Question

2.4 We have defined the simple linear regression model to be y =B1 + B2x+e. Suppose however that we knew, for a fact, that ßı = 0. (a) What does the linear regression model look like, algebraically, if ßı = 0? (b) What does the linear regression model look like, graphically, if ßı = 0? (c) If Bi=0 the least squares "sum of squares" function becomes S(R2) = Gyi - B2x;)?. Using the data, x 1 2 3 4 5 6 weet when y 4 DD 6 7 7 9 11 mah plot the value of the sum of squares function for enough values of B2 for you to o locate the approximate minimum. What is the significance of the value of B2 that minimizes S(B2)? (Hint: Your computations will be simplified if you s on algebraically expand S(B2) = 4(yi - B2x;)? by squaring the term in par- To som entheses and carrying the summation operator through.) (d) •Using calculus, show that the formula for the least squares estimate of B2 in this model is b2 = x;yi/x7. Use this result to compute by and compare this value to the value you obtained geometrically. (e) Using the estimate obtained with the formula in (d), plot the fitted (estimated) regression function. On the graph locate the point (1,7). What do you observe? (f) Using the estimates obtained with the formula in (d), obtain the least squares residuals, ê; = y; - b2x;. Find their sum. m (g) Calculate xiê;. ni dola (6) World

Answer 1

(a) if Bi = 0, the given linear regression model, algebraically, look like y = B2x + e (b) if B1 = 0, the regression line would start from the origin. It would result in a straight line from the origin. If B2 > 0, the line would be as follows:

(c) The sum of square function against B2 can be shown as follows: (to generate the plot, see the description at the end) Error Sum of square sum of squares 1.95, 11.6275 0.0 0.5 1.0 2.5 3.0 3.5 1.5 2.0 Beta_2 As can be seen, the value B2=1.95 is minimizing the error sum of square function. The significance of the B2 that minimizes this function is that it is the ordinary least square estimate of the B2 in our regression model.

(d) The objective function is to minimize error sum of square, i.e., Minimizcess = E(): – B2x;)2 Hence, the first order condition is = 2(y; - B2x;)(-x;) = 0 => xiyi - B22 x 2 = 0 => B2 = Using the above formula, we have estimated B2 as follows. The table below shows the calculated values. The value of B2 below is very close to the value shown in part (c) 1 xyxyx 11 4 4 1 26 12 37 21 4 7 28 945 25 6 11 66 36 16 176 9 1 9 Total 10 Beta_2

(e) The fitted regression function can be shown as below: Fitted The calculation can be shown below. A B C D E 1 xyxyx Fitted y 2 11 41 4 1 1.934066 3 2 6 12 4 3.868132 4 3 7 21 9 5.802198 5 4 7 28 16 7.736264 6 59 4525 9.67033 6 1166 36 11.6044 7 9 Total 10 Beta 2 17691 1.93

(f) é, has been calculated in column F and the sum is equal to 3.38 (should be ideally very close to zero) as shown below (g) Xe, has been calculated in column G and xê, = 0 as shown below

Note: The calculated values for all the expressions discussed above is shown in left hand side table and the right-hand side table shows the formula view of the calculations. A B C D A 6 1 x 2 3 4 5 6 7 B y 1 4 21 6 3 7 7 5 9 6 11 C xyx 4 12 21 28 45 661 D E F G Fitted y Residual XResidual 1 1.934066 2.065934 2.065934066 4 3.868132 2.131868 4.263736264 95.802198 1.197802 3.593406593 16 7.736264 -0.73626 -2.94505495 25 9.67033 -0.67033 -3.35164835 36 11.6044 -0.6044 -3.62637363 214 3 2 4 3 7 5 4 65 9 7 16 A2°B2 =A3 83 A4 84 =A5 BS =A6*16 =A7*87 A2 A2 =A3 A3 A4 A4 AS AS =A6 A6 -A7 A7 E Fitted y 1 =$C$10 A2 -$C$10*A3 |=$C$10*A4 -$C$10*A5 ESC$10%A6 $C$10*A7 F Residual =B2-E2 =B3-E3 =84-E4 =B5-ES =B6-E6 =B7-E7 G x Residual =F2A2 =F3*A3 =F4A4 =F5*AS I=F6"A6 =F7* A7 7 4 11 8 =SUM(F2:F7) -SUM(G2:57) 91 3.384615 0.0000 9 Total 10 Beta 2 176 1.93 9 Total 10 Beta 2 -SUM(C2:07) -SUM(D2:07) =C9/D9

