63 Pages Posted: 15 Aug 2011
Date Written: August 15, 2011
We develop results for the use of LASSO and Post-LASSO methods to form first-stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments, p, that apply even when p is much larger than the sample size, n. We rigorously develop asymptotic distribution and inference theory for the resulting IV estimators and provide conditions under which these estimators are asymptotically oracle-efficient. In simulation experiments, the LASSO-based IV estimator with a data-driven penalty performs well compared to recently advocated many-instrument-robust procedures. In an empirical example dealing with the effect of judicial eminent domain decisions on economic outcomes, the LASSO based IV estimator substantially reduces estimated standard errors allowing one to draw much more precise conclusions about the economic effects of these decisions.
Optimal instruments are conditional expectations; and in developing the IV results, we also establish a series of new results for LASSO and Post-LASSO estimators of non-parametric conditional expectation functions which are of independent theoretical and practical interest. Specifically, we develop the asymptotic theory for these estimators that allows for non-Gaussian, heteroscedastic disturbances, which is important for econometric applications. By innovatively using moderate deviation theory for self-normalized sums, we provide convergence rates for these estimators that are as sharp as in the homoscedastic Gaussian case under the weak condition that log p = o(n1=3). Moreover, as a practical innovation, we provide a fully data-driven method for choosing the user-specified penalty that must be provided in obtaining LASSO and Post-LASSO estimates and establish its asymptotic validity under non-Gaussian, heteroscedastic disturbances.
Keywords: Instrumental Variables, Optimal Instruments, LASSO, Post-LASSO, Sparsity, Eminent Domain, Data-Driven Penalty, Heteroscedasticity, Non-Gaussian Errors, Moderate Deviations for Self-Normalized Sums
Suggested Citation: Suggested Citation
Belloni, Alexandre and Chen, Daniel and Chernozhukov, Victor and Hansen, Christian, Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain (August 15, 2011). MIT Department of Economics Working Paper No. 11-19. Available at SSRN: https://ssrn.com/abstract=1910169 or http://dx.doi.org/10.2139/ssrn.1910169