Does adjusted R^2 have any physical interpretation?

[wrongsaidfred]

Hi David,

It has been stated many times that R^2 tells us the amount of variance in the dependent variable that is explained by the variance in the independent variable. Do the adjusted R^2 terms tell us anything about the actual regression or are they only used for comparison purposes?

Thanks,
Mike

 

[David Harper CFA FRM]

Hi Mike,

I'm don't know for certain. First, it seems like FRM our econometric assignments have generally recommended adjusted R^2 as superior to R^2 in the multivariate regression due to the R^2 tendency to increase unconditionally with the addition of more independent vars. Stock & Watson say, "The adjusted R^2 is useful because it quantifies the extent to which the regressors account for, or explain, the variation in the dependent variable ..." I read this as a direct analogy between R^2 in univariate and adjusted R^2 in multivariate; i.e., % variance in the dependent explained by independents. In this sense, i think the adjusted R^2 very much means to "tell us something" about the actual regression ...

As the same time, I check my Peter Kennedy text (my go to source http://www.amazon.com/Guide-Econometrics-Peter-Kennedy/dp/1405182571) and, clearly, it seems technically complicated. I noted too, here (http://stackoverflow.com/questions/2870631/what-is-the-difference-between-multiple-r-squared-and-adjusted-r-squared-in-a-sin ) the comment that "it [adjusted R^2] does not have the simple summarizing interpretation that R2 has." ... I do notice that, unlike R^2, adjusted R^2 can go negative which seems to support the idea that it is more of an "adjustment" than a pure fit estimate.

So, my naive check of three references (I looked at Gujarati too) does NOT give me access to a clean direct physical interpretation of adjusted R^2; only what appears to be an "adjustment with the intent to give an approximate estimate" that is similar to R^2 in the univariate case.

I hope that helps a little, thanks,

David

 

[wrongsaidfred]

It certainly does help.

Thanks,
Mike

 

[Deepak Chitnis]

Hi @David Harper CFA FRM CIPM, just thing about adjusted R^2, We can also derive the Adjusted R^2=1−(1−R^2) N −1/N −k −1. In our practice exam I found some question that had R^2 and asked for adjusted R^2. We can use this formula for that also. Correct me if I am wrong. And what do you think about this formula?
Thanks a lot

 

[S666]

@Deepak Chitnis that is the same adjusted R^2 formula that I always use to calculate it....so as far as I am concerned that formula is correct.

 

[Dr. Jayanthi Sankaran]

Hi @Deepak Chitnis and @S666,

There are two interpretations that follow from your formula for Adjusted R-Square below:

Adjusted R-Square is computed using the formula 1-((1-R^2)*(N-1)/(N-k-1)).

(1) When the number of observations (N) is small and the number of predictors (k) is large, there will be a much greater difference between R-Square and adjusted R-Square (because the ratio of (N-1)/(N-k-1) will be much less than 1).

(2) By contrast, when the number of observations is very large compared to the number of predictors, the value of R-Square and adjusted R-Square will be much closer because the ratio of (N-1)/(N-k-1) will approach 1.

The adjusted R(square) does not necessarily increase when a new regressor is added, unlike the R(square). Also, adding a regressor has two opposite effects on the adjusted R (square). On the one hand, the adjusted R (square) increases. On the other hand, the factor (n-1)/(n - k -1) increases. Whether the adjusted R (square) increases or decreases depends on which of these two effects is stronger.

Thanks
Jayanthi

 

[Dr. Jayanthi Sankaran]

Thanks Nicole for the star - appreciate it

Jayanthi