## Simple Linear Regression

**3.1 **For least squares, the first-order conditions of minimization, given by Eqs. (3.2) and (3.3), yield immediately the first two numerical properties of OLS esti-

n n n л n

mates, i. e., !>i = 0 and eiXi = 0. Now consider eiYi = a ei C

i=1 i=1 i=1 i=1

n

" P eiXi = 0 where the first equality uses Yi = a C "Xi and the second

i=1

equality uses the first two numerical properties of OLS. Using the fact that

/V n n n

ei = Yi — Yi, we can sum both sides to get ei = Yi — Yi, but

i=1 i=1 i=1

n n n л

P ei = 0, therefore we get £ Yi = P Yi. Dividing both sides by n, we get

i=1 _ i=1 i=1

Y = Y.

nn

**3.2 ** Minimizing P (Yi — a)2 with respect to a yields — 2 P(Yi — a) = 0. Solv-

ing for a yields aols = Y. Averaging Yi = a C ui we get Y = a C u.

n

Hence aols = a C u with E (cіols) = a s...

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