1. This is all one question and if mentions appendix 4.2 so I have provided that as well. please help
understandble but thats why im here...to get help 6.11 (Requires calculus) Consider the
regression model Yi=1X1i+2X2i+ui for i=1,,n. (Notice that there is no constant term in the
regression.) Following analysis like that used in Appendix 4.2: a. Specify the least squares
function that is minimized by OLS. b. Compute the partial derivatives of the objective function
with respect to b1 and b2. c. Suppose that i=1nX1iX2i=0. Show that ^1=i=1nX1iYi/i=1nX1i2. d.
Suppose that i=1nX1iX2i=0. Derive an expression for ^1 as a function of the data
(Yi,X1i,X2i),i=1,,n. e. Suppose that the model includes an intercept: Yi=0+1X1i+2X2i+ui. Show
that the least squares estimators satisfy ^0=Y^1X1^2X2. f. As in (e), suppose that the model
contains an intercept. Also suppose that i=1n(X1iX1)(X2iX2)=0. Show that
^1=i=1n(X1iX1)(YiY)/i=1n(X1iX1)2. How does this compare to the OLS estimator of 1 from
the regression that omits X2 ?
This appendix uses calculus to derive the formulas for the OLS estimators given in Key Concept
4.2. To minimize the sum of squared prediction mistakes i=1n(Yib0b1Xj)2 [Equation (4.4)], first
take the partial derivatives with respect to b0 and b1 :
b0i=1n(Yib0b1Xi)2=2i=1n(Yib0b1Xi)andb1i=1n(Yib0b1Xi)2=2i=1n(Yib0b1Xi)Xi The OLS
estimators, ^0 and ^1, are the values of b0 and b1 that minimize i=1n(YiIb0b1Xi)2 or,
equivalently, the values of b0 and b1, for which the derivatives in Equations (4.21) and (4.22)
equal 0 . Accordingly, setting these derivatives equal to 0 , collecting terms, and dividing by n
shows that the OLS estimators, ^0 and ^1, must satisfy the two equations
Y^0^1X=0andn1i=1nXiYi^0X^1n1i=1nXi2=0
Sampling Distribution of the OLS Estimator 131 Solving this pair of equations for ^0 and ^1
yields ^1=n1i=1nXi2(X)2n1i=1nXiYiXY=i=1n(XiX)2i=1n(XiX)(YiY)^0=Y^1X. Equations
(4.25) and (4.26) are the formulas for ^0 and ^1 given in Key Concept 4.2; the formula
^1=sXY/sX2 is obtained by dividing the numerator and denomin tor in Equation (4.25) by n1.