Quality of predictions

• In a “perfect” predictive model predicted value == actual value of the variable for every observation.

• We want the predictions to be reasonably close to the actual values.

• To quantify the quality of predictions we can use the difference between the predicted value and the actual valuecalled as residual.

For a continuous dependent variable $Y$, residual $r_i$ for the $i$-th observation in a dataset:

$$\begin{equation} r_i = y_i - f(\underline{x}_i) = y_i - \widehat{y}_i \end{equation}$$