Interpreting R-Squared in Robust Regression
Summary
The document explains that R-squared can be calculated for a robust linear regression, but its interpretation depends on how the fit statistic is defined. The conventional measure compares the sum of squared residuals with the total variation around the dependent variable’s mean, so it can be affected by extreme observations even when the regression method reduces their influence on coefficient estimates.
A weighted R-squared could reflect the weights used by a robust fitting procedure and might show a better fit when an outlier is downweighted. The document does not establish whether MATLAB’s robust regression output uses such a weighted measure, and recommends checking the software’s calculation. It also cautions that R-squared alone cannot establish model adequacy: a useful model can have a low value, while a high value does not guarantee a sound fit.
Key ideas
- R-squared summarizes fit by comparing residual variation with total variation in the response.
- The conventional squared-error calculation can remain sensitive to outliers in robust regression.
- A weighted R-squared may better reflect how a robust procedure downweights observations.
- The document does not confirm how MATLAB computes R-squared for robust fits.
- R-squared alone cannot determine whether a regression model is adequate.
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# Robust regressions: how to interpreter R^2
# Robust regressions: how to interpreter R^2
I am not sure if this is the right site, I hope so! But it is half coding half econometric so I guess the answer can only be given from finance professional.
In matlab it is possible to run robust regressions by putting the option 'RobustOpts','on' in fitlm.
In the output there is also the $R^2$ and adjusted-$R^2$. However, my question is: $R^2$ is a typical concept of min squared residuals, does this apply to robust regressions?
And what is exactly the $R^2$ in that specific context?
## Answer by John (score 1)
https://quant.stackexchange.com/a/31858
$R^{2}$ is a measure of goodness of fit. You can calculate it regardless of the type of linear regression model.
However, it may not always have value. For instance, if you have an extreme outlier in your data, then a classic $R^{2}$ will typically be lower than you expect (because there is variation in the data that your model is not picking up).
Alternately, you can calculate a weighted $R^{2}$ based on how the robust regression is performed. Assuming Matlab chooses weights to effectively ignore the outlier and treat the other data the same, then a weighted $R^{2}$ would be higher. That being said, I don't know if that's how Matlab calculates it or not. It would be simple enough to verify.
You might also find the discussion here informative (and how to calculated weighted $R^{2}$).
## Answer by user18663 (score 0)
https://quant.stackexchange.com/a/31859
R-squared is a statistical measure of how close the data are to the fitted regression line. It is also known as the coefficient of determination, or the coefficient of multiple determination for multiple regression.
It is also a statistical tool which measures the goodness of fit for a linear model.
R2=1−SSE/SST, where SSE is the sum of squared error (residuals or deviations from the regression line) and SST is the sum of squared deviations from the dependent's Y mean.
And yes you may apply it to robust regression but there are limitations to it as it doesn't indicate whether a regression model is adequate. You can have a low R-squared value for a good model, or a high R-squared value for a model that does not fit the data.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.