# Sparse Models and Methods for Optimal Instruments With an Application to Eminent Domain*

* This paper has been replicated

## Author

Listed:
• A. Belloni
• D. Chen
• V. Chernozhukov
• C. Hansen

## Abstract

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, $p$. Our results apply even when $p$ is much larger than the sample size, $n$. We show that the IV estimator based on using Lasso or Post-Lasso in the first stage is root-n consistent and asymptotically normal when the first-stage is approximately sparse; i.e. when the conditional expectation of the endogenous variables given the instruments can be well-approximated by a relatively small set of variables whose identities may be unknown. We also show the estimator is semi-parametrically efficient when the structural error is homoscedastic. Notably our results allow for imperfect model selection, and do not rely upon the unrealistic "beta-min" conditions that are widely used to establish validity of inference following model selection. 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 outperforms an intuitive benchmark. In developing the IV results, we establish a series of new results for Lasso and Post-Lasso estimators of nonparametric conditional expectation functions which are of independent theoretical and practical interest. We construct a modification of Lasso designed to deal with non-Gaussian, heteroscedastic disturbances which uses a data-weighted $\ell_1$-penalty function. Using moderate deviation theory for self-normalized sums, we provide convergence rates for the resulting Lasso and Post-Lasso estimators that are as sharp as the corresponding rates in the homoscedastic Gaussian case under the condition that $\log p = o(n^{1/3})$.
(This abstract was borrowed from another version of this item.)

## Suggested Citation

• A. Belloni & D. Chen & V. Chernozhukov & C. Hansen, 2012. "Sparse Models and Methods for Optimal Instruments With an Application to Eminent Domain," Econometrica, Econometric Society, vol. 80(6), pages 2369-2429, November.
• Handle: RePEc:ecm:emetrp:v:80:y:2012:i:6:p:2369-2429
DOI: ECTA9626
as

File URL: http://hdl.handle.net/10.3982/ECTA9626

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## References listed on IDEAS

as
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Full references (including those not matched with items on IDEAS)

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## Replication

This item has been replicated by:
• Martin Spindler, 2016. "Lasso for Instrumental Variable Selection: A Replication Study," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 31(2), pages 450-454, March.

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