Problem 2
You want to estimate 81 using the OLS method from the following multiple linear regression:
Y = Bu + 81X + B_Wa,
To do so, specify the following objective function (the sum of squares residuals) which you want to minimize:
max
Eo (Y - Bu - 81X - B_Wa)^2
There are parameters to estimate. You have the first-order conditions:
B: ∑(Y - Bu - 81X - B_Wa) = 0
B : ∑(X(Y - Bu - B1X - B2Wa)) = 0
B2: ∑(Wa(Y - Bu - B1X - B2Wa)) = 0
There are equations and unknowns, and you are asked to find 81. To slightly alleviate the algebra, I have done some of the steps for you:
Bu = Y - BX - B2Wa
GZ_WY_Y+W = BGEXW_X+W + 4EiWa
Where T- Ci=1X;
Hint: Remember that for generic variables,
XX = ∑X^2
4iX W = ∑XW
Cie1 W = ∑XW
Find B. Show all steps.