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The Two Variable Regression for the regression model y = α + β x + ε
(a) Show that the least squares normal equations imply Σiei = 0 and Σi xi ei = 0. (b) Show that the solution for the constant term is a = y – bx.
(c) Show that the solution for b is b = [Σni = 1(xi – x) (yi – y)]/[Σni=1(xi – x)2].
(d) Prove that these two values uniquely minimize the sum of squares by showing that the diagonal elements of the second derivatives matrix of the sum of squares with respect to the parameters are both positive and that the determinant is 4n[Σni=1 xi2) nx2] = 4n[Σni=1 (xi – x)2], which is positive unless all values of x are the same. |
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