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# lecture4solutions - ISYE 6414 Spring 2009 Solution lecture...

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ISYE 6414 - Spring 2009 Solution lecture 4 The Linear Model: OLS Regression - Diagnostics Problem 3.1 Pag.146 1. The residuals e i , i = 1 , 2 , ...n are defined as e i = Y i - ˆ Y i , and the semstudentized residuals are the standarized residuals by the inverse of the variance of the error term: e * i = e i p ( MSE . 2. E { i } = 0 is one of the assumptions of the model that make that the line β 0 + β 1 X to be the expectation of the random variable Y as a function of X . n i =1 e i = 0 is a property derived from the OLS estimation, and is not a consequence of the first statement. 3. The error term ( ) is an unobservable random varaible defined in the model Y = β 0 + β 1 X + . The residual is the observed difference between the Y and the estimated mean of Y using OLS. Problem 3.2 Pag.146 1. residual plot when variance decreases with X : figure 1. Figure 1: Residuals pattern when varaince decreases with X. 2. residual plot when the expectation of Y is a U shape funtion on X : figure 2. Problem 3.3 Pag.146 1. BoxPlot for X: figure 3. 2. Dot plot for the residuals: figure 4. 3. Plot residuals vs. estimated means: figure 5. 4. Q-Q plot for normality of the residuals: figure 6. 5. H 0 : residuals Normal vs. H a : residualsarenotNormal r = 0 . 97373. Given that r < 0 . 987, H 0 is rejected. 1

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Figure 2: Residuals pattern when the function is U shape.
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