To detect spatial structure, we may calculate Moran’s I tests for spatial autocorrelation. Data {zi} are said to be spatially autocorrelated if neighbouring values are more alike than those further apart.

Moran’s I statistic is defined as^{5}:

where Wj is the (i,j)^{th} element of the matrix W, describing the spatial contiguity of departements under study. There are different ways to define the spatial weight matrix: a binary contiguity matrix, a distance-based spatial weight matrix with or without a critical cut-off, and many others (Anselin, 1988; Fingleton, 2003). The one we use to calculate Moran’I statistics is the first-order spatial contiguity matrix, where Wj is equal to one if locations share at least a common border and zero otherwise.

Moran’s I has a sampling distribution which is approximately normal. The expected

value of Moran’s I is E(I) = —^{1} , and the interpretation is similar to that of the product

n -1

moment correlation coefficient. Informally, +1 indicates strong positive spatial autocorrelation (i.e. clustering of similar values), 0 indicates random spatial ordering, and -1 indicates strong negative spatial autocorrelation (i.e. a checkerboard pattern). Given I, E(I) and Var(I), we can easily test the null hypothesis (H_{0}) of no spatial autocorrelation against the two-tailed alternative (H_{1}) that the data are spatially autocorrelated. Note that the use of standardized variables makes the Moran’s I statistics comparable across time.

Figure 16.1 displays Moran’s I statistic for different agricultural production sectors (EUR Billion) for the period 1990-2006 for the 90 French departements of our sample. Inference is based on the permutation approach with 9999 permutations (Anselin 1995). It appears that all Moran’s I statistics differ in a statistically significant way from zero, and that all agricultural production sectors are positively spatially autocorrelated. This result suggests that the distributions of agricultural production sectors are by nature clustered over the whole period.

Comparing the results for 1990 and 2006, Moran’s I statistics are almost unchanged over the period, especially for dairy sector.

Figure 16.2. Evolution of Moran’s I applied to different agricultural production sectors

for the period 1990-2006

EUR Million

Source: Authors’ calculations from the budgetary accounts of agriculture 1990-2006 AGRESTE, France.

Moran’s I statistic is a global statistic and does not allow us to assess the regional structure of spatial autocorrelation. In order to gain more insight into how regions with high or low agricultural production are located in France, we now analyze local spatial autocorrelation using Local Indicators of Spatial Association (LISA) (Anselin 1995). Local spatial autocorrelation statistics provide a measure, for each unit in the region, of the unit’s tendency to have an attribute value that is correlated with values in nearby areas.

The LISA for each region i and year t is written as:

where z_{it} is the observation in region i and year t, and z_{t} is the mean of the observations across regions in year t.

The high-high and low-low locations (positive local spatial autocorrelation) are typically referred to as spatial clusters, while the high-low and low-high locations (negative local spatial autocorrelation) are termed spatial outliers. While outliers are single locations by definition, this is not the case for clusters, and the cluster itself likely extends to its neighbours as well.

Figure 16.3 shows cluster maps for the production value of the different agricultural sectors studied^{6}. Dairy production is clustered in North-Western France (Grand-Ouest). We note that other animal production activities (pig, beef, and poultry) are also clustered in the Grand-Ouest, suggesting the existence of agglomeration externalities in this region benefiting all animal productions. To these animal products is added the horticulture and market gardening sectors which are a new cluster in the region; this did not exist twenty years ago. Indeed, horticulture and market gardening are historically clustered in the south-east of France. The Bouches-du-Rhone departement remains the first departement producing horticultural and vegetable product worth EUR 345 million, being the driving force of the south-east region.

Spatial dynamics are more regional than sectoral; the region of Grand-Ouest benefits from positive dynamics in many agricultural sectors. This assumption is confirmed by the last cluster map (Figure 16.3, Map h) concerning the agricultural production (all sectors); indeed, we see the Grand-Ouest as a driving force in the agricultural production. However, in Map h we can also see two outliers (atypical regions). First, the Drome departement shows a high value of agricultural production (about EUR 700 million) compared to its neighbours which show rather low values. Indeed, the Drome is specialised in a quality agricultural production; it is the leading departement in organic farming (especially for fruits). Second, the Ardennes shows a negative autocorrelation (low-high spatial association).It accounts for production of about EUR 437 million, whereas it is contiguous to departements that show a high level of agricultural production (i.e. departement of Marne and Aisne), in particular thanks to the production of champagne.

To go further with these observations, we focus on the dairy sector — which shows the highest level of spatial autocorrelation — to determine which factors could influence the location of dairy farms and to verify the existence of agglomeration externalities. We then develop a spatially explicit, departement-level model of dairy inventories (number of farms).

Figure 16.3. Cluster maps (LISA) for production value of agricultural sectors in 2006

(significant at 5%)

Source: Authors' calculations from AGRESTE: Provisional budgetary accounts of agriculture in 2006.