Question

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1. Derive the OLS estimator for B1 2. Prove that it is unbiased 3. Given the data below Obs y 25 X 2 1 5 2 14 1 4 38 -2 4 3 7 4 23 50 0 5 Compute B using algebra. In other words you are god and you observe u. Now use your ols estimator, as a human who does not observe u, to compute B. Are they the same? Why?

Answer 1

Ordinary Least square method is the method of estimating regression coefficients such that the sum of squares of the error term is minimized.

| Page No. Date Answers Yi B, Xit lli W Deruation of Bi Method that We use Ordinary Least Square minimize the en sum of squares of essas tem to get best estimate for the population paramily! I thi Yi - Bixi - 42 (Ye-B, x)² Let Bl be the estimat Summating both sides for ad B, 34² = {CLE Rixe? Now, find or that minimiges Elli2 differentiale El with respect to Bl we get asu? = a[{lyi- Bix;)?] = QE(Yi - BiX ( ) x (-1) 267 OBI - -23( 4-Bixu! putting furst different deviative equat to gewo asui? 20 2BT 4 - 2 E(4i-Bixi)=0 y Slyo - Bi Xi) zo Eli-Bisxi-o Bi= {li dviding numerator | & denominator lyn ΣχI Al we get Therefore, By is equal to sui/sxo or qlx.

Page No. Date T in for B to be unbiased, E(B) = B. Now, Yi Bi xi thi ELY)= EB, X;) + Elli) i Elyi) = BIE(Xi) + Elui) (using multiplicatur rule) Now, ELY() = Ý , Elxi) = ň and Elli to Lby assumption 1 | Y = B. Ž = 18 - |x| No O 2 = E (B) 2 t = 7 E (718) 4 / X : Elconstant) =constant from eq (1 and 1 ELBID = Bil is an unbiased estimator of Bl.

* liv obs y 38 23 5 50 CY= 150 EX-15 Euro Bi= sy/Ex __ = 150/15 (6 = 10 Regussion equation : Y = Bi Xitli using 1st obs: 25=2 Bits 202 2B, i B= 20/2=10 using and obs: 14 = Bit4 o B =10 & Bi=10 and B) = 10 tot Yes, they are same because our sample statistic is an unbiased estimate of the populat parameter. However, the conclusion is inferential only to sample and not necessarily ture every tume. T herego aleans that ou au vang of cample It may also imply that the guen obscuado are is our true population and the fore y