An analyst adds 12 macro variables to a 3-variable return model estimated on 120 months, and R-squared rises from 8% to 17%. Compute adjusted R-squared before and after and an F-test on the added block. Did the twelve add anything?

Adjusted R-squared falls from 5.6% to 5.0%, and the F-test on the twelve added variables gives F = 0.94 against a 5% critical value of about 1.85, so the macro block adds nothing. R-squared rises mechanically with every variable; twelve noise series would have added about 9.5 points on 120 months, and these added 9. The p-value is about 0.51. Keep the three-factor model.

Give a student twelve extra guesses on a multiple-choice test and the score rises whether or not the student knows anything, because some guesses land. Least squares does the same: every added variable is fitted to the sample's quirks, so R-squared can only rise, and with n observations and k variables a noise variable removes about 1/(n - k - 1) of the remaining unexplained variance on average. Here twelve noise series would raise R-squared by roughly 12/116 of the 92% unexplained, 9.5 points. The observed rise is 9.0 points. On its face, the macro block has done exactly what random numbers would have done.

Adjusted R-squared charges a price for each variable: 1 - (1 - R-squared) x (n - 1)/(n - k - 1). Before, it is 1 - 0.92 x 119/116 = 5.62%; after, 1 - 0.83 x 119/104 = 5.03%, lower than before, so by this measure the twelve variables cost more than they explain. The fall is small because the penalty is a crude one, but the direction is the finding. A report that quotes R-squared doubling while adjusted R-squared fell is the kind of thing a reviewer catches in one line.

The F-test asks whether the improvement in fit per added variable is larger than the unexplained variance per remaining degree of freedom. F = ((0.17 - 0.08)/12) / ((1 - 0.17)/104) = 0.00750 / 0.00798 = 0.94; under the null of no effect F averages about 1, the 5% critical value with 12 and 104 degrees of freedom is about 1.85, and the p-value is about 0.51. A result this size or larger happens half the time when the twelve variables are pure noise. For the block to clear the 5% bar, R-squared would have needed to rise by about 16 points, to 24%, not 9.

Drop the block, and say why in one sentence: on 120 months, twelve extra variables buy nine points of R-squared for free, and these bought nine. If the analyst believes one or two of the macro series matter, the honest route is to pick them before looking at the fit, add them alone, and test them on months the model has not seen. Adding a dozen and keeping whichever looks best is the same mistake with a smaller block. State the limitation: the F-test assumes independent, equal-variance errors, which monthly returns often break, so a robust version should confirm the result; but a block that cannot pass the ordinary test will not pass a stricter one.