In this note, we propose to use sparse methods (e.g. LASSO, Post-LASSO, sqrt-LASSO, and Post-sqrt-LASSO) to form first-stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments in the canonical Gaussian case. The methods apply even when the number of instruments is much larger than the sample size. We derive asymptotic distributions for the resulting IV estimators and provide conditions under which these sparsity-based IV estimators are asymptotically oracle-efficient. In simulation experiments, a sparsity-based IV estimator with a data-driven penalty performs well compared to recently advocated many-instrument-robust procedures. We illustrate the procedure in an empirical example using the Angrist and Krueger (1991) schooling data.
In a linear instrumental variables (IV) setting for estimating the causal effects of multiple confounded exposure/treatment variables on an outcome, we investigate the adaptive Lasso method for selecting valid instrumental variables from a set of available instruments that may contain invalid ones. An instrument is invalid if it fails the exclusion conditions and enters the model as an explanatory variable. We extend the results as developed in Windmeijer et al. (2019) for the single exposure model to the multiple exposures case. In particular we propose a median-of-medians estimator and show …
Many estimators of the average effect of a treatment on an outcome require estimation of the propensity score, the outcome regression, or both. It is often beneficial to utilize flexible techniques such as semiparametric regression or machine learning to estimate these quantities. However, optimal estimation of these regressions does not necessarily lead to optimal estimation of the average treatment effect, particularly in settings with strong instrumental variables. A recent proposal addressed these issues via the outcome-adaptive lasso, a penalized regression technique for estimating the pr…
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 variab…
The method of instrumental variables provides a fundamental and practical tool for causal inference in many empirical studies where unmeasured confounding between the treatments and the outcome is present. Modern data such as the genetical genomics data from these studies are often high-dimensional. The high-dimensional linear instrumental-variables regression has been considered in the literature due to its simplicity albeit a true nonlinear relationship may exist. We propose a more data-driven approach by considering the nonparametric additive models between the instruments and the treatment…
Empirical researchers are increasingly faced with rich data sets containing many controls or instrumental variables, making it essential to choose an appropriate approach to variable selection. In this paper, we provide results for valid inference after post- or orthogonal $L_2$-Boosting is used for variable selection. We consider treatment effects after selecting among many control variables and instrumental variable models with potentially many instruments. To achieve this, we establish new results for the rate of convergence of iterated post-$L_2$-Boosting and orthogonal $L_2$-Boosting in a…
Consider the problem of estimating the local average treatment effect with an instrument variable, where the instrument unconfoundedness holds after adjusting for a set of measured covariates. Several unknown functions of the covariates need to be estimated through regression models, such as instrument propensity score and treatment and outcome regression models. We develop a computationally tractable method in high-dimensional settings where the numbers of regression terms are close to or larger than the sample size. Our method exploits regularized calibrated estimation, which involves Lasso …
Learning a causal effect from observational data is not straightforward, as this is not possible without further assumptions. If hidden common causes between treatment $X$ and outcome $Y$ cannot be blocked by other measurements, one possibility is to use an instrumental variable. In principle, it is possible under some assumptions to discover whether a variable is structurally instrumental to a target causal effect $X \rightarrow Y$, but current frameworks are somewhat lacking on how general these assumptions can be. A instrumental variable discovery problem is challenging, as no variable can …
In observational studies, instrumental variables estimation is greatly utilized to identify causal effects. One of the key conditions for the instrumental variables estimator to be consistent is the exclusion restriction, which indicates that instruments affect the outcome of interest only via the exposure variable of interest. We propose a likelihood-free Bayesian approach to make consistent inferences about the causal effect when there are some invalid instruments in a way that they violate the exclusion restriction condition. Asymptotic properties of the proposed Bayes estimator, including …
Instrumental and Proximal Causal Inference with Gaussian Processes
Department of Statistics and Actuarial Science, University of Hong Kong（香港大学统计与精算科学系）；Rational Intelligence Lab, CISPA Helmholtz Center for Information Security（CISPA霍普夫信息安全中心理性智能实验室）；Australian Institute for Machine Learning, Adelaide University（阿德莱德大学澳大利亚机器学习研究所）；College of Computing & Data Science, Nanyang Technological University（南洋理工大学计算与数据科学学院）
Mixed-effects Outcome-Adaptive Lasso for Propensity Score Estimation under Partial Interference
Joint Graduate School of Mathematics for Innovation, Kyushu University（九州大学数学创新联合研究生院）；Institute of Statistical Mathematics, Tokyo（东京统计数学研究所）；Faculty of Mathematics, Kyushu University（九州大学数学学院）
