Page 196 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes 214 = 40a + 340b ... (iv)
Deducting Eqn. (iv) from (iii)
– 20b = 26
∴ b = – 1.3
Substituting the value of b in Eqn. (i)
30 = 5a + 40 (– 1.3)
5a = 30 + 52 = 82
a = 16.4
Putting the values of a and b in the equation, the regression line of X on Y is
X = 16.4 – 1.3 Y.
Deviations taken from Arithmetic Means of X and Y

The calculations can be very much simplified if instead of dealing with the actual values of X and Y
we take the deviations of X and Y series from their respective means. In such a case the equation Y =
c
a + bX is changed to
Y – Y = ( b ) X – X

or simply y = bx

where y = ( ) Y – Y and x = ( ) X – X
The value of b can be easily obtained as follows:

∑ xy
b = 2
∑ x
The two normal equations which we had written earlier when changed in terms of x and y become
∑ y = N + a b ∑ x ... (i)

∑ xy = ∑ a x ∑ + b x 2 ... (ii)

Since ∑x = ∑y = 0 (deviations being taken from means)
Equation (i) reduces to Na = 0 ∴ a = 0

∑ y
x
Equation (ii) reduces to ∑ xy = ∑b x 2 ∴ b =
∑ x 2
After obtaining the value of b the regression equation can easily be written in terms of X and Y by
substituting for y, ( ) Y – Y and for x,( ) X – X .

Similarly the regression equation X = a + b Y is reduced to x = 0 and the value of b is obtained as
c
follows:
∑ xy
b = 2
∑ y
Example 2: From the following data obtain the regression equation of X on Y, and also that of Y on X:

X 6 2 10 4 8
Y 9 11 5 8 7

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