The mandated-wage studies also have the advantage of better treatment of some data complexities. Several of them try to relax the assumption of a small price-taking U.S. economy, as will be discussed below in greater detail. In addition, these studies also account for intermediate inputs as well as primary factors.
Mandated-wage regressions might appear odd because the exogenous variable is the regressand rather than the regressor while the dependent variable of interest (factor-price changes) is estimated rather than the regressand. The most important reason a “standard” regression cannot be used is the dimensionality of the data prevents inversion of the 0 matrix. For example, the NBER Productivity Data Base used by LS, Leamer, and FH contain 450 four-digit SIC manufacturing industries but only three primary factors plus two intermediate inputs. With more products than factors, in equation (2) the 0 matrix is not square and thus cannot be inverted to obtain a set of equations equating wage changes with product-price changes multiplied by an inverted 0 matrix.
This lack of invertibility suggests that the warranted-wage regressions can be interpreted as an accounting exercise rather than one identifying causation in the way regressions are usually presumed to. Warranted-wage regressions estimate what changes in factor prices are mandated from the observed changes in technology and/or product prices. With these mandated changes one can determine what share of actual wage changes is accounted for by the driving exogenous change. Note that because the exogenous change enters the regression as the dependent variable, the mandated-wage methodology cannot analyze two or more exogenous forces in the same regression–it can process only one exogenous force at a time.
Overall, the product-price methodology has advanced from consistency checks to warranted-wage regressions. I argue that this progression has moved empirical work closer to the motivating SS theory in a number of important ways.