Methodology

We employ a spatial lag of X model (SLX) to assess the existence and role of spatial spillovers in shaping SCC of dairy farms. We choose the SLX model because it is the simplest and most straightforward way (both in terms of estimation and interpretation) of producing such spillovers without imposing a priori restrictions on the magnitude of spillover effects (Halleck Vega and Elhorst Reference Halleck Vega and Elhorst2015).Footnote ² The SLX model can be written as follows:

(1)$$\hbox{SC}\hbox{C}_t = \hbox{X}_t{\bf \beta } + \hbox{W}\hbox{X}_t{\bf \gamma } + {\bf \alpha } + {\rm \epsilon }_t$$

where SCC is an N × 1 vector of log-transformed farm-specific somatic cell count scores observed in period t (with N being the number of observations), X is an N × K matrix of K farm-specific characteristics that may affect SCC, W is an N × N spatial weighting matrix (defined below), β and γ are the associated K × 1 vectors of parameters to be estimated, α is an N × 1 vector of unobserved farm-specific heterogeneity effects, and ε is an N × 1 vector of disturbance terms. The WX term, which is referred to in spatial econometrics as “the spatial lags of the explanatory variables”, represents the characteristics and production decisions of neighboring farms. Therefore, γ, which is known in spatial econometrics as the “indirect or spillover effect”, captures how a farm's level of mastitis changes when a particular explanatory variable in neighboring farms increases. On the other hand, β captures the so-called “direct effect”, i.e. the change in a farm's level of mastitis attributable to an increase in an explanatory variable of that farm itself. Note that if γ = 0, the model in (equation 1) reduces to a non-spatial regression model that does not allow any spatial spillovers to affect a farm's SCC level. Such a model is also estimated for comparison purposes. Finally, the Hausman test is used to check whether α represents a fixed or a random effect.

We note here that more spatial econometric models are able to produce spatial spillover effects without imposing restrictions on the magnitude of these effects in advance. As mentioned above, the SLX model is the simplest of these flexible models (Halleck Vega and Elhorst Reference Halleck Vega and Elhorst2015). A comparison of different spatial econometric models is out of the scope of the current study and is left for future research.

Spatial Weight Matrix Specifications

In order to estimate the SLX model we need to specify a spatial weighting matrix W, whose elements reflect the intensity of the interdependence between neighboring farms. Two different spatial weight matrix specifications are considered in this article.

The first specification is one in which neighborhood is defined only in the dimension of geographic space. More specifically, we use an inverse distance matrix W ^inv that takes the following form:

(2)$$\lpar W^{inv}\rpar _{ij} = \bigg\{ \matrix{{1/d_{ij}\comma }& {{\rm if \ i}\ne {\rm j\comma \ and}\;d_{ij} \le d^\ast} \cr {0\comma} & {\rm otherwise} } $$

where d _ij is the Eucledian distance between two farms i and j and d* is the minimum distance requirement or cut-off. Following Skevas and Grashuis (Reference Skevas and Grashuis2020) and Marasteanu and Jaenicke (Reference Marasteanu and Jaenicke2016), we set d* to the minimum distance that guarantees at least one neighbour to each farmer. The diagonal elements of the inverse distance matrix are set equal to zero since no farm can be viewed as its own neighbor. Finally, following common practice, the inverse distance matrix is scaled by dividing it by its maximum eigenvalue (Elhorst Reference Elhorst2001; Kelejian and Prucha Reference Kelejian and Prucha2010). Note that the inverse distance matrix places a higher weight on closer than more distant neighbors, it is in line with Tobler's First Law of Geography that near units are more likely to be related than distant ones (Tobler Reference Tobler1970), and it is commonly used in studies that examine the role of peer effects in farm decision-making and performance (e.g., Läpple et al. Reference Läpple, Holloway, Lacombe and O'Donoghue2017; Skevas, Skevas, and Swinton Reference Skevas, Skevas and Swinton2018).

The second spatial weights matrix specification, assumes that farmers follow other farmers close to them both geographically and with higher farm profit levels. This matrix is constructed as the Hadamard productFootnote ³ of the inverse distance matrix (before normalization) defined in equation 2 and a time-varying matrix that reflects the economic distance between a given farmer and his/her more economically successful neighborsFootnote ⁴: $W_t^{inv{\rm \_}s} = W^{inv}\circ W_t^s$, where

(3)$${\rm (}W_t^s {\rm )}_{ij} = {\rm }\Bigg\{ {\matrix{ {{\rm \mid }{\tilde{\Pi }}_{it}-{{\tilde{\Pi }}_{jt}{\rm \mid }} \over {Max{\rm \{ }{\rm \mid }{\tilde{\Pi }}_{it}-{\tilde{\Pi }}_{1t}{\rm \mid },{\rm \mid }\tilde{\Pi} _{it}}-{\tilde{\Pi }}_{2t}{\rm \mid }, \ldots ,{\rm \mid }{\tilde{\Pi} _{it}-{\tilde{\Pi }}_{Jt}{\rm \mid }{\rm \} }} }& { {\rm if }\, i\ne j\,{\rm and}\,\tilde{\Pi }_{it}{\rm \lt }\tilde{\Pi }_{\,jt},\,j = 1,..,J} \cr {0\comma \,}& {\rm otherwise}}}$$

