The theorem is so-named because (i) in the very fist volume of

*Econometrica*Frisch and Waugh (1933) established it in the particular context of "de-trending" time-series data; and (ii) Lovell (1963) demonstrated that the same result establishes the equivalence of "seasonally adjusting" time-series data (in a particular way), and including seasonal dummy variables in an OLS regression model. (Also, see Lovell, 2008.)

We'll take a look at the statement of the FWL Theorem in a moment. First, though, it's important to note that it's purely an algebraic/geometric result. Although it arises in the context of regression analysis, it has no

*statistical*content,

*per se*.

What's not generally recognized, however, is that the FWL Theorem doesn't rely on the geometry of OLS. In fact, it relies on the geometry of the Instrumental Variables (IV) estimator - of which OLS is a special case, of course. (OLS is just IV in the just-identified case, with the regressors being used as their own instruments.)

Implicitly, this was shown in an old paper of mine (Giles, 1984) where I extended Lovell's analysis to the context of IV estimation. However, in that paper I didn't spell out the generality of the FWL-IV result.

Let's take a look at all of this.