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

Abstract

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 wellapproximated 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.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 which 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.

Machine Learning: An Applied Econometric Approach

“…These techniques are applicable to the first stage of instrumental variable analysis as well. In particular, a set of papers has already introduced regularization into the first stage in a high-dimensional setting, including the LASSO (Belloni, Chen, Chernozhukov, and Hansen 2012) and ridge regression (Carrasco 2012;Hansen and Kozbur 2014). More recent extensions include nonlinear functional forms, all the way to neural nets (Hartford, Leyton-Brown, and Taddy 2016).…”

Dissecting the Effect of Credit Supply on Trade: Evidence from Matched Credit-Export Data

“…In Subsection 4.1, we analyze non-parametrically the relationship between bank foreign funding and credit supply. And, in Subsection 4.2, we use the Least Absolute Shrinkage and Selection Operator (LASSO) method in Belloni et al (2012) to select the optimal parametrization of the instrument and to avoid choosing the functional form of the instrument in an ad hoc manner. 14 We test the robustness of the results to alternative definitions of the instrument in Subsection 4.6.…”

Inference in high dimensional panel models with an application to gun control

“…A useful structure which has been employed in the recent econometrics literature focusing on inference in high dimensional settings is approximate sparsity; see, for example, , Belloni, Chen, Chernozhukov, and Hansen (2012), and . A leading example is the approximately sparse linear regression model which is characterized by having many covariates of which only a small number are important for predicting the outcome.…”

“…data even when perfect variable selection is not feasible; see, e.g., Candès and Tao (2007), Meinshausen and Yu (2009), Bickel, Ritov, and Tsybakov (2009), Huang, Horowitz, and Wei (2010, and the references therein. Such methods have also been shown to extend to nonparametric and non-Gaussian cases as in Bickel, Ritov, and Tsybakov (2009) and Belloni, Chen, Chernozhukov, and Hansen (2012), the latter of which also allows for conditional heteroscedasticity.…”

“…It is thus important to offer procedures that are robust to such model selection mistakes. To address this concern, we follow the approach of Belloni, Chen, Chernozhukov, and Hansen (2012) in the IV model and in the partially linear model making use of the Cluster-Lasso to accommodate within-individual dependence and partialing out of fixed effects. We show that standard inference following these procedures results in inference about model parameters of interest that is uniformly valid within a large class of approximately sparse regression models as long as a clustered covariance estimator is used in estimating the parameters' asymptotic variance.…”