For part (c), the table with calculated numbers is shown below. (please make sure to sufficiently drag the formula. I have dragged the formula for 40 rows. Then you can plot sum of squares (column H) in y- axis and beta_2 from column A to get the plot I have shown in part (c). 2 6 3 7 4 7 5 9 1 4 3 Beta 2 1 9 1.05 8.7025 1.1 8.41 7 1.15 8.1225 1.2 7.84 1.25 7.5625 1. 3 7 .29 11 1.357.0225 16 15.21 1 4.44 13.69 12.96 12.25 11.56 10.89 16 14.8225 13.69 12.6025 11.56 105635 10.5625 9.61 8.7025 9 7.84 6.75 5.76 4.84 4 3.24 2.56 16 14.0625 12.25 10.5625 9 7.5625 6.25 5.0625 6 11 Error Sum of square 25 22.09 82.2275 19.36 74,91 16.81 67.5475 14.44 12.25 54.1875 10.24 48.19 8.41 42.6475 And the formula view of the table is as follows: 2 Y 3 Beta_2 5 =A4+0.05 6 A5+0.05 7 A6+0.05 8 A7+0.05 9 =A8+0.05 10 =A9+0.05 11 =A10+0.05 12 A1140.05 13 =A1240.05 Error Sum of square =(B$2-$A4B$1)^2 =(C$2-$A4°C$1)^2 =(D$2-$A4"D$1)^2 =(E$2-$44E$1)^2 =(F$2-$44"F$1)^2 =[G$2-$44"G$1)^2 =SUM(B4:G4) =(B$2-$A5*B$1)^2 =(C$2-$AS C$1)^2 =(D$2-$A5"D$1)^2 =(E$2-$45 E$1)^2 =(F$2-$45*F$1)^2 =[G$2-$45*G$1)^2 =SUM(B5:65) =(B$2-$A6*B$1)^2 =(C$2-$A6*C$1)^2 =(D$2-5A6*D$1)^2 =(E$2-$A6*E$1)^2 =F$2-5A6*F$1)^2 =(G$2-5A6*G$1)^2 =SUM(B6:56) (B$2-SA7*B$1)^2 -(C$2-SA7C$1)^2 -(D$2-SA 7*D$1)^2 -(E$2-SA 7*E$1)^2 -(F$2-SA 7*F$1)^2 -(G$2-5A7*G$1)^2 SUM(B7:47) (B$2-SAS*B$1)^2 -(CS2-SAS C$1)^2 -(D$2-SA8*D$1)^2 =(E$2-SAB*E$1)^2 -(F$2-SAB*F$1)^2 =($2-SA8*G$1)^2 =SUM(B8-G8) ={B$2 SA9B$1)^2 =(C$2 SA9°C$1)^2 =(D$2-SASD$1)^2 =(E$2-$A9 E$1)^2 =(F$2-SA9*F$1)^2 =(G$2-$A9*G$1)^2 =SUM(B9:39) ={B$2SA10"B$1)^2 =(C$2-SA10°C$1)^2 =(D$2-SA10'D$1)^2 = E$2- SA10"E$1)^2 =(F$2- SA10"F$1)^2 =[G$2-$410G$1)^2 =SUM(B10-G10) ={B$2-SA11"B$1)^2 =(C$2-SA11"C$1)^2 ={D$2.SA11'D$1)^2 = E$2.$A11°E$1)^2 =[F$2-SA11"F$1)^2 =($2$A11*G$1)^2 =SUM(B11:11) =(B$2-$A12PB$1)^2 =(C$2-SA12°C$1)^2 ={D$2-SA12"D$1)^2 = E$2-SA12E$1)^2 =[F$2-$A12"F$1)^2 =[G$2-$A12"G$1)^2 =SUM(B12:412) =(B$2-$A13*B$1)^2 =(C$2-SA13*C$1)^2 =D$2-$A13"D$1)^2 = E$2-$A13E$1)^2 =[F$2-$A13*F$1)^2 =[G$2-$A13*G$1)^2 =SUM(B13:13)