On the Computational Complexity of MCMC-based Estimators in Large Samples

In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasi-Bayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the Laplace-Bernstein-Von Mises central limit theorem, which states that in large samples the posterior or quasi-posterior approaches a normal density. Using the conditions required for the central limit theorem to hold, we establish polynomial bounds on the computational complexity of general Metropolis random walks methods in large samples. Our analysis covers cases where the underlying log-likelihood or extremum criterion function is possibly non-concave, discontinuous, and with increasing parameter dimension. However, the central limit theorem restricts the deviations from continuity and log-concavity of the log-likelihood or extremum criterion function in a very specific manner. Under minimal assumptions required for the central limit theorem to hold under the increasing parameter dimension, we show that the Metropolis algorithm is theoretically efficient even for the canonical Gaussian walk which is studied in detail. Specifically, we show that the running time of the algorithm in large samples is bounded in probability by a polynomial in the parameter dimension $d$, and, in particular, is of stochastic order $d^2$ in the leading cases after the burn-in period. We then give applications to exponential families, curved exponential families, and Z-estimation of increasing dimension.Comment: 36 pages, 2 figure

Least squares after model selection in high-dimensional sparse models

In this article we study post-model selection estimators that apply ordinary least squares (OLS) to the model selected by first-step penalized estimators, typically Lasso. It is well known that Lasso can estimate the nonparametric regression function at nearly the oracle rate, and is thus hard to improve upon. We show that the OLS post-Lasso estimator performs at least as well as Lasso in terms of the rate of convergence, and has the advantage of a smaller bias. Remarkably, this performance occurs even if the Lasso-based model selection "fails" in the sense of missing some components of the "true" regression model. By the "true" model, we mean the best s-dimensional approximation to the nonparametric regression function chosen by the oracle. Furthermore, OLS post-Lasso estimator can perform strictly better than Lasso, in the sense of a strictly faster rate of convergence, if the Lasso-based model selection correctly includes all components of the "true" model as a subset and also achieves sufficient sparsity. In the extreme case, when Lasso perfectly selects the "true" model, the OLS post-Lasso estimator becomes the oracle estimator. An important ingredient in our analysis is a new sparsity bound on the dimension of the model selected by Lasso, which guarantees that this dimension is at most of the same order as the dimension of the "true" model. Our rate results are nonasymptotic and hold in both parametric and nonparametric models. Moreover, our analysis is not limited to the Lasso estimator acting as a selector in the first step, but also applies to any other estimator, for example, various forms of thresholded Lasso, with good rates and good sparsity properties. Our analysis covers both traditional thresholding and a new practical, data-driven thresholding scheme that induces additional sparsity subject to maintaining a certain goodness of fit. The latter scheme has theoretical guarantees similar to those of Lasso or OLS post-Lasso, but it dominates those procedures as well as traditional thresholding in a wide variety of experiments.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ410 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Post-l1-penalized estimators in high-dimensional linear regression models

In this paper we study post-penalized estimators which apply ordinary, unpenalized linear regression to the model selected by first-step penalized estimators, typically LASSO. It is well known that LASSO can estimate the regression function at nearly the oracle rate, and is thus hard to improve upon. We show that post-LASSO performs at least as well as LASSO in terms of the rate of convergence, and has the advantage of a smaller bias. Remarkably, this performance occurs even if the LASSO-based model selection 'fails' in the sense of missing some components of the 'true' regression model. By the 'true' model we mean here the best s-dimensional approximation to the regression function chosen by the oracle. Furthermore, post-LASSO can perform strictly better than LASSO, in the sense of a strictly faster rate of convergence, if the LASSO-based model selection correctly includes all components of the 'true' model as a subset and also achieves a sufficient sparsity. In the extreme case, when LASSO perfectly selects the 'true' model, the post-LASSO estimator becomes the oracle estimator. An important ingredient in our analysis is a new sparsity bound on the dimension of the model selected by LASSO which guarantees that this dimension is at most of the same order as the dimension of the 'true' model. Our rate results are non-asymptotic and hold in both parametric and nonparametric models. Moreover, our analysis is not limited to the LASSO estimator in the first step, but also applies to other estimators, for example, the trimmed LASSO, Dantzig selector, or any other estimator with good rates and good sparsity. Our analysis covers both traditional trimming and a new practical, completely data-driven trimming scheme that induces maximal sparsity subject to maintaining a certain goodness-of-fit. The latter scheme has theoretical guarantees similar to those of LASSO or post-LASSO, but it dominates these procedures as well as traditional trimming in a wide variety of experiments.

