between first- and second-level cooperatives on their productivity growth. With this purpose, we built a spatial connectivity matrix W , which each element w _ij defines with values different from zero if cooperative i and cooperative j are connected. Once W was defined, we evaluated the significance of the interconnection among cooperatives in their productivity growth. There are different specifications to evaluate this effect. The first specification we considered is known as the endogenous spatial interaction effect, where the productivity growth of each cooperative depends on the productivity growth (G _Y ) of their associated cooperatives i depends on the productivity growth of their associated cooperatives j . The second specification considers the interconnection effect among cooperatives in the error term. Finally, the third specification allows the co-existence of endogenous and exogenous interaction effects among cooperatives. The difference between previous specifications is explained by the source of interdependence. In the first case, the spatial effect is caused by the structural character of the analysed variable. In this case, if this structure is significant, then we can conclude that the particular characteristics of a company influence the productivity growth values of their associated peer companies. In the second case, the spatial interactions in the error term are explained by the omission of relevant variables in the model that generate this result. The last case contains endogenous and exogenous interaction effects in the model. In this sense, spatial effects are caused by the structural character of the productivity growth and the explanatory variables of other units associated with them.

To show these specifications, we parted from a general panel data model that includes both individual and temporal heterogeneity in the specification. In particular, this model takes the following specification for i = 1,..., N individuals and t = 1,..., T periods:

where G _{Y NT×1} is the vector of the dependent variable; l _NT×1 is a vector of ones associated with the constant term parameter to be estimated; X _NT×K denotes the vector of explanatory variables; G _L and G _K . φ is the vector of ( K × 1) associated parameters (α, ß), and µ = ( µ ₁ , µ ₂ ,…, µ _N ) ^T represents the invariant time effects of each company i . This effect can be considered fixed or random. In the first case, we assume a different parameter to be estimated for each firm i , which is correlated with the explicative variables of the model, while the random specification considers that the individual effect is an independent random variable with average zero and variance s _µ ² (Wooldridge, 2012). The application of fixed or random specifications depends on the characteristics of the empirical application. The fixed specification assumes dependence between the unobserved heterogeneity and the explicative variables, while the random specification assumes independence in this term. However, this last assumption may be highly restrictive when we are analysing the different dimensions in a company (Martínez-Sola et al ., 2014). Therefore, we based our analysis on a fixed effects model. ζ = ( ζ ₁ , ζ ₂ ,…, ζ _T ) ^N represents time-period-specific effects invariant across individuals. Finally, u _NT×1 is the error terms independent and identically distributed with mean zero and variance σ ² .

Based on the previous specification [7], the model that contains a spatial lagged dependent variable is known as the spatial lag model (SLM) expressed in [8]:

where W _NT×NT is the spatial weight matrix, and WG _Y denotes the spatial lag effect of dependent variable G _Y . ρ is the spatial autoregressive coefficient, which tests the significance and the value of the spatial interaction among individuals in the dependent variable.

The spatial panel data specification with spatial interdependences in the error term is known as the spatial error model (SEM) expressed in [9]:

where W _NT×NT is the spatial weight matrix; W _u denotes the spatial lag effect of error term u; λ is the spatial autoregressive coefficient in the error term.

The final specification that takes into account the co-existence of separable endogenous and exogenous interaction effects is known as the spatial Durbin model (SDM) and is defined as [10]:

where _WNT×NT is the spatial weight matrix; WY denotes the spatial lag effect of the endogenous term; WX is the exogenous interaction for explicative variables G _L and G _K ; is the spatial autoregressive coefficient, which tests the significance and the value of the spatial interactions among individuals in the dependent variable; represents unknown parameters to be estimated.

Previous models [7]-[10] have been estimated by applying maximum likelihood (ML) (Ord, 1975; Elhorst, 2010), instrumental variables (IV) or the generalised method of moments (GMM) (Fingleton & Le Gallo, 2008). The ML estimation is the most commonly used method based on the maximisation of the log-likelihood function.

The information to develop this study was obtained from the Iberian Balance Analysis System (SABI) database ³ , which provides a wide range of information on the different business dimensions of Spanish firms. We selected Spanish agri-food companies based on the criterion established in the National Classification of Economic Activities (NACE, 2007). Once we obtained all the information about agri-food second-level cooperatives, we eliminated the companies with anomalies in their financial statements. Our final sample comprised information for 265 agri-food companies over the period 2012-2015. Of these 265 cooperatives, 218 were first-level cooperatives that were integrated into 45 second-level organisations. In addition, to determine the cooperatives associated to each of them, we hand collected this information from the webpages of each of these cooperatives. Table 1 shows the composition of our sample. Table 1 shows, in general terms, the distribution of the sample in categories relative to the age and the size of the firm for first- and second-level cooperatives. The high number of micro and small cooperatives shows an important issue, the limited dimension of the cooperatives.

Inputs and outputs to compute Malmquist productivity index

Based on previous agri-food studies, this analysis used one output and two inputs to compute Malmquist productivity index. The output was defined as the turnover volume, which represents the operating revenue from selling the products produced; this allows an adequate evaluation of the activity of the productive unit examined (Soboh et al ., 2012; 2014). The inputs ⁴ were represented by the labour costs and fixed assets (Galdeano-Gómez, 2006; Guzmán & Arcas, 2008; Soboh et al ., 2012).

Variables to estimate the productivity growth model

Productivity growth was evaluated through the Malm­quist productivity index. The explicative variables, labour and capital growth, were built as the growth rates of employees and capital intensity defined as fixed assets on total assets. Finally, we defined the interconnection among first- and second-level companies by a binary weight matrix W _c , where its elements w _ij were valued one if companies i and j are connected and zero otherwise. The elements of the main diagonal were valued zero by definition (Anselin, 1988). This matrix was row standardized.