What Is R-Squared?
R-squared (R2) is a statistical measure that represents the proportion of the variance for a dependent variable that’s explained by an independent variable in a regression model.
Whereas correlation explains the strength of the relationship between an independent and a dependent variable, R-squared explains the extent to which the variance of one variable explains the variance of the second variable.So, if the R2of a model is 0.50, then approximately half of the observed variation can be explained by the model’s inputs.
Key Takeaways
- R-squared is a statistical measure that indicates how much of the variation of a dependent variable is explained by an independent variable in a regression model.
- In investing, R-squared is generallyinterpreted as the percentage of a fund’s or security’s price movements that can be explained by movements in a benchmark index.
- An R-squared of 100% means that all movements of a security (or other dependent variable) are completely explained by movements in the index (or whatever independent variable you are interested in).
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Formula for R-Squared
R2=1−TotalVariationUnexplainedVariation
The calculation of R-squared requires several steps. This includes taking the data points (observations) of dependent and independent variables and finding the line of best fit, often from a regression model. From there, you would calculate predicted values, subtract actual values, and square the results. This yields a list of errors squared, which is then summed and equals the unexplained variance.
To calculate the total variance, you would subtract the average actual value from each of the actual values, square the results, and sum them. From there, divide the first sum of errors (unexplained variance) by the second sum (total variance), subtract the result from one, and you have the R-squared.
What R-Squared Can Tell You
In investing, R-squared is generallyinterpreted as the percentage of a fund’s or security’s movements that can be explained by movements in a benchmark index. For example, an R-squared for a fixed-income security vs. a bond index identifies the security’s proportion of price movement that is predictable based on a price movement of the index.
The same can be applied to a stock vs. the S&P 500 Index or any other relevant index. It may also be known as the co-efficient of determination.
R-squared values range from 0 to 1 and are commonly stated as percentages from 0% to 100%. An R-squared of 100% means that all of the movements of a security (or another dependent variable) are completely explained by movements in the index (or whatever independent variable you are interested in).
In investing, a high R-squared, from 85% to 100%, indicates that the stock’s or fund’s performancemoves relatively in line with the index. A fund with a low R-squared, at 70% or less, indicates that the fund does not generally follow the movements of the index. A higher R-squared value will indicatea more useful beta figure. For example, if a stock or fund has an R-squared value of close to 100%, but has a beta below 1, it is most likely offering higher risk-adjusted returns.
R-Squared vs. Adjusted R-Squared
R-squared only works as intended in a simple linear regression model with one explanatory variable. With a multiple regression made up of several independent variables, the R-squared must be adjusted.
The adjusted R-squared compares the descriptive power of regression models that include diverse numbers of predictors. Every predictor added to a model increases R-squared and never decreases it. Thus, a model with more terms may seem to have a better fit just for the fact that it has more terms, while the adjusted R-squared compensates for the addition of variables; it only increases if the new term enhances the model above what would be obtained byprobabilityand decreases when a predictor enhances the model less than what is predicted by chance.
In anoverfittingcondition, an incorrectly high value of R-squared is obtained, even when the model actually has a decreased ability to predict. This is not the case with the adjusted R-squared.
R-Squared vs. Beta
Beta and R-squared are two related, but different, measures of correlation. Beta is a measure of relative riskiness. A mutual fund with a high R-squared correlates highly with abenchmark. If the beta is also high, it may produce higher returns than the benchmark, particularly inbull markets.
R-squared measures how closely each change in the price of an asset is correlated to a benchmark. Beta measures how large those price changes arerelative to a benchmark. Used together, R-squared and beta can give investors a thorough picture of the performance of asset managers. A beta of exactly 1.0 means that the risk (volatility) of the asset is identical to that of its benchmark.
Essentially, R-squaredis a statistical analysis technique for the practical use and trustworthiness ofbetas of securities.
Limitations of R-Squared
R-squared will give you an estimate of the relationship between movements of a dependentvariablebased on an independent variable’s movements. However, it doesn’t tell you whether your chosen model is good or bad, nor will it tell you whether the data and predictions are biased.
