Ordinary least squares

OLS is the basic linear regression method used to describe how an outcome (for example, farm income) changes on average when one or more explanatory variables change (for example, farm size, subsidies, or input use).

OLS chooses the straight line (or more generally, the linear combination of variables) that passes ‘as close as possible’ to all observed data points, where ‘closeness’ is measured by the sum of squared differences between the observed values and the values predicted by the model (the squared residuals).

In a standard application, OLS is used with a linear model such as:

Y_i=α+β_1 D_i+X_i^' γ+ε_i

where Y_i is the outcome (e.g. income of farm i), D_i is the policy variable (e.g. CAP support received), X_i is a vector of other observed characteristics (e.g. farm size, region), ε_i (error term) is everything not observed. OLS chooses the coefficients α, β_1 and γ so that the sum of squared residuals∑_i ε ̂_i^2 is minimized.

What it does and the assumptions behind:

OLS with single‑year (cross‑section) data is used to describe the statistical relationship between CAP support (e.g. direct payments, investment aids) and farm outcomes (e.g. income, productivity) in that year. It shows how these outcomes differ, on average, between farms receiving different levels or types of support, after accounting for other observable farm characteristics that may affect the results, such as farm size, region, type of production, labour input and capital. In practice, this means that farms are compared with others that are similar in these respects, so that the remaining differences in outcomes are more directly associated with differences in CAP support rather than with structural differences between farms. Technically, the regression includes these characteristics as control variables, and the coefficient on CAP support is interpreted as the average difference in the outcome when CAP support changes, while the other included variables are held fixed (a ceteris‑paribus interpretation).

In other words, OLS answers questions such as:

OLS can be applied to assess the effect of CAP support in various situations.

It is particularly useful for screening interventions and for providing an initial, indicative assessment of how support is associated with income or other agricultural outcomes, especially when only a single cross-sectional dataset is available.

OLS can also be used for preliminary assessments to identify general patterns and groups of farms or interventions that may merit more detailed causal analysis.

Furthermore, it offers a pragmatic option when more demanding methods (such as panel models, matching, instrumental variables or differences-in-differences) are not feasible, typically due to data limitations.

However, in all these cases, OLS results should be interpreted as associations rather than causal effects, because the method remains vulnerable to omitted-variable bias, simultaneity, endogeneity and selection bias.

OLS can be used to analyse a wide range of result and impact indicators of the CAP strategic plan.

Typical examples include economic indicators at the national (macro) level, such as agricultural net income, household agricultural income, gross margin per hectare or per annual work unit, labour productivity and capital intensity. a

It can also be applied to environmental and climate indicators measured at farm level, such as fertiliser and pesticide use per hectare, stocking density, soil organic matter content or greenhouse gas emission indicators.

Further examples include structural indicators, such as farm size, degree of specialisation, investments per hectare, adoption of ecological schemes or rural development measures, and participation in risk management tools.

In all cases, OLS can be used to describe how these indicators differ, on average, between farms receiving different levels or types of CAP support, after controlling for relevant observed characteristics (e.g. region, type of production, farm size).

Step 1 – Clarify what relationship is of interest (e.g. association between direct payments and farm income, controlling for structural characteristics). Select an appropriate outcome indicator (e.g. farm net income, gross margin, labour use).

Step 2 – Identify the policy variables (e.g. payment delivered under specific CAP instruments) and a set of control variables capturing farm structure, specialisation, region and other observable factors that are likely to affect the outcome.

Step 3 – Using cross‑sectional data, estimate a linear regression of the outcome on the policy variables and controls. Use robust standard errors where appropriate. Then assess the robustness of the results by identifying extreme outliers that may influence the estimates, checking that the key coefficients (e.g. on CAP support and main controls) have economically plausible signs and magnitudes, and examining simple goodness-of-fit measures (such as graphs, R-squared values and residuals) to verify whether the linear specification provides a reasonable description of the data.

Step 4 – Present and interpret coefficients as conditional associations, not as causal effects. For example, farms with higher decoupled payments tend to have higher income, after controlling for farm size and type. Clearly note that other unobserved factors may still drive these patterns.

Step 5 – Use OLS findings to identify patterns, raise questions and prioritise further analyses (e.g. panel data, DiD, matching, IV, qualitative case studies). Explain in the report how OLS fits within a broader mixed‑methods evaluation strategy.

In the evaluation project ‘Evaluation of income effects of direct support’, OLS has been used to examine the possible role of some CAP policy measures (including decoupled direct payments) on farm income. Specifically, the farm net value added per annual work unit has been regressed on the level of support received by a large sample of FADN farms as well as on farm- and region-specific characteristics that may affect the level of income.