Orthogonalized regressors and spurious precision, with an application to currency exposures

Regressions often use pre-orthogonalized regressors: prior to the main regression, an independent variable xi is regressed upon the other regressor(s), and its residuals are used in the right-hand side of the main regression instead of the raw variable itself. For example, the exposure of a stock's return to exchange rate changes is conventionally estimated by a regression, and often the market return is included as an additional regressor. By first orthogonalizing the market return on the exchange rate, in a regression separate from the main one, one seems to have the best of both worlds: the market factor cannot subsume part of the exposure present in a stock's return, and the standard error (se) of the estimate beats both the simple- and the multiple-regression se's. This last effect is illusory: since the simple regression and its two-step variant, with the orthogonalization, produce the same exposure estimate, given the sample, their precision must be identical too. Technically, the source of the problem is that the uncertainty about the market's exposure estimate is left out of the calculated se. In published work, the calculated error variances should be corrected upward by 20–100 percent.