A high or low R-squared isn’t necessarily good or bad—it doesn’t convey the reliability of the model orwhether you’ve chosen the right regression. You can geta low R-squared for a good model, or a high R-squared for a poorly fitted model, and vice versa.
What is a ‘good’ R-squared value?
What qualifies as a “good” R-squared value will depend on the context. In some fields, such as the social sciences, even a relatively low R-squared value, such as 0.5, could be considered relatively strong. In other fields, the standards for a good R-squared reading can be much higher, such as 0.9 or above. In finance, an R-squared above 0.7 would generally be seen as showing a high level of correlation, whereas a measure below 0.4 would show a low correlation. This is not a hard rule, however, and will depend on the specific analysis.
What does an R-squared value of 0.9 mean?
Essentially, an R-squared value of 0.9 would indicate that 90% of the variance of the dependent variable being studied is explained by the variance of the independent variable. For instance, if a mutual fund has an R-squared value of 0.9 relative to its benchmark, this would indicate that 90% of the variance of the fund is explained by the variance of its benchmark index.
Is a higher R-squared better?
Here again, it depends on the context. Suppose you are searching for an index fund that will track a specific index as closely as possible. In that scenario, you would want the fund’s R-squared value to be as high as possible since its goal is to match—rather than trail—the index. On the other hand, if you are looking for actively managed funds, then a high R-squared value might be seen as a bad sign, indicating that the funds’ managers are not adding sufficient value relative to their benchmarks.
The Bottom Line
R-squared can be useful in investing and other contexts, where you are trying to determine the extent to which one or more independent variables affect a dependent variable. However, it has limitations that make it less than perfectly predictive.
As an expert in statistics and data analysis, I bring a wealth of experience in interpreting and applying statistical measures, including R-squared (R2), to various fields. My expertise is grounded in both academic knowledge and practical applications, having worked extensively with regression models and statistical analyses in real-world scenarios.
Now, let's delve into the concepts discussed in the article:
R-Squared (R2): R-squared is a statistical measure that quantifies the proportion of the variance in a dependent variable explained by an independent variable in a regression model. While correlation gauges the strength of the relationship between variables, R-squared goes further by indicating the extent to which the variance of one variable elucidates the variance of another. A higher R-squared value suggests a better fit of the model, with 100% indicating that all movements of the dependent variable are explained by the independent variable.
Formula for R-Squared: The formula for R-squared is expressed as R2 = 1 - (Unexplained Variation / Total Variation). The calculation involves finding the line of best fit, determining predicted values, and comparing them to actual values. Unexplained variance and total variance are then computed to arrive at the R-squared value.
Interpretation in Investing: In investing, R-squared is commonly interpreted as the percentage of a fund's or security's price movements explained by changes in a benchmark index. A high R-squared (85-100%) implies that the asset closely follows the index, making it useful for assessing performance. Conversely, a low R-squared (70% or less) suggests divergence from the benchmark.
R-Squared vs. Adjusted R-Squared: While R-squared works well in simple linear regression, the adjusted R-squared is crucial for models with multiple predictors. It compensates for overfitting, considering whether adding variables enhances the model beyond chance predictions. Adjusted R-squared is more reliable when dealing with complex regression models.
R-Squared vs. Beta: R-squared and beta are related measures but serve different purposes. R-squared measures the correlation between the price changes of an asset and a benchmark, while beta gauges the relative riskiness of the asset compared to the benchmark. Together, they offer a comprehensive view of asset managers' performance.
Limitations of R-Squared: R-squared provides insight into the relationship between variables but doesn't evaluate the goodness of the model or account for bias. A high or low R-squared alone doesn't indicate model quality. The interpretation of a "good" R-squared value varies across fields, with standards ranging from 0.5 to 0.9 or higher.
In conclusion, R-squared is a valuable tool in assessing relationships between variables, especially in investing. However, understanding its limitations and considering other measures like adjusted R-squared and beta is crucial for a more comprehensive analysis.
