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I have a very simple question. I know that the R-squared is the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model.

I also know that the p-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.

My question essentially is, is the value of the R-squared only dependent on the coefficients? Or also on the significance of the coefficients? In other words, does an increase in the p-value of a Beta estimate, ceteris paribus (everything else equal) affect the R-squared?

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  • $\begingroup$ "Does an increase in the p-value of a Beta estimate, ceteris paribus (everything else equal) affect the R-squared?" this seems me a bad posed question. If "everything else equal" is not clear why p-value increase. $\endgroup$
    – markowitz
    Nov 25 '20 at 12:09
  • $\begingroup$ @markowitz Please feel free to suggest a better posed question.. $\endgroup$
    – Tom
    Nov 25 '20 at 12:59
  • $\begingroup$ From your reply It seems me that a better title would be: “usual vs robust standard error, affect also $R^2$ value in regression estimated with OLS?”. As you noted the reply is no. $\endgroup$
    – markowitz
    Nov 25 '20 at 13:11
  • $\begingroup$ @markowitz Well, that was not really the point of the question though. I wanted to know if the p-value was somehow connected the R2. My answer, was simply a way I thought I could check it with. Your are suggesting that I base my question on the answer. That would not really help anyone who has the same question as I have right? $\endgroup$
    – Tom
    Nov 25 '20 at 13:18
  • $\begingroup$ If it is so, my initial doubt come back. P-value is an output of estimate not an input. You have to declare what we can move and what not in the comparison. Otherwise you can ask "explain relations between p-value of coefficients and $R^2$ in OLS regression" $\endgroup$
    – markowitz
    Nov 25 '20 at 13:26
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I realised shortly after posting, that my question can be quite easily answered by running a regression with and without robust standard errors.

I found a Stata example that does just that here: https://www.statology.org/robust-standard-errors-stata/

The R-squared stays the same, so the p-value (affected by robust standard errors) does not change the R-squared.

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  • $\begingroup$ If you've changed from OLS to robust standard errors, you're no longer using the same model, so it's certainly not the case that all else is equal. $\endgroup$
    – Eoin
    Nov 25 '20 at 12:01
  • $\begingroup$ Well, the estimates do not change, the R squared does not change. Could you explain then it what ways it matters? $\endgroup$
    – Tom
    Nov 25 '20 at 12:54
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Is there a relation between the p-values of coefficients and the R2 in an OLS regression?

In variables selection context if we add one regressor the $R^2$ increase, or at worst remain the same. So if you add and add regressors, $R^2$ approach to 1.

Moreover, is less known that if the regressor added have a p-value less than $0,32$ (about) also the $R_{adj}^2$ increase. So, this indicator tell us that more significative the regressors are more the $R^2$ increase.

Related: Should $ R^2$ be calculated on training data or test data? How to calculate out of sample R squared?

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