In the context of regression, what is the purpose of the error term ε?

In regression analysis, the error term, often denoted as ε (epsilon), plays a crucial role by representing the unexplained variation in the dependent variable that cannot be accounted for by the independent variables included in the model. This term captures all other factors that may influence the dependent variable but are not explicitly included in the regression equation.

For instance, in a simple linear regression model that predicts a person's salary based on their years of experience, the error term would account for deviations in salary that are caused by other factors such as education, industry, or individual negotiation skills which are not measured in the model. Hence, the error term is essential for acknowledging that the regression model may not fully capture all aspects of the relationship, thus indicating variance that remains unexplained.

While correlation measures the strength and direction of a relationship between variables, it does not address the adequacy of the regression model. Establishing causation goes beyond the scope of what regression can prove, as correlation does not imply causation, and asserting causal relationships requires more robust evidences. Verifying assumptions pertains to ensuring that the data satisfies certain conditions necessary for the validity of regression analyses, rather than addressing the unknowns represented by the error term. Thus, the representation of unexplained variation is