In the field of research design, researchers often want to know whether there is a relationship between an observed variable (say, y) and another observed variable (say, x). To answer the question, researchers may construct the model in which y depends on x. Although y is not necessarily explained only by x, a discrepancy always exists between the observed value of y and the predicted value of y obtained from the model. The discrepancy is taken as a disturbance term or an error term.

Suppose that n sets of data, (x1, y1), (x2, y2), …, (xn, yn), are observed, where yi is a scalar and xi is a vector (say, 1 × k vector). We assume that there is a relationship between x and y, which ...