i. Consider the case where you want to estimate the following model: i = o + 1 + 2xi2 + 3xi3 where x3 = (x1 + x2) and a is a known parameter. For which value(s) of a is this model violating any OLS assumption? Explain. [6 pts] ii. Consider the case where you want to estimate the following model: i = Bo+Bxi How do you expect Bo and B to change if you multiply x; by 100? [6 pts] iii. Consider the following OLS regression, i = 0 + xi,1 + 2Xi2 + 3Xi3 + 4Xi4- where x,4 = 2x,1 xi2+3x,3. Is this model violating any OLS assumption? Explain. [6 pts] iv. Using the sample {(Yi, x1,i, x2,i, . . ., xk,i) : i = 1, 2, ., n}, consider the estimate model Yi = o + 1,i + 22,i + + kXki+i where ; with j = 0,1,..., k is estimated using ordinary least squares (OLS). Show that the sample average of the observations of the dependent variable (yi) is equal to the sample average of the fitted values (i), that is, n 17 n Yi = = = yi n n i=1 i=1 where n is the number of observations. [6 pts] v. Using the sample {(Yi, x1,i, x2,i, ..., xk,i) : i = 1, 2, ..., n}, consider the estimated model Yi = o + 1x1,i + 2X2i + + kXki +i where B, with j = 0, 1, ..., k is estimated using ordinary least squares (OLS). Show that the point (1, 2,...,k) is always on the OLS regression line. That is, = o + 1 + 22 + + kk, where = -1 j,i for j = 1, 2,...,k. [6 pts]