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Register a new userSimultaneous Confidence Intervals for High-dimensional Linear Models with Many Endogenous Variables
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High-dimensional linear models with endogenous variables play an increasingly important role in recent econometric literature. In this work we allow for models with many endogenous variables and many instrument variables to achieve identification. Because of the high-dimensionality in the second stage, constructing honest confidence regions with asymptotically correct coverage is non-trivial. Our main contribution is to propose estimators and confidence regions that would achieve that. The approach relies on moment conditions that have an additional orthogonal property with respect to nuisance parameters. Moreover, estimation of high-dimension nuisance parameters is carried out via new pivotal procedures. In order to achieve simultaneously valid confidence regions we use a multiplier bootstrap procedure to compute critical values and establish its validity.
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We study high-dimensional linear models with error-in-variables. Such models are motivated by various applications in econometrics, finance and genetics. These models are challenging because of the need to account for measurement errors to avoid non-vanishing biases in addition to handle the high dimensionality of the parameters. A recent growing literature has proposed various estimators that achieve good rates of convergence. Our main contribution complements this literature with the construction of simultaneous confidence regions for the parameters of interest in such high-dimensional linear models with error-in-variables. These confidence regions are based on the construction of moment conditions that have an additional orthogonal property with respect to nuisance parameters. We provide a construction that requires us to estimate an additional high-dimensional linear model with error-in-variables for each component of interest. We use a multiplier bootstrap to compute critical values for simultaneous confidence intervals for a subset $S$ of the components. We show its validity despite of possible model selection mistakes, and allowing for the cardinality of $S$ to be larger than the sample size. We apply and discuss the implications of our results to two examples and conduct Monte Carlo simulations to illustrate the performance of the proposed procedure.
We propose a new estimator for the high-dimensional linear regression model with observation error in the design where the number of coefficients is potentially larger than the sample size. The main novelty of our procedure is that the choice of penalty parameters is pivotal. The estimator is based on applying a self-normalization to the constraints that characterize the estimator. Importantly, we show how to cast the computation of the estimator as the solution of a convex program with second order cone constraints. This allows the use of algorithms with theoretical guarantees and reliable implementation. Under sparsity assumptions, we derive $ell_q$-rates of convergence and show that consistency can be achieved even if the number of regressors exceeds the sample size. We further provide a simple to implement rule to threshold the estimator that yields a provably sparse estimator with similar $ell_2$ and $ell_1$-rates of convergence. The thresholds are data-driven and component dependents. Finally, we also study the rates of convergence of estimators that refit the data based on a selected support with possible model selection mistakes. In addition to our finite sample theoretical results that allow for non-i.i.d. data, we also present simulations to compare the performance of the proposed estimators.
This was a revision of arXiv:1105.2454v1 from 2012. It considers a variation on the STIV estimator where, instead of one conic constraint, there are as many conic constraints as moments (instruments) allowing to use more directly moderate deviations for self-normalized sums. The idea first appeared in formula (6.5) in arXiv:1105.2454v1 when some instruments can be endogenous. For reference and to avoid confusion with the STIV estimator, this estimator should be called C-STIV.
Instrumental variables (IV) regression is a popular method for the estimation of the endogenous treatment effects. Conventional IV methods require all the instruments are relevant and valid. However, this is impractical especially in high-dimensional models when we consider a large set of candidate IVs. In this paper, we propose an IV estimator robust to the existence of both the invalid and irrelevant instruments (called R2IVE) for the estimation of endogenous treatment effects. This paper extends the scope of Kang et al. (2016) by considering a true high-dimensional IV model and a nonparametric reduced form equation. It is shown that our procedure can select the relevant and valid instruments consistently and the proposed R2IVE is root-n consistent and asymptotically normal. Monte Carlo simulations demonstrate that the R2IVE performs favorably compared to the existing high-dimensional IV estimators (such as, NAIVE (Fan and Zhong, 2018) and sisVIVE (Kang et al., 2016)) when invalid instruments exist. In the empirical study, we revisit the classic question of trade and growth (Frankel and Romer, 1999).
Introductory texts on statistics typically only cover the classical two sigma confidence interval for the mean value and do not describe methods to obtain confidence intervals for other estimators. The present technical report fills this gap by first defining different methods for the construction of confidence intervals, and then by their application to a binomial proportion, the mean value, and to arbitrary estimators. Beside the frequentist approach, the likelihood ratio and the highest posterior density approach are explained. Two methods to estimate the variance of general maximum likelihood estimators are described (Hessian, Jackknife), and for arbitrary estimators the bootstrap is suggested. For three examples, the different methods are evaluated by means of Monte Carlo simulations with respect to their coverage probability and interval length. R code is given for all methods, and the practitioner obtains a guideline which method should be used in which cases.
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