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On the sample distribution of the adjusted coefficient of determination (R2Adj) in OLS

Thomas Cool
Organization: Thomas Cool Consultancy & Econometrics
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In specification search, one invariably meets with the coefficient of determination R2. Minimizing the sum of squares of the errors is equivalent to maximizing R2. Including more explanatory variables increases the value of R2. Thus there is a temptation to include variables, even at random. The "adjustedR2", denoted as R2Adj, is a first step for correction for inclusion of variables. Maximizing R2Adj is equivalent to minimizing the estimated variance of the errors. The paper first clarifies the definition of the sample parameter Rho2 and then derives the densities of R2 and R2Adj and their expected values as depending upon Rho2. From this follows the confidence interval for Rho2, and it is also shown that R2Adj is "almost" an unbiased estimator of Rho2. The formal deductions are supplemented with algebraic, numerical and graphical routines that allow a quick grasp of the relationships.

*Mathematics > Probability and Statistics

RSquared, AdjustedRSquared, Regression, Goodness of fit, F-Distribution, Confidence interval, Specification search
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Distribution-R2-adj.nb (1.5 MB) - Mathematica Notebook [for Mathematica 5.0]