Errors and residuals in Regression A residual is defined as (r_i = y_i - X_i \hat{\beta}). That is, a residual is a tangible thing in the sense that it describes observables (cf. Gelman 2021, chap. 11.3, p. 161). That is, the residuals are the difference between observed and predicted values. In contrast, the error term is defined as the difference between the observed value and the true (unobserved) …