Introduction to econometrics II eco 356 faculty of social sciences course guide course Developers: Dr. Adesina-Uthman

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Introduction to Econometrics ECO 356 Course Guide and Course Material

2.1.3.3.2 Alternative Expressions for b 2 INTRODUCTION TO ECONOMETRICS II
2.1.4.0 SUMMARY

INTRODUCTION TO ECONOMETRICS II

ECO 306

NOUN
50
∑
∑
∑
∑
∑
…[2.19] Noting that
̅
∑
̅
∑
…[2.20] maybe rewritten as
̅
̅
…[2.21] and hence
̅
̅
…[2.22] Substituting for and again noting that ∑
̅ we obtain
∑
∑
( ̅
̅) ̅
…[2.23] Separating the terms involving b
2
and not involving b
2
on opposite sides of the equation, we have
,(∑
) ̅
- ∑
̅̅̅̅
…[2.24] Dividing both sides by 2n,
0
(∑
) ̅
1
(∑
)
̅̅̅̅
…[2.25] Using the alternative expressions for sample variance and covariance, this maybe rewritten as
( ) ( )

( )
( )
…[2.26]
b
2
is from [2.23], b
1
is equally from [2.22]. Those who know about the second-order conditions will have no difficulty confirming that we have minimized RSS.
2.1.3.3.2 Alternative Expressions for b
2

INTRODUCTION TO ECONOMETRICS II

ECO 306

NOUN
51 From the definitions of Cov(X, Y) and Var(X) one can obtain alternative expressions for b
2
( )
( )
∑
(
̅)(
̅)
∑
(
̅)
∑
(
̅)(
̅)
∑
(
̅)
1 2
1
n
i
n
i
xy
x



 

…[2.27] where,
x
X
X
 
y
Y
Y
 

2.1.4.0 SUMMARY
In this unit, you are expected to have learnt the essentials and applications of the concept of simple regression analyses and its estimation.

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