Provide solutions for the questions attached below.
Use the willingness-to-pay information about the buyers (Ariel, Bridget, and Connie) and the willingness-to-accept information about the sellers (Daniel, Etienne, and Franklin) below to construct a "stepped" demand and supply diagram like this one from my notes on Unit #7. (You'll also have one question to answer below.) Willingness-To-Pay information Ariel Bridget Connie willingness-to-pay $5 $7 $9 for the 1 widget willingness-to-pay $4 $6 $7 for the 2" widget willingness-to-pay $3 $4 $5 for the 3 widget willingness-to-pay $2 $3 $4 for the 4 widget willingness-to-pay $1 $2 $3 for the 5" widget Willingness-To-Accept information A Daniel Etienne Franklin willingness-to-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) : Econometrics questions 1. Labor economists often study the returns on investment in education (see, e.g., Card 1999). Suppose we have data on salaries of a set of people. some of whom went to college and some who did not. A simple model linking education to salary is Salary, = Bo + By College graduate, + er where the value of Salary, is the salary of person i and the value of College graduate; is 1 if person i graduated from college and is O' if person i did not. (a) What does Bo mean? What does , mean? (b) What is in the error term? (c) What are the conditions for the independent variable X to be endogenous? (d) Is the independent variable likely to be endogenous? Why or why not? (e) Explain how endogeneity could lead to incorrect inferences
