PRACTITIONERS' CORNER: Coefficient Sign Changes when Restricting Regression Models Under Instrumental Variables Estimation *

It is now well known that deleting a variable from (or otherwise restricting the coefficients of) a least squares regression model has implications for the signs of the remaining variables' coefficients which are predictable under certain conditions. This topic was discussed by Leamer (1975) who showed, among other things, that if the absolute f-value associated with the deleted variable is less than that assodated with another variable in the model, then the sign of the latter's estimated coefficient cannot change as a result of the deletion. Leamer's necessary condition was extended to include a sufficient condition by Visco (1978); was presented in a simple alternative form by Oksanen (1987); and was generalized by McAleer et al. (1986) to apply to situations where the deleted variables are combined in arbitrary linear combinations (such as in certain distributed lag models). See, also, Visco (1988). The results obtained by these authors are extremely useful to the applied econometrician — certain possibilities may be ruled out, a priori, simply by inspecting the values of appropriate r-values or F-statistics, However, all of these results are derived only in the context of Ordinary Least Squares (O.L.S.) estimation. Given the widespread use of estimators in the Instrumental Variables (I.V.) family, it is interesting to ask whether similar results hold in this more general context. If so, then prescriptions could be made to cover the estimation of structural equations from simultaneous systems, as well as dynamic models with autocorrelated errors, for example. This question is readily answered. As Oksanen (1987, p. 229) notes, the existing results relate to a problem in the algebra (not statistics) of least squares. The algebra of LV. estimation can be written in a form analogous to that of O.L.S., and so the results of interest hold in the I.V. case too. To see this, let the model be