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) : HW 1 (1) - Word (Product Activation Failed) SERT DESIGN PAGE LAYOUT REFERENCES MAILINGS REVIEW VIEW 1. Towards the end of the 20th century, the U.S. government wanted to save money by closing a small portion of its domestic military installations. While many people agreed that saving money was a desirable goal, people in areas potentially affected by a closing soon reacted negatively. Congress finally selected a panel whose task was to develop a list of installations to close, with the legislation specifying that Congress could not alter the list. Since the goal was to save money, why was this problem so hard to solve? 2. Your car gets 29 miles per gallon (mpg) at 60 miles per hour (mph) and 25 mpg a 70 mph. At what speed should you make a 525-mile trip: a. If gas costs $3 per gallon and your time is worth $18 per hour b. If gas costs $4 per gallon and your time is worth $12 per hour c. If gas costs $5 per gallon and your time is worth $9 per hour 3. A firm is planning to manufacture a new product. As the selling price is increased, the quantity that can be sold decreases. Numerically the sales department estimates: P =$475-0.250 Where P = selling price per unit and Q = quantity sold On the other hand, management estimates that the average unit cost of manufacturing and selling the product will decrease as the quantity sold increases, They estimate C = $480 + $22,500 Where C = cost to produce and sell Q per year The firm's management wishes to maximize profit. What quantity should be sold? How much profit will be made? ENGLISH (CANADA)Use the information above to get expressions for the consumption function and the AE equation. The vertical intercept for the consumption function is . 9. The slope of the consumption function is . 10. The vertical intercept of the AE equation is . 11. The slope of the AE equation is . 12. Equilibrium output is equal to . 13. Equilibrium consumption is equal to . 14. Suppose the investment demand function changes and is now |=700 - 50(r). The new value of equilibrium output is . 15. The new value of equilibrium consumption is . 16.Recall that the consumption function is C = 800 +.75(Y - T) - 30 (r). The Keynesian spending multiplier in this economy is