where $\tilde{\Pi }_{it}$ is the inverse hyperbolic sine transformation of profit per cow of farmer i at time t (see section “Results” for the reasoning behind this transformation), | ⋅ | denotes absolute value, and J is the number of observations in the sample less the farm under investigation (i.e., i). In $W_t^s$, the absolute difference in profit levels between two farmers i and j is divided by the maximum absolute difference in profit between farmer i and all other farmers in the sample. This procedure allows weights in $W_t^s$ to range from close to zero (little interaction or influence between two farmers) to one (maximum interaction or influence between two farmers), facilitating influence contrasts between farmer i and his/her more economically successful neighbors. This matrix was then multiplied (in a Hadamard way) by the inverse distance matrix to assure that a higher weight is assigned to the closest (in distance terms) more profitable neighbors. Hence, the final matrix $W_t^{inv{\rm \_}s}$ assumes that farmers follow other farmers close to them both geographically and with higher profit levels. The latter assumption is motivated by the finding that farmers, both in developing (Conley and Udry Reference Conley and Udry2010) and developed countries (Chatzimichael, Genius, and Tzouvelekas Reference Chatzimichael, Genius and Tzouvelekas2014), tend to imitate the behavior of more successful farmers. Following Elhorst (Reference Elhorst2001) and Kelejian and Prucha (Reference Kelejian and Prucha2010), each element of $W_t^{inv{\rm \_}s}$ is divided by its largest characteristic root. An example showing how the $W_t^{inv{\rm \_}s}$ matrix works is provided in the supplementary material.

To the best of our knowledge, the employed approach to construct a spatial weight matrix that captures decision makers' proximity to more economically successful peers (i.e., $W_t^{inv{\rm \_}s}$) is novel and no implementations of such a matrix have been identified previously. We note, however, that the use of spatial weights that are based on both geographic contiguity and economic distance (using different economic variables and applied to non-agricultural settings) is not uncommon in applied spatial econometrics analysis (see for e.g., Delgado, Lago-Peñas, and Mayor Reference Delgado, Lago-Peñas and Mayor2018; Qu, Wang, and Lee Reference Qu, Wang and Lee2016).

Endogeneity of the Spatial Economic Weight Matrix

Spatial weight matrices that are based on a theory of distance, such as our inverse distance matrix above, are considered exogenous to any system (Getis Reference Getis2009). When spatial weights are determined by economic variables, such as the profit variable in the case of $W_t^{inv{\rm \_}s}$ above, then the exogeneity assumption of the spatial weight matrix will likely be violated (Qu, Wang, and Lee Reference Qu, Wang and Lee2016). In our case, farm profitability and SCC might be affected by the same time-varying factors, such as hired labor quality, and cleanliness of the housing, bedding, and milking equipment, which are hard to observe or measure. As a result, it is very likely that $W_t^{inv{\rm \_}s}$ is correlated with the error term in the SCC model, leading to endogeneity bias.Footnote ⁵

To deal with the potential endogeneity problem, we use a two-stage instrumental variables (2SIV) approach, which was developed by Qu, Wang, and Lee (Reference Qu, Wang and Lee2016). In the first step, we estimate the following regression model:

(4)$$\tilde{\Pi }_t = \hbox{Z}_t{\bf \delta } + {\bf \eta } + {\bf \xi }_t$$

where $\tilde{\Pi }$ is an N × 1 vector of the inverse hyperbolic sine transformation of profits per cow observed in period t, Z is an N × L matrix of instruments, η is an N-dimensional vector of unobserved farm-specific heterogeneity effects,Footnote ⁶ and ξ is a N-dimensional vector of disturbance terms. The residuals from estimating equation 4 are then included in the following SLX model:

(5)$$\hbox{SC}\hbox{C}_t = \hbox{X}_t{\bf \beta } + \hbox{W}_t^{inv{\rm \_}s} \hbox{X}_t{\bf \gamma } + {\bi \xi }_t\phi + {\bf \alpha } + {\bi u}_t$$

where ϕ is a parameter to be estimated, u_t is an N × 1 vector of disturbances, and all other variables are as previously defined. Once ξ has been added in equation 5, the spatial weight matrix could be treated as exogenous (Qu, Wang, and Lee Reference Qu, Wang and Lee2016). This is because the unobserved factors that may affect both the spatial weight matrix and SCC are now captured by ξ, implying that the spatial weight matrix is not anymore correlated with u_t. Therefore, we would expect that ξ significantly affects SCC.

