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Properties of Regression Coefficients

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The fundamental aim of regression analysis is to determine a regression equation (line) that makes sense and fits the representative data such that the error of variance is as small as possible.

Let us consider the line of regression of y on x, that is, y = a + bx The coefficient ‘b’ which is the slope of the line of regression of y on x is called the coefficient of regression of y on x. It represents the increment in the value of the dependent variable y for a unit change in the value of the independent variable x. In other words, it represents the rate of change of y with respect to x. For notational convenience, the slope b, i.e., coefficient of regression of y on x is written as b_{yx}.

Similarly in the regression equation of x on y, that is,x = a + by, the coefficient b represents the change in the value of dependent variable x for a unit change in the value of independent variable y and is called the coefficient of regression of x on y. For notational convenience, it is written as b_{xy}. We now list out some of the properties of these regression coefficients.

Properties of Regression Coefficients:

  1. The correlation coefficient is the geometric mean of two regression coefficients, that is, r = \sqrt{b_{yx} × b_{xy}}
  2. If one regression coefficient is greater than one, then the other regression coefficient must be less than one, because the value of correlation coefficient r cannot exceed one. However, both the regression coefficients may be less than one.
  3. Both regression coefficients must have the same sign (either positive or negative). This property rules out the case that the two regression coefficients have opposite signs.
  4. The correlation coefficient will have the same sign (either positive or negative) as that of the two regression coefficients.
  5. The arithmetic mean of regression coefficients b_{xy} and b_{yx} is more than or equal to the correlation coefficient r, that is, \frac{b_{yx} + b_{xy}}{2}\geq r.
  6. Regression coefficients are independent of origin but not of scale.
  7. The correlation coefficient between two variables x and y is a symmetrical function between x and y, that is, r_{xy}=r_{yx}. However, the regression coefficients are not symmetric functions of x and y, that is, b_{xy}\neq b_{yx}.

References:

Fundamentals of Business Statistics – JK Sharma

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