Let two data sets (Xi,Yi) and (xị,yi) be related by xị = c1 + c2x¡ and y = c3 + c4Yi for all i = 1,..,n. This means that the

only differences between the two data sets are the location and the scale of measurement. Such data transformations are often applied in economic studies. For example, Yi may be the total variable production costs in y; maybe the total variable production costs in dollars of a firm in month i and y; the total production costs in million of dollars. Then c3 are the total fixed costs (in millions of dollars) and c4 = 10^-6. \text { (a) Suppose that } y_{i} \& x_{i} \text { satisfy the linear relation } y_{i}=\alpha+\beta x_{i} \forall i=1, \ldots, n \text { . Show } \text { that this implies that the relationship between } y_{i}^{*} \& x_{i}^{*} \text { is } y_{i}^{*}=\alpha^{*}+\beta^{*} x_{i}^{*} \text { with } \alpha_{i}^{*}=c_{3}+c_{4} \alpha-\frac{\beta c_{1} c_{4}}{c_{2}} \quad \& \quad \beta^{*}=\frac{\beta c_{4}}{c_{2}}