1. Consider the linear regression models Y; = Bo+B;Fij i = 1,...,n, (1) j=1 12 Y; = Bo+8, (tij -7.j) + i = 1,....n, wher

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1 Consider The Linear Regression Models Y Bo B Fij I 1 N 1 J 1 12 Y Bo 8 Tij 7 J I 1 N Wher 1 (51.09 KiB) Viewed 36 times

1. Consider the linear regression models Y; = Bo+B;Fij i = 1,...,n, (1) j=1 12 Y; = Bo+8, (tij -7.j) + i = 1,....n, where Tj = = -Is Lij ) (2) j=1 Model (2) is known as the centred model. (a) What assumptions are usually made about the error terms €1,..., En and 1...., n? [3] (b) Write down the design matrices X and X for models (1) and (2) and specify their dimensions. [3] (e) Calculate XTX and XT X analytically. (d) Suppose the predictor variables (Fis) can take only strictly positive values. Which among Bo and Bo is more interpretable in this scenario and why? [2] (e) Suppose we have p = 1. The models can therefore be simplified as E(Y) = Bo + Bıri and E(Y) = 30 +31 (1; –ī), where 7 = ri. n Recall from Problem Sheet 2 that in this case, 1 (XTX)-- η Στ - (n)2 (Σετ -nt) -NI 12 0 (XX)-?- (4-7) (1) Derive expressions for the least square estimates 3 = (Bo, Bi) and § =(Bo, Ŝi) in this setting. Use the notation = (1/n) Ei=1Yif you like. Simplify your expressions as much as you can. [4] (ii) Derive expressions for the sum of residuals y-XŹ and y- X3. Simplify your expressions as much as you can. y-