R-Squared vs. Adjusted R-Squared: An Overview

R-squared and adjusted R-squared enable investors to measure the performance of a mutual fund against that of a benchmark. Investors ma꧋y also use them to calculate the performance of their portfolio against a given benchmark.

In the world of investing, R-squared is expressed as a percentage between 0 and 100, with 100 signaling perfect correlation and zer🐼o no correlation at all. The figure does not indicate how well a particular group of securities is performing. It only measures how closely the returns align with those of the measured benchmark. It is also backwards-looking—it is not a predictor of future results.

Adjusted R-squared can provide a more precise view of that correlation by also taking into account how many independent variables are added to a particular model against which the 澳洲幸运5开奖号码历史查询:stock index is measured. This is done because such additions of independent variables usually increase the reliability of that model. For investors, 🦋that means the correlation with the index.

R-Squared

澳洲幸运5开奖号码历史查询: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 or variables in a 澳洲幸运5开奖号码历史查询:regression model. R-squared explains how the variance of one variable explains the variance of the second variable. So, if the R² of a model is 0.50, then approximately half of the observed variation can be explained 🌠by the model's inputs.

An R-squared result of 70 to 100 indicates that a given portfolio closely tracks the stock index in question, while a score between 0 and 40 indicates a very low correlation with the index. Higher R-squared values also indicate the reliability of beta readings. Beta 澳洲幸运5开奖号码历史查询:measures the volatility of a security or a portfolio.

While R-squared can return a figure that indicates a level of correlation with an index, it has certain limitations when it comes to measuring the impact of indepenღdent variables on the correlation. This is where adjusted R-squared is useful in measuring correlation.

Adjusted R-Squared

Adjusted R-squared is a modified version of R-squared that has been adjusted for the number of predictors in the model. The adjusted R-squared increases when the new term improves the model more than would be expected by chance. It decreases when a p🔴redictor improves the model by less than expected. The adjusted R-squared is positive, not negative. It is always lower than the R-squared.

Adding more independent variables or predictors to a regression model tends to increase the R-squared value, which tempts makers of the model to add even more variables. This is called overfitting and can return an unwarranted high R-squared value. Adjusted R-squared is used to determine how reliable the model is and how much of its correlation is improved with additional independent variables.

In a portfolio model that has more independent variables, adjusted R-squared will help determine how much of the correlation with the index is du🎃e to the addition of those variables. The adjusted R-squared compensates for the addition of variables and only increases if the new predictor enhances the model above what woꦉuld be obtained by chance. Conversely, it will decrease when a predictor improves the model less than what is predicted by chance.

Key Differences

The most obvious dif♈ference between adjusted R-squared and R-squared is simply that adjusted R-squared considers and tests different independent variables against the stock index and R-squared does not. Because of this, many in♔vestment professionals prefer using adjusted R-squared because it has the potential to be more accurate. Furthermore, investors can gain additional information about what is affecting a stock by testing various independent variables using the adjusted R-squared model.

R-squared, on the other hand, does have its limitations. One of the most essential limits to using this model is that R-squared cannot be used to determine whether or not the coefficient estimates and predictions are biased. Furthermore,  in multip🎐le linear regression, the R-squared ꧙cannot tell us which regression variable is more important than the other.

Adjusted R-Squared vs. Predicted R-Squared

The predicted R-squared, unlike the adjusted R-squared, is used to indicate how well a regression model predicts responses for new observations. So where the adjusted R-squared𓂃 can provide an accurate model that fits the current data, the predicted R-squared determines how likely it is that this model will be accurate for future data.

Example of R-Squared vs. Adjusted R-Squared

Imagꦏine a hedge fund that wants to predict the performance of different stock prices. The fund starts by constructing a model based on known v🌠alues like the stock price, revenue, earnings per share, and so on. Then, the fund tests the model against actual data and calculates the R-squared to determine how well it explains the variation of stock prices. They might also tweak the variables to try to improve the R-squared.

The problem with this approach is that it encourages the researchers to overfit the model by including irrelevant variables. Every time an additional term is introduced, the R-squared of the model either increases or stays the same.

To counteract this tendency, the fund can also calculate adjusted R-squared, a measure that decreases if new variables are added that do not improve the model. Using adjusteꩵd R-squared effectively penalizes the researchers for adding irrelevant variables, 𝓀allowing them to zero in on the ones with the most predictive power.

Special Considerations

R-Squared and Goodness-of-Fit

The basic idea of regression analysis is that if the deviations between the observed values and the predicted values of the linear model are small, the model has well-fit data. Goodness-of-fit is a mathematical model that helps to explain and account for the difference between this observed data and the predicted data. In other words, goodness-of-fit is a statistical hypothesis test to see how well sample data fit a distribution from a population with a 澳洲幸运5开奖号码历史查询:normal distribution.

Low R-Squared vs. High R-Squared Value

One misconception about regression analysis is that a low R-squared value is always a ba🌌d thing. This is not so. For examp🐟le, some data sets or fields of study have an inherently greater amount of unexplained variation. In this case, R-squared values are naturally going to be lower. Investigators can make useful conclusions about the data even with a low R-squared value.

In a different case, such as in investing, a high R-squared value—typically between 85% and 100%—indicates the stock or fund's performance moves relatively in line with the index. This is very useful information to investors, thus a higher R-squared value is necessary for a successful project.

Explain Like I'm Five

Statisticians and economists often con✱struct mathematical models to explain the relationships between different va🦩riables. R-squared measures how well a model explains the variation in actual observed data.

The problem with R-squared is that it doesn't tell you if a variable fails to improve the model. Some researchers might be tempted to add extra variables, even if they don't add much explanatory power. The adjusted R-squared addresses this problem by penalizing variables that do not help explain the data.

The Bottom Line

R-squared and adjusted R-squared enable investors to measure the performance of a mutual fund against that of a benchmark. Many investors have found success using adjuste𒁏d R-squared over R-squared because of its ability to make a more accurate view of the correlation between one variable and another.