Page 105 - Basic Statistics

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were,

 n  X 1 i  X i 2 . . .  X p i 
 
 2 
  X 1 i  X 1 i  X 1 i X i 2 . . .  X 1 i X p i 
 2 
  X i 2  X 1 i X i 2  X i 2 . . .  X i 2 X p i 

X’X =  . . . . 
 
 . . . . 
 
 . . . . 
 2 
  X p i  X 1 i X p i  X i 2 X p i . . .  X p i 



 

  Y 
 i 
  X 1 i Y i 
 
  X i 2 Y i 
 
X’Y =  . 
 . 
 
 . 
 X Y 
 

 p i i 

Unique solution to the normal equations (if any) is

ˆ
β = ( X’X ) ( X’Y ) (G.4)
-1

Unique solution of the normal equations exist, if the inverse of X’X exists

(nonsingular). Nonsingular matrix X achieved if its full rank, scilicet there is no

linear dependence between the independent variables (the columns are linearly
independent vectors).

Having obtained the estimates of the regression coefficients or estimated

ˆ
values for vector , i.e β the estimation of the regression equation has been
ˆ
ˆ
obtained, i.e Y = X β .

~~* CHAPTER 5 THE MULTIPLE LINEAR REGRESSION MODEL *~~