R-Squared

R-squared is built from three sums of squares. Total variation, the sum of squared deviations of each observed dependent-variable value from its mean, splits cleanly into explained variation, the sum of squared deviations of the fitted values from that same mean, and unexplained variation, the sum of squared residuals. Because the two components add to the total, the ratio of explained to total variation is bounded by zero and one, and one minus the ratio of residual variation to total variation gives the identical figure.

Every quantity needed sits in a standard ANOVA table: the regression row carries explained variation, the error row carries residual variation, and their sum is the total row. Candidates are frequently handed such a table with one cell missing and asked to reconstruct the fit, so working comfortably between sums of squares, mean squares and degrees of freedom is essential.

In a simple linear regression with one independent variable, R-squared is exactly the square of the sample correlation coefficient between the two variables. Squaring discards the sign, so a fit of 0.64 is consistent with a correlation of either plus or minus 0.8; the slope coefficient, not R-squared, reveals direction. In multiple regression the interpretation generalises: R-squared becomes the squared correlation between the observed and fitted values.

R-squared is also mechanically tied to the overall F-test. Both are monotone functions of the ratio of explained to unexplained variation, so a large R-squared in a small model corresponds to a large F-statistic. The standard error of the estimate moves in the opposite direction: it is derived from residual variation, so tighter scatter around the regression line simultaneously raises R-squared and lowers the standard error.

In equity analysis, regressing a security's excess returns on a market or factor index produces both beta and an R-squared that measures how much of the return series is systematic. The complement, one minus R-squared, approximates the proportion attributable to idiosyncratic risk, which is why concentrated portfolios and market-neutral strategies display low fits against broad benchmarks. Style analysis and benchmark selection lean directly on this logic: an inappropriate benchmark reveals itself through a weak fit.

Elsewhere, R-squared appears in hedge-effectiveness testing, in credit and macro models linking spreads to growth or inflation variables, and in fixed-income duration regressions. It also carries a warning: two trending, unrelated series can produce a very high fit, the classic spurious regression, so economic reasoning must precede statistical comfort.

At Level I, candidates should be able to compute R-squared from an ANOVA table or from raw sums of squares, recover it from a reported correlation in the single-variable case, and move between R-squared, the F-statistic and the standard error of the estimate. They should also convert the figure into words without overreaching: the model accounts for a stated share of the variation in the dependent variable across the sample used to fit it.

Equally important is knowing what the statistic cannot support. It says nothing about whether individual slope coefficients are significant, whether residuals are homoskedastic or independent, whether the functional form is correct, or whether the relationship will persist out of sample. Testing individual coefficients requires t-statistics; assessing added regressors requires adjusted R-squared.