View source for Coefficient of determination

'''[[Coefficient of determination]]''' is a statistical measure that indicates the proportion of the variation in a dependent variable that is explained by a statistical model. It is commonly denoted by <math>R^2</math> and is widely used in [[Linear regression|linear regression]].

For a regression model with an intercept, the coefficient of determination can be expressed as:

<math>
R^2
=
1-
\frac{\mathrm{SS}_{\mathrm{res}}}
{\mathrm{SS}_{\mathrm{tot}}}
</math>

where:

* <math>R^2</math> = coefficient of determination;
* <math>\mathrm{SS}_{\mathrm{res}}</math> = residual sum of squares;
* <math>\mathrm{SS}_{\mathrm{tot}}</math> = total sum of squares.

==Sum of squares==

The total sum of squares measures the total variation of the observed dependent variable around its mean:

<math>
\mathrm{SS}_{\mathrm{tot}}
=
\sum_{i=1}^{n}
(Y_i-\bar{Y})^2
</math>

The residual sum of squares measures the variation that remains unexplained by the fitted model:

<math>
\mathrm{SS}_{\mathrm{res}}
=
\sum_{i=1}^{n}
(Y_i-\hat{Y}_i)^2
</math>

where:

* <math>Y_i</math> = observed value;
* <math>\hat{Y}_i</math> = fitted value;
* <math>\bar{Y}</math> = mean of the observed values;
* <math>n</math> = number of observations.

The explained sum of squares is:

<math>
\mathrm{SS}_{\mathrm{explained}}
=
\sum_{i=1}^{n}
(\hat{Y}_i-\bar{Y})^2
</math>

For ordinary least-squares regression with an intercept:

<math>
\mathrm{SS}_{\mathrm{tot}}
=
\mathrm{SS}_{\mathrm{explained}}
+
\mathrm{SS}_{\mathrm{res}}
</math>

Therefore:

<math>
R^2
=
\frac{
\mathrm{SS}_{\mathrm{explained}}
}{
\mathrm{SS}_{\mathrm{tot}}
}
</math>

==Interpretation==

An <math>R^2</math> of 0 indicates that the model does not explain any of the variation in the dependent variable relative to a model containing only an intercept.

An <math>R^2</math> of 1 indicates that the fitted model explains all of the observed variation in the dependent variable.

For example:

<math>
R^2=0.70
</math>

can be interpreted, in the corresponding regression setting, as the model accounting for 70% of the sample variation in the dependent variable around its mean.

The remaining 30% is represented by the residual component:

<math>
1-R^2=0.30
</math>

The interpretation should be made in the context of the model and data; <math>R^2</math> does not establish causation.

==Coefficient of determination in simple linear regression==

In a simple linear regression with an intercept, the coefficient of determination is equal to the square of the [[Pearson correlation coefficient]]:

<math>
R^2
=
r_{XY}^2
</math>

where <math>r_{XY}</math> is the Pearson correlation coefficient between the dependent variable and explanatory variable.

Because the correlation coefficient ranges from −1 to +1:

<math>
0\leq R^2\leq1
</math>

for this standard simple-regression setting.

The squaring operation removes the sign of the relationship. Therefore, <math>R^2</math> alone does not indicate whether the relationship is positive or negative.

==Example==

Suppose a regression model produces:

<math>
\mathrm{SS}_{\mathrm{tot}}=1000
</math>

and:

<math>
\mathrm{SS}_{\mathrm{res}}=250
</math>

The coefficient of determination is:

<math>
R^2
=
1-\frac{250}{1000}
=
0.75
</math>

Thus, the model explains 75% of the sample variation in the dependent variable relative to the mean-only benchmark.

The unexplained proportion is:

<math>
1-R^2
=
0.25
</math>

or 25%.

==Adjusted coefficient of determination==

The [[Adjusted R-squared|adjusted coefficient of determination]] adjusts <math>R^2</math> for the number of explanatory variables included in a regression model.

A commonly used formula is:

<math>
\bar{R}^2
=
1-
(1-R^2)
\frac{n-1}{n-k-1}
</math>

where:

* <math>\bar{R}^2</math> = adjusted coefficient of determination;
* <math>R^2</math> = coefficient of determination;
* <math>n</math> = number of observations;
* <math>k</math> = number of explanatory variables.

Unlike ordinary <math>R^2</math>, adjusted <math>R^2</math> can decrease when additional explanatory variables are added if they do not provide sufficient explanatory power relative to the additional model complexity.

==R-squared in financial models==

In finance, <math>R^2</math> is commonly used when evaluating regression models of asset returns.

For example, a market model can be written as:

<math>
R_i
=
\alpha_i
+
\beta_iR_m
+
\epsilon_i
</math>

where:

* <math>R_i</math> = asset return;
* <math>\alpha_i</math> = regression intercept;
* <math>\beta_i</math> = market sensitivity;
* <math>R_m</math> = market return;
* <math>\epsilon_i</math> = residual return.

The resulting <math>R^2</math> indicates the proportion of the variation in the asset's returns explained by the fitted market model over the specified sample period.

A higher <math>R^2</math> indicates that the model explains a larger proportion of the observed variation, while a lower <math>R^2</math> indicates that a larger proportion remains in the residual component.

==R-squared and beta==

<math>R^2</math> and [[Beta (finance)|beta]] measure different aspects of a regression relationship.

Beta measures the estimated sensitivity of the dependent variable to changes in an explanatory variable:

<math>
\beta_i
=
\frac{
\operatorname{Cov}(R_i,R_m)
}{
\operatorname{Var}(R_m)
}
</math>

By contrast, <math>R^2</math> measures the proportion of variation in the dependent variable explained by the regression model.

Therefore:

<math>
\text{Beta}
\rightarrow
\text{Sensitivity}
</math>

<math>
R^2
\rightarrow
\text{Explained Variation}
</math>

==Limitations==

The coefficient of determination has several limitations:

* a high <math>R^2</math> does not establish causation;
* a low <math>R^2</math> does not necessarily mean that a model is useless;
* <math>R^2</math> can increase when additional variables are added to a regression;
* <math>R^2</math> does not by itself indicate whether estimated coefficients are statistically significant;
* <math>R^2</math> does not establish that a model is correctly specified;
* a high <math>R^2</math> can result from spurious relationships in some settings;
* the value of <math>R^2</math> is dependent on the data and model specification.

Consequently, <math>R^2</math> is generally considered alongside other measures such as coefficient estimates, standard errors, statistical tests, residual diagnostics, and economic or substantive reasoning.

==See also==

* [[Linear regression]]
* [[Ordinary least squares]]
* [[Adjusted R-squared]]
* [[Pearson correlation coefficient]]
* [[Correlation]]
* [[Covariance]]
* [[Coefficient of correlation]]
* [[Beta (finance)]]
* [[Alpha (finance)]]
* [[Regression analysis]]
* [[Factor model]]
* [[Statistical significance]]

[[Category:Regression analysis]]
[[Category:Statistics]]
[[Category:Econometrics]]
[[Category:Financial mathematics]]
[[Category:Quantitative finance]]