Inference with Many Weakly Invalid Instruments ∗ Michal Kolesar † Raj Chetty ‡ John Friedman § Edward Glaeser ¶ Guido W. Imbens ‖ February 2011 Abstract Building on the literature on many weak instruments we analyze the properties of con- ventional estimators including tsls, liml, btsls, and jive in the presence of various forms of misspecification that allow for systematic direct effects of the instruments on the outcome. We find that liml is sensitive to particular departures of the standard assumptions that do not effect the btsls estimator. We also extend the random effects set up to develop new estimators that are robust to additional forms of misspecification. JEL Classification: Keywords: ∗ Financial support for this research was generously provided through. We are grateful for comments by participants in the econometrics lunch seminar at Harvard University, and in particular for discussions with Gary Chamberlain. † Dept of Economics, Harvard University. Electronic correspondence: mkolesar@fas.harvard.edu. ‡ Department of Economics, Harvard University, M-24 Littauer Center, 1805 Cambridge Street, Cambridge, MA 02138, and NBER. Electronic correspondence: chetty@fas.harvard.edu, http://www.economics.harvard.edu/faculty/chetty. § , Harvard University. Electronic correspondence: john friedman@harvard.edu, http://www.hks.harvard.edu/fs/jfriedm/. ¶ Department of Economics, Harvard University, M-24 Littauer Center, 1805 Cambridge Street, Cambridge, MA 02138, and NBER. Electronic correspondence: eglaeser@harvard.edu, http://www.economics.harvard.edu/faculty/glaeser. ‖ Department of Economics, Harvard University, M-24 Littauer Center, 1805 Cambridge Street, Cambridge, MA 02138, and NBER. Electronic correspondence: imbens@harvard.edu, http://www.economics.harvard.edu/faculty/imbens/imbens.html. [1]