On the computational complexity of MCMC-based estimators in large samples

In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasi-Bayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the Laplace-Bernstein-Von Mises central limit theorem, which states that in large samples the posterior or quasi-posterior approaches a normal density. Using this observation, we establish polynomial bounds on the computational complexity of general Metropolis random walks methods in large samples. Our analysis covers cases, where the underlying log-likelihood or extremum criterion function is possibly nonconcave, discontinuous, and of increasing dimension. However, the central limit theorem restricts the deviations from continuity and log-concavity of the log-likelihood or extremum criterion function in a very specific manner.

L1-Penalized quantile regression in high-dimensional sparse models

We consider median regression and, more generally, quantile regression in high-dimensional sparse models. In these models the overall number of regressors p is very large, possibly larger than the sample size n, but only s of these regressors have non-zero impact on the conditional quantile of the response variable, where s grows slower than n. Since in this case the ordinary quantile regression is not consistent, we consider quantile regression penalized by the L1-norm of coefficients (L1-QR). First, we show that L1-QR is consistent at the rate of the square root of (s/n) log p, which is close to the oracle rate of the square root of (s/n), achievable when the minimal true model is known. The overall number of regressors p affects the rate only through the log p factor, thus allowing nearly exponential growth in the number of zero-impact regressors. The rate result holds under relatively weak conditions, requiring that s/n converges to zero at a super-logarithmic speed and that regularization parameter satisfies certain theoretical constraints. Second, we propose a pivotal, data-driven choice of the regularization parameter and show that it satisfies these theoretical constraints. Third, we show that L1-QR correctly selects the true minimal model as a valid submodel, when the non-zero coefficients of the true model are well separated from zero. We also show that the number of non-zero coefficients in L1-QR is of same stochastic order as s, the number of non-zero coefficients in the minimal true model. Fourth, we analyze the rate of convergence of a two-step estimator that applies ordinary quantile regression to the selected model. Fifth, we evaluate the performance of L1-QR in a Monte-Carlo experiment, and provide an application to the analysis of the international economic growth.

Inference for Extremal Conditional Quantile Models, with an Application to Market and Birthweight Risks

Quantile regression is an increasingly important empirical tool in economics and other sciences for analyzing the impact of a set of regressors on the conditional distribution of an outcome. Extremal quantile regression, or quantile regression applied to the tails, is of interest in many economic and financial applications, such as conditional value-at-risk, production efficiency, and adjustment bands in (S,s) models. In this paper we provide feasible inference tools for extremal conditional quantile models that rely upon extreme value approximations to the distribution of self-normalized quantile regression statistics. The methods are simple to implement and can be of independent interest even in the non-regression case. We illustrate the results with two empirical examples analyzing extreme fluctuations of a stock return and extremely low percentiles of live infants' birthweights in the range between 250 and 1500 grams.Comment: 41 pages, 9 figure

Inference for High-Dimensional Sparse Econometric Models

This article is about estimation and inference methods for high dimensional sparse (HDS) regression models in econometrics. High dimensional sparse models arise in situations where many regressors (or series terms) are available and the regression function is well-approximated by a parsimonious, yet unknown set of regressors. The latter condition makes it possible to estimate the entire regression function effectively by searching for approximately the right set of regressors. We discuss methods for identifying this set of regressors and estimating their coefficients based on $\ell_1$-penalization and describe key theoretical results. In order to capture realistic practical situations, we expressly allow for imperfect selection of regressors and study the impact of this imperfect selection on estimation and inference results. We focus the main part of the article on the use of HDS models and methods in the instrumental variables model and the partially linear model. We present a set of novel inference results for these models and illustrate their use with applications to returns to schooling and growth regression