What is an unbiased estimate of population R-square? I am interested in getting an unbiased estimate of $R^2$ in a multiple linear regression.
On reflection, I can think of two different values that an unbiased estimate of $R^2$ might be trying to match. 


*

*Out of sample $R^2$: the r-square that would be obtained if the regression equation obtained from the sample (i.e., $\hat{\beta}$) were applied to an infinite amount of data external to the sample but from the same data generating process.

*Population $R^2$: The r-square that would be obtained if an infinite sample were obtained and the model fitted to that infinite sample (i.e., $\beta$) or alternatively just the R-square implied by the known data generating process.


I understand that adjusted $R^2$ is designed to compensate for the overfitting observed in sample $R^2$. Nonetheless, it's not clear whether adjusted $R^2$ is actually an unbiased estimate of $R^2$, and if it is an unbiased estimate, which of the above two definitions of $R^2$ it is aiming to estimate.
Thus, my questions:


*

*What is an unbiased estimate of what I call above out of sample $R^2$?

*What is an unbiased estimate of what I call above population $R^2$?

*Are there any references that provide simulation or other proof of the unbiasedness?

 A: Evaluation of analytic adjustments to R-square
@ttnphns referred me to the Yin and Fan (2001) article that compares different analytic methods of estimating $R^2$. As per my question they discriminate between two types of estimators.
They use the following terminology:


*

*$\rho^2$: Estimator of the squared population multiple correlation coefficient 

*$\rho_c^2$: Estimator of the squared population cross-validity coefficient


Their results are summarised in the abstract:

The authors conducted a Monte Carlo experiment to investigate the
  effectiveness of the  analytical formulas for estimating $R^2$ shrinkage,
  with 4 fully crossed factors (squared  population multiple correlation
  coefficient, number of predictors, sample size, and  degree of
  multicollinearity) and 500 replications in each cell. The results
  indicated  that the most widely used Wherry formula (in both SAS and
  SPSS) is probably not  the most effective analytical formula for
  estimating $\rho^2$. Instead, the Pratt formula  and the Browne formula
  outperformed other analytical formulas in estimating $\rho^2$ and $\rho_c^2$,
  respectively.

Thus, the article implies that the Pratt formula (p.209) is a good choice for estimating $\rho^2$:
$$\hat{R}^2=1 - \frac{(N-3)(1 - R^2)}{(N-p-1)} \left[ 1 + \frac{2(1-R^2)}{N-p-2.3} \right]$$
where N is the sample size, and p is the number of predictors.
Empirical estimates of adjustments to R-square
Kromrey and Hines (1995) review empirical estimates of $R^2$ (e.g., cross-validation approaches). They show that such algorithms are inappropriate for estimating $\rho^2$. This makes sense given that such algorithms seem to be designed to estimate $\rho_c^2$. However, after reading this, I still wasn't sure whether some form of appropriately corrected empirical estimate might still perform better than analytic estimates in estimating $\rho^2$.
References


*

*Kromrey, J. D., & Hines, C. V. (1995). Use of empirical estimates of shrinkage in multiple regression: a caution. Educational and Psychological Measurement, 55(6), 901-925.

*Yin, P., & Fan, X. (2001). Estimating $R^2$ shrinkage in multiple regression: A comparison of different analytical methods. The Journal of Experimental Education, 69(2), 203-224. PDF
A: We made some progress regarding this. Thus here an updated answer.
Assumptions
All existing comparisons for this  (thus also the results summarized by Jeromy, and all claims I make dependent on all regression assumptions being met, and the predictors being multivariate normally distributed. For a more formal treatment, see Shieh (2007) or Karch (2020)
1. Population $R^2$
As Jeremey notes, this is commonly called squared population multiple correlation coefficient $\rho^2$.  It is defined as the amount of variance explained by the true regression model. The unbiased estimator has been derived theoretically in Olkin & Pratt (1958) and is correspondingly known as Olkin-Pratt estimator. Until recently, this estimator was not available in any software as it's nontrivial to compute it. However, I showed how to do that (Karch, 2020) and provide an R package for extracting it from a fitted regression model.
Note that unbiasedness might not be what you want. Especially Bayesians tend to get angry at the Olkin-Pratt estimator as it can return negative values, which of course has a posterior probability $0$. At the same time, sometine returning negative values is needed for unbiasdness. If you consider other optimality criteria, most notably, lowest MSE, the results change dramatically, see Karch (2020),  Which is better: r-squared or adjusted r-squared?, and Would the real adjusted R-squared formula please step forward?.
2. Out of sample $R^2$
First a warning: The squared population cross-validity coefficient $\rho_c^2$ is not equivalent to the description in the question

the r-square that would be obtained if the regression equation obtained from the sample (i.e., $\hat{\beta}$) were applied to an infinite amount of data external to the sample but from the same data generating process.

If we call the value described in the quote $\rho_c(\hat{\beta})$, $\rho_c^2$ is actually the expectation $E[\rho_c(\hat{\beta})]$ (Shieh,2007). After consulting the latest paper on the issue (Shieh, 2007), it seems that no unbiased estimator for this exists yet.
