Coefficient of determination

Okuns law quarterly differences

Thiel-Sen estimator

Coefficient of Determination

Screen shot proj fig

Bias and variance contributing to total error

Geometric R squared

Coefficient of determination, commonly denoted as \(R^2\) or \(r^2\), is a statistical measure that assesses the goodness of fit of a model. In regression analysis, it quantifies the proportion of the variance in the dependent variable that is predictable from the independent variable(s). The coefficient of determination is a key output of regression analysis and is used to analyze how well a regression model fits the actual data.

Definition

The coefficient of determination, \(R^2\), is calculated as the square of the correlation coefficient (\(r\)) in simple linear regression, which represents the linear correlation between the actual and predicted values of the dependent variable. In multiple regression, \(R^2\) is computed as 1 minus the ratio of the residual variance to the total variance of the dependent variable, mathematically represented as:

\[R^2 = 1 - \frac{SS_{res}}{SS_{tot}}\]

where \(SS_{res}\) is the sum of squares of residuals (the variance in the dependent variable not explained by the model), and \(SS_{tot}\) is the total sum of squares (the total variance in the dependent variable).

Interpretation

A higher \(R^2\) value indicates a better fit between the model and the data. An \(R^2\) of 1 implies that the regression model perfectly fits the data, with all points lying on the regression line, whereas an \(R^2\) of 0 indicates that the model does not explain any of the variability of the response data around its mean.

However, it is important to note that a high \(R^2\) does not necessarily mean that the model is appropriate or accurate. Other statistical tests should be used in conjunction to assess the model's validity and reliability.

Limitations

One limitation of \(R^2\) is that it can only increase as more predictors are added to a regression model, which can lead to overfitting. This means that the model becomes too complex and starts to capture the random noise in the data rather than the underlying relationship. To address this issue, adjusted \(R^2\) is often used in multiple regression analyses, as it adjusts for the number of predictors in the model, providing a more accurate measure of the goodness of fit.

Applications

The coefficient of determination is widely used in various fields such as economics, finance, engineering, and the social sciences to quantify the explanatory power of regression models. It helps researchers and analysts understand the strength of the relationship between the dependent and independent variables and to compare the performance of different models.

See Also

• Regression analysis
• Correlation coefficient
• Adjusted R-squared
• Sum of squares

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