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# Appendix D

# Coefficient of Determination (R-Squared)

The coefficient of determination () is used in the context of statistical models which we wish to use to predict future outcomes. The is defined as the proportion of the variability in the sample data that is accounted for by the statistical model. The serves as a goodness of fit measure.

For a given data set with observed values and an associate model's values ,

the variability of the data set is measure as the sum of squared differences.

Where

- = the sample average of the observed values
- = the total sum of squares
- = the model (e.g. regression) sum of squares
- = the sum of the squares of the residuals (residuals sum of squares)
- = number of observations

To factor in the number of explanatory variables in the model, the adjusted (or ) is used as a modification.

is defined as follow:

Where:

- = number of explanatory variables
- = number of explanatory variables

**Remarks**

- The adjusted is not a test of the model in the sense of hypothesis testing, but can be used as a tool for model selection

**References**

- Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
- Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740