# The coefficient of determination in multiple regression

## Abstract

In the case of simple regression analysis, the coefficient of determination measures the proportion of the variance in the dependent variable explained by the independent variable. This coefficient is computed using either the variance of the errors of prediction or the variance of the predicted values in relation to the variance of the observed values on the dependent variable as follows:
$$r_{yx}^2 = \left( {\frac{{Var\left( {\hat y} \right)}} {{Var\left( y \right)}}} \right) = 1 - \left( {\frac{{Var\left( e \right)}} {{Var\left( y \right)}}} \right)$$
This equation for the coefficient of determination in simple regression analysis can easily be extended to the case of multiple regression analysis. The variances of the predicted values and the errors of prediction in simple regression have direct counterparts in multiple regression. In the case of two independent variables, for example, the following equations obtain:
$$\begin{gathered} \hat y = {\mathbf{ }}ua{\mathbf{ }} + {\mathbf{ }}x_1 b_{y1.2} {\mathbf{ }} + {\mathbf{ }}x_2 b_{y2.1} \hfill \\ e = {\mathbf{ }}y{\mathbf{ }} - {\mathbf{ }}\hat y \hfill \\ \end{gathered}$$

In short, the addition of independent variables to the regression model does not affect the equations for computing either the predicted values or the errors of prediction.

## Keywords

Regression Model Multiple Regression Analysis Economic Openness Multiple Regression Model Public Expenditure
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