Reliability of the Data
With causal relationship forecasting we are concerned with how much of the changes in the dependent variable are being “explained” by changes in the independent variable. This is measured by the variance. The greater the proportion of the variance that can be explained by the independent variable. the stronger the relationship. The coefficient of determination (2) measures the proportion of the variability in the dependent variable that can be explained by changes in the independent variable, and is calculated as follows:
v, = Actual value of Y that has been observed for a given value of X
Y = Arithmetic mean for all values of y
Si = Value of r corresponding to a given value of X that has been calculated from the
regression equation The relationship between these variables is shown in Exhibit 9.16. In the equation above. the first term in the numerator and the term in the denominator are the same
This term represents the total variation of the Y variable around the arithmetic mean Y. The second term in the numerator represents the error or that variation in the Y variable that cannot be explained by the regression equation. Thus the numerator represents that amount of variation that can be explained by the regression equation. and the denominator represents the total variation. As stated above. the coefficient of determination therefore measures the proportion of variation in Y that can be explained by changes in X. Another measure for evaluating the reliability of a regression forecast is the mean squared error (MSE). Using the same notation as above. the MSE is calculated as follows:
where n = the number of observations. The following example will demonstrate the use of both of these terms.