Oblique projectors are useful in many contexts, especially in the instrumental variable estimation in regression models in econometrics, where the disturbance term tends to be correlated with predictor variables. This paper addresses two important questions regarding oblique projectors: (1) products of two oblique projectors and (2) decompositions of oblique projectors. In (1), we investigate what determines the onto-and the along-spaces of an oblique projector defined as a product of two oblique projectors under a variety of conditions. In (2), we examine various decompositions of oblique projectors when both predictor variables and instrumental variables consist of two distinct sets of variables. These decompositions are analogous to those of orthogonal projectors. © 1999 Elsevier Science Inc. All rights reserved.
Takane, Y., & Yanai, H. (1999). On oblique projectors. Linear Algebra and Its Applications, 289(1–3), 297–310. https://doi.org/10.1016/S0024-3795(98)10180-5