Transcription of Chapter 3 Multiple Linear Regression Model ... - IIT Kanpur
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Chapter 3 Multiple Linear Regression Model ... - IIT Kanpur
Regression Analysis | Chapter 3 | Multiple Linear Regression Model | Shalabh, IIT Kanpur 1 1 1 Chapter 3 Multiple Linear Regression Model We consider the problem of Regression when the study variable depends on more than one explanatory or independent variables, called a Multiple Linear Regression Model . This Model generalizes the simple Linear Regression in two ways. It allows the mean function ()Ey to depend on more than one explanatory variables and to have shapes other than straight lines, although it does not allow for arbitrary shapes. The Linear Model : Let y denotes the dependent (or study) variable that is linearly related to k independent (or explanatory) variables 12, ,..,kXXX through the parameters 12, ,..,k and we write 11 2 This is called the Multiple Linear Regression Model . The parameters 12, ,..,k are the Regression coefficients associated with 12.
X XXX XX XX XXX XX X XXXX X X Theorem: (i) Let yˆ Xb be the empirical predictor of y. Then yˆ has the same value for all solutions b of X ''.Xb X y (ii) S() attains the minimum for any solution of X ''.Xb X y Proof: (i) Let b be any member in bXXXy I XXXX (') ' (') ' . Since X(') ' ,XX XX X so then
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