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

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, that is, 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 that the estimator is semiparametrically 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 (see also Belloni, Chernozhukov, and Hansen (2011b)). 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 Lassobased IV estimator outperforms an intuitive benchmark.Optimal instruments are conditional expectations. 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 that uses a data-weighted 1 -penalty function. By innovatively 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 ). We also provide a data-driven method for choosing the penalty level that must be specified in obtaining Lasso and post-Lasso estimates and establish its asymptotic validity under non-Gaussian, heteroscedastic disturbances.KEYWORDS: Inference on a low-dimensional parameter after model selection, imperfect model selection, instrumental variables, Lasso, post-Lasso, data-driven penalty, heteroscedasticity, non-Gaussian errors, moderate deviations for self-normalized sums.

Big Data and Audience Measurement: A Marriage of Convenience?

“…An illustration of the potential interactions between the two communities can be found, for example, in Belloni et al (2010Belloni et al ( , 2012, in the context of the choice of instrument in a regression. Using the data produced by Angrist & Krueger (1991) relating to an academic achievement problem, they show how to effectively implement instrumental econometric techniques when 1,530 instruments are available (a recurring problem with the increase in the volume of data).…”

lassopack: Model selection and prediction with regularized regression in Stata

“…A test that has been developed and is available in rlasso corresponds to the test for joint significance of regressors using F or χ 2 tests that is common in regression analysis. Specifically, Belloni et al (2012) suggest using the Chernozhukov et al (2013, Appendix M) sup-score test to test for the joint significance of the regressors, i.e., H 0 : β 1 = . .…”

“…Belloni et al 2012), the post-double-selection (PDS) estimator(Belloni et al 2014a) and the post-regularization estimator (CHS)(Chernozhukov et al 2015); all of which are implemented in our sister package pdslasso (Ahrens et al 2018).…”

Health Effects of Caring for and About Parents and Spouses

“…These administrative data include annual outpatient prescription drug use at the ATC 3 level, eligibility for publicly funded formal home care and nursing home care, and various types of health insurance spending such as GP care, hospital care and nursing care. As these data contain a very large number of potential indicators that would reduce any omitted variable bias, we use LASSO regression to select the most relevant variables (Belloni et al, 2012). The caregiving effect on mental health persists after adding these additional health variables, which suggests that the effect in the main analysis is robust to more extensive control for the family effect (Table A12).…”