SIMPLE LINEAR REGRESSION

When we identify the fact that the correlation exists between two variables, we shall develop an estimating equation, known as regression equation or estimating line, i.e., a methodological formula, which helps us to estimate or predict the unknown value of one variable from known value of another variable. In the words of Ya-Lun-Chou, “regression analysis attempts to establish the nature of the relationship between variables, that is, to

study the functional relationship between the variables and thereby provide a mechanism for prediction, or forecasting.” For example, if we confirmed that advertisement expenditure (independent variable), and sales (dependent variable) are correlated, we can predict the required amount of advertising expenses for a given amount of sales or vice-versa. Thus, the statistical method which is used for prediction is called regression analysis. And, when the relationship between the variables is linear, the technique is called simple linear regression.




Hence, the technique of regression goes one step further from correlation and is about relationships that have been true in the past as a guide to what may happen in the future. To do this, we need the regression equation and the correlation coefficient. The latter is used to determine that the variables are really moving together. The objective of simple linear regression is to represent the relationship between two variables with a model of the form shown below:

Y =  β0 + β1 X+ ei

wherein

Y  =  value of the dependent variable,

β0  =  Y-intercept,

β1  =  slope of the regression line,

X  =  value of the independent variable,

ei  =  error term (i.e., the difference between the actual Y value and the value of Y predicted by the model.

i  =  represents the observation number, ranges from 1 to n.  Thus Y3 is the third observation of the dependent variable and X6 is the sixth observation of the independent variable.

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