The purpose of the regression equation is to estimate

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are the parameters of the regression equation. The purpose of the regression equation is to estimate the values of one variable (dependent variable) based on the known values of the other variable (independent variable). Expressed in another way, the regression equation may be used to predict the values of one variable (dependent), given the values of the other variable (independent). 18.3.3.1. The Least Square Method The least squares method is used to fit a straight line to the points on the scatter diagram. If it were possible to draw one straight line passing through every point on the scatter diagram, then the fit would be perfect. This is almost impossible, as most of the points do not fall directly on the line. The vertical distance, from a point to the line is called the deviation (both positive or negative) of that point from the line. The method of least squares chooses a line such that the sum of the squares of these deviations is minimum. The general form of a linear regression model is i i i Y a bX e where i e , 1,2,3, ..... i n , is the random error term corresponding to the observation and a and b are the regression parameter. The error term represents the amount of the observed value of i Y which deviates from its estimated value. The regression model discussed here is built upon the following underlying assumptions: i. The regression function is linear. If X and Y are not linearly related, it is sometimes possible to transform one variable or both variables, so that a linear relationship is established. This aspect has been discussed later in the lesson. Further, non-linear regression analysis, which is an extension of the linear regression analysis, also uses the concepts of the least squares method. ii. The conditional probability distributions of Y (dependent variable), given X (independent variable), are normally distributed for all X . As the independent variable, X , in the regression analysis is treated as predetermined, no assumption is required with respect to the conditional probability distribution of X given Y iii. All the conditional standard deviations are equal. This characteristics, in regression analysis, is referred to as ‘homeoscedasticity’, meaning equal variance.
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164 iv. The error terms, i e , are independent random variables, with ( ) 0 i E e and variance 2 ( ) i e for all 1,2,3, ..... i n . The least squares regression method has the following characteristics: i. The sum of the deviations of the observed values of Y from the estimated straight line is equal to zero. That is, 0 i i Y Y ii. The sum of the squared deviations from the least squares straight line is less than the sum of the squared deviations that would result if computed from any other line of that type fitted to the same sample data. That is, 2 0 i i Y Y is the minimum under the least squares straight line, where i Y is the estimated value of i Y using the least squares method. From the straight line model of expression, we can write 2 2 1 1 n n i i i i i e Y a bX We estimate the values of a
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