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R-Squared
see also: alpha and beta
          
       

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  1. Definition
  2. Examples, Types, or Variations
  3. Formula
  4. Related Terms
  5. As Used in the Hedge Fund World
  6. Applications
  7. Misused & Abused
  8. Additional Sources of Information
    1. Books
    2. News
    3. Scholarly Papers
       
 

1.
 

Definition
 
 

In statistics, the coefficient of determination R2 is the proportion of variability in a data set that is accounted for by a statistical model. In this definition, the term "variability" is defined as the sum of squares. There are equivalent expressions for R2 based on an analysis of variance decomposition.

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2.
 

Examples, Types, or Variations
 
  Adjusted R-square is a modification of R-square that adjusts for the number of terms in a model. R-square always increases when a new term is added to a model, but adjusted R-square increases only if the new term improves the model more than would be expected by chance.
 

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3.
 

Formula
 
  A general version, based on comparing the variability of the estimation errors with the variability of the original values, is
{R^{2} = {1-{SS_E \over SS_T}}}.

Another version is common in statistics texts but holds only if the modeled values are obtained by ordinary least squares regression (which must include a fitted intercept or constant term): it is

{R^{2} = {SS_R \over SS_T} }.

In the above definitions,

SS_T=\sum_i (y_i-\bar{y})^2, SS_R=\sum_i (\hat{y_i}-\bar{y})^2, SS_E=\sum_i (y_i - \hat{y_i})^2,

where {y_i},\hat{y_i} are the original data values and modeled values respectively. That is, SST is the total sum of squares, SSR is the regression sum of squares, and SSE is the sum of squared errors. In some texts, the abbreviations SSR and SSE have the opposite meaning: SSR stands for the residual sum of squares (which then refers to the sum of squared errors in the upper example) and SSE stands for the explained sum of squares (another name for the regression sum of squares).

In the second definition, R2 is the ratio of the variability of the modeled values to the variability of the original data values. Another version of the definition, which again only holds if the modeled values are obtained by ordinary least squares regression, gives R2 as the square of the correlation coefficient between the original and modeled data values.

 

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  • i Six Sigma: A mathematical term describing how much variation is being explained by the X. More…
     
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4.
 

Related Terms
 
 

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5.
 

As Used in the Hedge Fund World
 
 

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  • Venture Japan: A measure of the degree to which a hedge fund's returns are correlated to the broader financial market. The result is used to determine whether a hedge fund follows a market-neutral investment strategy. More…


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6.
 

Applications
 
 

R2 is a statistic that will give some information about the goodness of fit of a model. In regression, the R2 coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R2 of 1.0 indicates that the regression line perfectly fits the data.

In some (but not all) instances where R2 is used, the predictors are calculated by ordinary least-squares regression: that is, by minimizing SSE. In this case R-squared increases as we increase the number of variables in the model (R-squared will not decrease). This illustrates a drawback to one possible use of R2, where one might try to include more variables in the model until "there is no more improvement". This leads to the alternative approach of looking at the adjusted R2. The explanation of this statistic is almost the same as R-squared but it penalizes the statistic as extra variables are included in the model. For cases other than fitting by ordinary least squares, the R2 statistic can be calculated as above and may still be a useful measure. However, the conclusion that that R-squared increases with extra variables no longer holds, but downward variations are usually small. If fitting is by weighted least squares or generalized least squares, alternative versions of R2 can be calculated appropriate to those statistical frameworks, while the "raw" R2 may still be useful if it is more easily interpreted. Values for R2 can be calculated for any type of predictive model, which need not have a statistical basis.

Other Resources:

  • Yahoo! Finance: R-squared ranges from 0 to 100 and reflects the percentage of a fund's movements that are explained by movements in its benchmark index. More…
     
  • Commonfund: R-squared can also be used to ascertain the significance of a particular beta or alpha. Generally, a higher R-squared indicates a more reliable beta figure. If the R-squared is lower, then the beta is less relevant to the fund's performance. More…
     

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7.
 

Misused & Abused
 
 

If an R-Squared value is low, and it is used as an inappropriate benchmark in finding Beta, the concluding figure cannot be trusted; and since Alpha is directly calculated using Beta, the Alpha figure will also be erroneous.


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8.
 

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