National open university of nigeria introduction to econometrics I eco 355

2.0.
 
OBJECTIVES 
At the end of this unit, you should be able to:
 

understand the meaning of regression analysis and variance 

know how to calculate the regression analysis and analysis of variance 
3.0.
 
MAIN CONTENT 
3.1. Regression Analysis 
In this unit we will study regression analysis from the point of view of the analysis of 
variance and introduce the reader to an illuminating and complementary way of looking 
at the statistical inference problem. 
In previous discussion, we developed the following identity: 
∑
∑ ̂
∑ ̂
̂
∑
∑ ̂
that is, TSS = ESS + RSS, which decomposed the total sum of squares (TSS) into two 
components: explained sum of squares (ESS) and residual sum of squares (RSS). A study 
of these components of TSS is known as the analysis of variance (ANOVA) from the 
regression viewpoint. 
Associated with any sum of squares is its df, the number of independent observations on 
which it is based. TSS has 
df because we lose 1 df in computing the sample mean 
̅
. RSS has n – 2 df. (Why?) (Note: This is true only for the two-variable regression 
model with the intercept 
present.) ESS has 1 df (again true of the two-variable case 
only), which follows from the fact that ESS 
̂
∑
is a function of 
̂
only, since 
∑
is known. 
However, we can also state that; 

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125 
̂
∑
∑ ̂
 
̂
∑
̂
If we assume that the disturbances u, are normally distributed, which we do under the 
CNLRM, and if the null hypothesis (
) is that 
= 0, then it can be shown that the F 
variable of (4.9.1) follows the F distribution with 1 df in the numerator and (n – 2) df in 
the denominator.
What use can be made of the preceding F ratio? It can be shown that 
( ̂
∑
)
∑