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Statistical Methods in Economics

Notes (b) Calculation of Regression coefficients from the assumed means 2 and 18

X {X– ) 2 Y (Y – 18)
d x d x 2 d y d y 2 dd
x y
1 – 1 1 10 – 8 64 + 8
2 0 0 20 + 2 4 0
3 + 1 1 15 – 3 9 – 3
4 + 2 4 25 + 7 49 + 14
5 + 3 9 30 + 12 144 + 36
2
2
Σ X = 15 Σd = 5 Σd = 15 Σ Y = 100 Σ d = 10 Σ d = 270 Σdd = 55
y
x y
x
x
y
N Σ dd – ( x )Σ ( d d ) Σ
b xy = y x y
N Σ 2 – ( y d y ) Σ d 2
2
Σdd x y y = 270, N = 5
x y = 55, Σd = 5, Σ d = 10, Σ d
()( ) ()( ) 275 – 50 225
555 – 5 10
= 2 = = = 0.18
) ( )
()(270–10 1,350 – 100 1,250
5
N Σ dd y – ( x x ) d ( Σ Σ y ) d
b yx = 2
N Σ d 2 ( x d x ) – Σ

(
)
( 555 – 5 ) ( )10 275 – 50 225
= 2 = = = 4.5
515 –
()( ) () 5 75 – 25 50
Thus the value of regression coefficients is the same by both the methods.
Example 5: Find the regression coefficient of X on Y and Y on X for the following data:
X 3 2 – 1 6 4 – 2 5
Y 5 13 12 – 1 2 20 0

Solution: Calculation of Regression Coefficients
X (X – 2) Y (Y – 7)

d x d x 2 d y d y 2 dd
x y
3 + 1 1 5 – 2 4 – 2
2 0 0 13 + 6 36 0
– 1 – 3 9 12 + 5 25 – 15
6 + 4 16 – 1 – 8 64 – 32
4 + 2 4 2 – 5 25 – 10
– 2 – 4 16 20 +13 169 – 52
5 + 3 9 0 – 7 49 – 21
2
2
x y = – 132
Σ X = 17 Σ d = + 3 Σ d = 55 Σ Y = 51 Σ d y = + 2 Σ d y = 372 Σ dd
x
x
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