Adjusted-R^2 takes into account the number of regressors. In a linear regression, the more regressors you add, the higher the R^2 is (since adding regressors cannot explain less of the variations in the dependent variable). Adjusted-R^2 adjusts the original R^2 by the degrees of freedom to arrive at a goodness of fit measure that’s independent of the number of independent variables.
R^2 is only useful when you have an unconstrained linear regression. It’s no good in all other regressions: nonlinear, constrained, censored, truncated, discrete, etc.
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