The coefficient of determination ($R^2$) is used in the context of statistical models which we wish to use to predict future outcomes. The R^2 is defined as the proportion of the variability in the sample data that is accounted for by the statistical model. The $R^2$ serves as a goodness of fit measure.

For a given data set with observed values ${y_i}$ and an associate model's values $\{ \widehat{y_i} \}$, the variability of the data set is measure as the sum of squared differences.

$$R^2 =\frac{SS_{reg}}{SS_{tot}}=1-\frac{SS_{err}}{SS_{tot}}$$Where

$$SS_{tot} = \sum_{i=1}^N{(y_i-\overline{y})^2}$$ $$SS_{reg} = \sum_{i=1}^N{(\widehat{y_i}-\overline{y})^2}$$ $$SS_{err} = \sum_{i=1}^N{(y_i - \widehat{y_i})^2}$$- $\overline{y}$= the sample average of the observed values
- $SS_{tot}$= the total sum of squares
- $SS_{reg}$= the model (e.g. regression) sum of squares
- $SS_{err}$= the sum of the squares of the residuals (residuals sum of squares)
- $N$= number of observations

To factor in the number of explanatory variables in the model, the adjusted $R^2$ (or $\overline{R}^2$) is used as a modification.

$\overline{R}^2$ is defined as follow:

$$\overline{y}^2=1-(1-R^2)\frac{N-1}{N-p-1}=1-\frac{(N-1)SS_{err}}{(N-p-1)SS_{tot}}$$Where:

- $p$ = number of explanatory variables
- $N$ = number of explanatory variables

**Remarks**

- The adjusted $R^2$ is not a test of the model in the sense of hypothesis testing, but can be used as a tool for model selection

## References

- D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
- James Douglas Hamilton; Time Series Analysis; Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896
- Tsay, Ruey S.; Analysis of Financial Time Series; John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740
- Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848
- Walter Enders; Applied Econometric Time Series; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568

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