Data and Empirical Issues

The data set used in this study was drawn from the Agricultural Financial Advisor (AgFA) database managed by the University of Wisconsin—Madison Center for Dairy Profitability. It is a balancedFootnote ⁷ panel of 83 specialized dairy farms for the period 2014–2017. These farms are operating in three contiguous counties, namely Sheboygan, Manitowoc, and Kewaunee, which stand out as having by far the highest proportion of farms in the database. Focusing on these counties ensures that each sample farmer has a considerable number of neighboring farmers.Footnote ⁸

Data available include financial and production information as well as information on the exact location of the dairy farms in the form of farm coordinates. Information on SCC is available at the herd level (herd average) and measured in cells per ml. As explained earlier, SCC is an indicator of mastitis. It is widely accepted that most contagious mastitis is transmitted by fomites (e.g., milkers' hands, the milking unit, and udder wash cloths) from infected to non-infected animals during the milking process. The recommendations to decrease/control mastitis normally include management practices within the farm such as equipment cleaning, proper milking procedure, good beds for the cows, cleanliness of facilities, etc. Our selection of factors (X) that may influence SCC is based on data availability and previous research on determinants of SCC (Dong, Hennessy, and Jensen Reference Dong, Hennessy and Jensen2012; Volpe et al. Reference Volpe, Park, Dong and Jensen2015). These factors include the following: herd size, summer precipitation and temperature, debt-to-asset ratio, non-farm income, and farm subsidies. Herd size is measured by the number of cows in an operation. The summer temperature and precipitation variables were constructed using the Parameter-Elevation Regressions on Independent Slopes Model (PRISM) maps (http://www.prism.oregonstate.edu/). Following Qi, Bravo-Ureta, and Cabrera (Reference Qi, Bravo-Ureta and Cabrera2015), summer in Wisconsin was defined as the months of June through September. Farm coordinates were then used to generate farm-specific average temperature and precipitation for the summer months that a farm is present in the panel. Debt-to-asset ratio is measured as the total debt a farm holds divided by its total assets. Non-farm income is a dummy variable assigned a value of one if the farmer earned income from non-agricultural activities, and zero otherwise. Finally, subsidies include all government payments received by farmers, including those relating to crop and livestock production and conservation. All SCC determinants (X) are used in logarithmic form in equations 1 and 5, except for non-farm income.

The profit per cow variable that is used in the construction of $W_t^{inv{\rm \_}s}$ and in the instrumental variable approach (equation 4) is defined as the value of production less total operating costs and is expressed per livestock unit. Twenty two percent of the observations in our sample have negative profit per cow. To avoid deleting these observations from the analysis, the inverse hyperbolic sine transformation (arcsinh) is used: $\tilde{\Pi }_{it} = arcsinh\lpar {\Pi_{it}} \rpar = ln\left({\Pi_{it} + \sqrt {\Pi_{it}^2 + 1} } \right)$.Footnote ⁹ The instruments used in equation 4 are the natural logarithm of the farm-level milk price and the use of a total mixed ration (TMR) feeding system. Milk price is measured in dollars per hundredweight. TMR is a dummy variable that takes the value of one if a farmer uses TMR to feed his/her cows, and zero otherwise. TMR is normally a long-standing farm decision. Farmers who elected it, will go with it for the foreseeable future. Simple descriptive statistics shows that farmers using TMR do not switch to a non-TMR feeding strategy during the sample period. Therefore, the use of TMR appears unaffected by unobserved factors and stochastic shocks. The employed instruments are assumed to determine farm profitability and, at the same time, have no direct impact on SCC.Footnote ^10, Footnote ¹¹

Milk price is a valid instrument because (a) it is not a choice variable and therefore unambiguously exogenous, (b) has a strong and statistically significant effect on farm profitability (Table A1), and (c) it does not have a direct effect on SCC (i.e., it affects SCC only through its effect on farm profitability (e.g., higher milk prices may lead to greater profitability which, in turn, increases a farmer's ability to invest in technologies and practices that reduce SCC)). While SCC levels can affect milk price through premiums and penalties imposed by milk buyers, such an effect is not confirmed in the present study. Correlation analysis revealed an insignificant relationship between SCC and milk price (r = −0.026, p = 0.633). Consistently, linear regression did not show an influence of SCC on milk price (β = −1.408 × 10⁻⁶, p = 0.633).

Note that the employed estimation technique that corrects for the potential endogeneity of $W_t^{inv{\rm \_}s}$ is not a conventional two-stage least squares (2SLS) approach, where the predicted value of the endogenous variable is used in the second-stage model. Under such an approach, and in the event that SCC affects milk price, the predicted value of profit from equation 4 (which would contain the milk price variable) could cause a simultaneity problem when included in equation 5 as an additional explanatory variable. In the contrary, the error term of the first-stage equation (equation 4) is used as an additional explanatory variable in the second-stage model (equation 5) to account for unobserved time-varying factors that might affect both SCC and farm profitability. Hence this procedure as described in Qu, Wang, and Lee (Reference Qu, Wang and Lee2016) provides a valid way of correcting for endogeneity.

Concerning the W ^inv matrix, the distance cut-off d* was set to 25 km which is the minimum distance that guarantees at least one neighbor to each farmer. To address concerns about the robustness of our results to changes in d*, we estimate models with 35, 45, and 55 km cut-off.

Table 1 presents a summary statistics of the variables used in the analysis. All monetary variables were deflated to 2014 $ using appropriate price indices retrieved from the National Agricultural Statistics Service database.

Table 1. Summary Statistics of the Variables Used in the Analysis