Solve the following

1. We use the added variable technique to derive the variance ination factor (VIP). Consider a linear model of the form 91' =50+l31$1+l3213922+-"+}3p$a'p+zr, 5'3: 1:"'ana (1) where the errors are uncorrelated with mean zero and variance 02. Let X denote the n X p' predictor matrix and assume X is of full rank. We will derive the VIP for ip. The same derivation applies to any other coefcient simply by rearranging the columns of X. Let U denote the matrix containing the rst p' 1 columns of X and let z denote the the last column of X so that X = [U 2]. Then we can write the model in (1) as 50 x91 Y=[U z](,:J)+t-:=Ua+z6p+e with a: (2) x810. 1 Let 2 denote the vector of tted values from the least squares regression of z on the columns of U (Le. the regression of X.p on all the other variables), and let T : z 2 denote the residuals from that regression. Note that 'r' and 3 are not random, they are constant vectors obtained by linear transformations of z. (a) Show that the regression model in (2) can be rewritten in the form for some constant vector 6 of the same length as a. (Hint: z : i l 'r and 2? = U(UTU)_1UTz). (b) Show that UT? 2 0, a zero vector. (0) Obtain simplied expressions for the least squares estimators of 5 and 5?, showing, in particular, that 5,, : 'rTY/rT'r. (d) Based on Part (c) and the model assumptions, show that 0.2 ELK\"? _ is)? where :Eg-p is the LS tted value from regression X,D on the all the other predictor variables with an intercept. var(,p) : 73. Comparing a competitor's product unfavorably to your product establishes liability for tortious interference with prospective advantage. True or False? 74. What is negligence? 75. Describe a fact situation in which a defendant can be charged with negligence per se. 76. In trying out a bat in the gift shop at Coors Field, defendant broke a display case. That scared the shop cat, who knocked over a candle. Is defendant liable for the resulting damage to the gift shop? Why or why not? 77. Warren Buffett offered you a job as a stock analyst for a salary he will decide after he reviews your resume. Do you have an employment contract with Mr. Buffett? Why or why not?_ 78. If you move to Omaha in reasonable reliance on your conversation with Mr. Buffett, what theory could you plead to recover the cost of your move? 79. An offer to pay a $100 reward for the return of my lost cat is an offer to enter into which type of contract? a. A bilateral contract b. An implied contract C. A unilateral contract d. A quasi-contract 80. A contract for the sale of goods that were later destroyed in a hurricane may be defended based on #1. Obligations under a performance contract may be assigned to another party unless [Circle off that apply) 82. The elements needed to prove a prima facie case of employment discrimination are (Circle all that apply) a. Plaintiff is a member of a protected class b. Plaintiff engaged in a protected activity C. Employer took tangible employment action against the plaintiff d. Employer's action was taken because of plaintiff's protected activity