You are working with the following linear model: y = beta_0 + beta_1 x_1 + beta_

Question

You are working with the following linear model: y = beta_0 + beta_1 x_1 + beta_2 x_2 + epsilon, You know nine pieces of information from the "sample moment matrix" [(y - y)' (y -y) (y - y)' (x_1 - x_1) (y - y)' (x_2 - x_2) (y - y)' (x_1 - x_1) (x_1 - x_1)' (x_1 - x_1) (x_1 - x_1)' (x_2 - x_2) (y - y)' (x_2 - x_2) (x_1 - x_1)' (x_2 - x_2) (x_2 - x_2)' (x_2 - x_2)] = [2000 100 90 100 10 5 90 5 5] And finally, you also know: y = 1200, x_1 = 100, x_2 = 50, and n = 100 observations. Given all this information find the Ordinary Least Squares estimates b = [b_0 b_1 b_2].

Explanation / Answer

let our model be

y -Ybar = b1(X1 - X1bar) +b2 (X2 -X2bar)

b =(X'X)-1*X'y

X'X = [10 5 ; 5 10];     {from matrix given above}

(X'X)-1 =

0.1333   -0.0667
-0.0667    0.1333

X'y = [100 ;90];

hence

[b2 ; b3] =   (X'X)-1*X'y

7.3333
5.3333

not b0 = ybar - b1X1bar - b2 Xbar =

= 1200 - 7.3333 *100 - 5.3333*50

= 200

hence

b0 = 200 , b1 = 7.33333 b2 = 5.333333

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