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We derive mean-unbiased estimators for the structural parameter in instrumental variables models with a single endogenous regressor where the sign of one or more ﬁrst stage coeﬀicients is known. In the case with a single instrument, there is a unique non-randomized unbiased estimator based on the reduced-form and ﬁrst-stage regression estimates. For cases with multiple instruments we propose a class of unbiased estimators and show that an estimator within this class is eﬀicient when the instruments are strong. We show numerically that unbiasedness does not come at a cost of increased dispersion in models with a single instrument: in this case the unbiased estimator is less dispersed than the 2SLS estimator. Our ﬁnite-sample results apply to normal models with known variance for the reduced-form errors, and imply analogous results under weak instrument asymptotics with an unknown error distribution.
Andrews, Isaiah and Armstrong, Timothy B., "Unbiased Instrumental Variables Estimation under Known First-Stage Sign" (2015). Cowles Foundation Discussion Papers. 2410.