Page 184 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes (4) Useful in Economic and Business Research : Regression analysis is very useful in business
and economic research. With the help of regression, business and economic policies can be
formulated.
12.2 Line of Regression

If the variables in a bivariate frequency distribution are correlated, we observe that the points in a
scatter diagram cluster around a straight line called the line of regression. In a bivariate study, we
have two lines of regression, namely :
1. Regression of Y on X.
2. Regression of X on Y.
Regression of Y on X

The line of regression of Y on X is used to predict or estimate the value of Y for the given value of the
variable X. Thus, Y is the dependent variable and X is an independent variable in this case. The algebraic
form of the line line of regression of Y on X is of the form :
Y= a + bX ... (1)
where, a and b are unknown constants to be determined by observed data on the two variables X and
Y. Let (X , Y ), (X , Y )..., (X , Y ) be N pairs of observations on the variable X and Y. Then, for determining
2
1
1
N
N
2
a and b in equation (1) we make use of the following normal equations :
ΣY = Nab ... (2)
+ΣX
ΣXY = Σ+ ΣXa b X 2 ... (3)
The values ΣY , ΣX , ΣX 2 and ΣXY can be obtained from the given data.
These normal equations are obtained by minimising the error sum of squares according to the principle
of least squares. Solving equations (2) and (3) for a and b, the line of regression of Y on X is completely
determined.
Alternatively
There is another way of finding the algebraic form of line of the regression of Y on X. Line of regression
of Y on X can also be written in the following form :
( YY ) − = r σ Y ( XX ) − ... (4)
σ X

or ( YY ) − = YX ( b − X ) X ... (5)

Here, Y = the mean of Y

X = the mean of X
σ Y = the S.D. of Y

σ X = the S.D. of X
r = the correlation coefficient between X and Y
σ
b YX = r σ Y X = the regression coefficient of Y on X

From observed bivariate data [(X , Y ); i = 1, 2, ... N] the regression coefficient of Y on X, b , can be
i i YX
computed from any of the following formula :

178 LOVELY PROFESSIONAL UNIVERSITY