R-Squared: Definition, Formula, Uses, and Limitations (2024)

What Is R-Squared?

R-squared (R²) is a number that tells you how well the independent variable(s) in a statistical model explain the variation in the dependent variable. It ranges from 0 to 1, where 1 indicates a perfect fit of the model to the data.

Understanding R-Squared

R-squared represents the proportion of the variance in the dependent variable that is predictable from the independent variables. A value of 1 implies that all the variability in the dependent variable is explained by the independent variables, while a value of 0 suggests that the independent variables do not explain any of the variability. R-squared should be interpreted alongside other statistics and context, as high R-squared values can sometimes be misleading if the model is overfitted.

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 R-squaredof a model is 0.50, then approximately half of the observed variation can be explained by the model’s inputs.

Formula for R-Squared

$\begin{aligned} &\text{R}^2 = 1 - \frac{ \text{Unexplained Variation} }{ \text{Total Variation} } \\ \end{aligned}$​R2=1−TotalVariationUnexplainedVariation​​

The calculation of R-squared requires several steps. steps. This includes taking the data points (observations) of dependent and independent variables and conducting regression analysis to find the line of best fit, often from a regression model. This regression line helps to visualize the relationship between the variables. From there, you would calculate predicted values, subtract actual values, and square the results. These coefficient estimates and predictions are crucial for understanding the relationship between the variables. 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. This process helps in determining the total sum of squares, which is an important component in calculating R-squared. 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. This is often assessed using measures like R-squared to evaluate the goodness of fit. 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.

Tips for Improving R-Squared

Improving R-squared often requires a nuanced approach to model optimization. One potential strategy involves careful consideration of feature selection and engineering. By identifying and including only the most relevant predictors in your model, you can increase the likelihood of explaining relationships. This process may involve conducting thorough exploratory data analysis or using techniques like stepwise regression or regularization to select the optimal set of variables.

Another way of enhancing R-squared is addressing multicollinearity. Multicollinearity is when independent variables are highly correlated with each other. However, they can distort coefficient estimates and reduce the accuracy of the model. Techniques like variance inflation factor analysis or principal component analysis can help identify and mitigate multicollinearity.

You can also improve r-squared by refining model specifications and considering nonlinear relationships between variables. This may involve exploring higher-order terms, interactions, or transforming variables in different ways to better capture the hidden relationships between data points. In some cases, you'll have to have strong domain knowledge to get able to get this type of insight outside of the model.

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.