Introduction
  • In this chapter, we extend the simple linear 
   regression model. Any number of independent 
   variables is now allowed.
  • We wish to build a model that fits the data better 
   than the simple linear regression model.
   • Computer printout is used to help us: 
     – Assess/Validate the model
      • How well does it fit the data?
      • Is it useful?
      • Are any of the required conditions violated?
     – Apply the model
      • Interpreting the coefficients
      • Estimating the expected value of the dependent variable
        Model and Required Conditions
        • We allow for k independent variables to 
           potentially be related to the dependent variable
                                    Coefficients          Random error variable
                   Y =   +  X +  X  + …+  X  + 
                           0     1  1     2  2            k  k
        Dependent variable          Independent variables
                   Multiple Regression for k = 2,
                           Graphical Demonstration
                                  Y         The simple linear regression model
                                            allows for one independent variable, “X”
                                                           Y = 0 + 1X + 
                                                   X               X
                                               +
                                                  1              1
                                         =                     +
                                                               
     Note how the straight line       Y      0              
                                                          =  0                             X2
                                                       Y                               + 2
                                                                                    X 
      becomes a plane                                                            1 1
                                                                               +
                                                                            
                                                                         =  0
                                                                       Y
                                                     X2                   X
                                                  + 2                        2
                                                                         2
                                               X                      +
                                            1 1                   X 
                                         +                           1
                                                                1
                                                             +
                                    =  0                   
                                 Y                      =  0                  X
                                                      Y                         1
                                  The multiple linear regression model
                                  allows for more than one independent variable.
                                  Y =   +  X  +  X  + 
                                        0    1 1    2 2 
          X
            2
     Required Conditions for the Error Variable
     • The error is normally distributed.
     • The mean is equal to zero and the standard 
       deviation is constant ( for all possible values 
                             
       of the Xis.
     • All errors are independent.