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1.
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Definition
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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.
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Examples, Types, or
Variations
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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.
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Formula
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A
general version, based on comparing the variability of the
estimation errors with the variability of the original values, is
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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
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In
the above definitions,
-
where
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.
Other Resources:
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i Six Sigma:
A mathematical term describing how much variation is being
explained by the X.
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4.
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Related Terms
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5.
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As
Used in the Hedge Fund World
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Other Resources:
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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.
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6.
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Applications
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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:
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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.
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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.
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7.
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Misused & Abused
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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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Additional Sources of Information
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Books
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News
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Scholarly Papers
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Back to Terms
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