In this paper, we propose a propensity score adapted variable selection procedure to select covariates for inclusion in propensity score models, in order to eliminate confounding bias and improve statistical efficiency in observational studies. Our variable selection approach is specially designed for causal inference, it only requires the propensity scores to be $\sqrt{n}$-consistently estimated through a parametric model and need not correct specification of potential outcome models. By using estimated propensity scores as inverse probability treatment weights in performing an adaptive lasso…
We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors $p$ is large, possibly much larger than $n$, but only $s$ regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation $σ$ or nor does it need to pre-estimate $σ$. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate $σ\{(s/n)\log p\}^{1/2}$ in …
We consider the task of estimating a Gaussian graphical model in the high-dimensional setting. The graphical lasso, which involves maximizing the Gaussian log likelihood subject to an l1 penalty, is a well-studied approach for this task. We begin by introducing a surprising connection between the graphical lasso and hierarchical clustering: the graphical lasso in effect performs a two-step procedure, in which (1) single linkage hierarchical clustering is performed on the variables in order to identify connected components, and then (2) an l1-penalized log likelihood is maximized on the subset …
We propose a self-tuning $\sqrt{\mathrm {Lasso}}$ method that simultaneously resolves three important practical problems in high-dimensional regression analysis, namely it handles the unknown scale, heteroscedasticity and (drastic) non-Gaussianity of the noise. In addition, our analysis allows for badly behaved designs, for example, perfectly collinear regressors, and generates sharp bounds even in extreme cases, such as the infinite variance case and the noiseless case, in contrast to Lasso. We establish various nonasymptotic bounds for $\sqrt{\mathrm {Lasso}}$ including prediction norm rate …
Given data $\mathbf{y}$ and $k$ covariates $\mathbf{x}_j$ one problem in linear regression is to decide which if any of the covariates to include when regressing the dependent variable $\mathbf{y}$ on the covariates $\mathbf{x}_j$. In this paper three such methods, lasso, knockoff and Gaussian covariates are compared using simulations and real data. The Gaussian covariate method is based on exact probabilities which are valid for all $\mathbf{y}$ and $\mathbf{x}_j$ making it model free. Moreover the probabilities agree with those based on the F-distribution for the standard linear model with i…
High dimensional Poisson regression has become a standard framework for the analysis of massive counts datasets. In this work we estimate the intensity function of the Poisson regression model by using a dictionary approach, which generalizes the classical basis approach, combined with a Lasso or a group-Lasso procedure. Selection depends on penalty weights that need to be calibrated. Standard methodologies developed in the Gaussian framework can not be directly applied to Poisson models due to heteroscedasticity. Here we provide data-driven weights for the Lasso and the group-Lasso derived fr…
Confidence intervals based on penalized maximum likelihood estimators such as the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the known-variance case, the finite-sample coverage properties of such intervals are determined and it is shown that symmetric intervals are the shortest. The length of the shortest intervals based on the hard-thresholding estimator is larger than the length of the shortest interval based on the adaptive LASSO, which is larger than the length of the shortest interval based on the LASSO, which in turn is larger than the standard interval based on the ma…
The time-evolving precision matrix of a piecewise-constant Gaussian graphical model encodes the dynamic conditional dependency structure of a multivariate time-series. Traditionally, graphical models are estimated under the assumption that data is drawn identically from a generating distribution. Introducing sparsity and sparse-difference inducing priors we relax these assumptions and propose a novel regularized M-estimator to jointly estimate both the graph and changepoint structure. The resulting estimator possesses the ability to therefore favor sparse dependency structures and/or smoothly …
Stability of a Generalized Debiased Lasso with Applications to Resampling-Based Variable Selection
Department of Statistics, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校统计系）；Department of Electrical and Computer Engineering, the Grainger College of Engineering（格拉inger工程学院电子与计算机工程系）
In a linear regression model of fixed dimension $p \leq n$, we construct confidence regions for the unknown parameter vector based on the Lasso estimator that uniformly and exactly hold the prescribed in finite samples as well as in an asymptotic setup. We thereby quantify estimation uncertainty as well as the "post-model selection error" of this estimator. More concretely, in finite samples with Gaussian errors and asymptotically in the case where the Lasso estimator is tuned to perform conservative model selection, we derive exact formulas for computing the minimal coverage probability over …
After selection with the Group LASSO (or generalized variants such as the overlapping, sparse, or standardized Group LASSO), inference for the selected parameters is unreliable in the absence of adjustments for selection bias. In the penalized Gaussian regression setup, existing approaches provide adjustments for selection events that can be expressed as linear inequalities in the data variables. Such a representation, however, fails to hold for selection with the Group LASSO and substantially obstructs the scope of subsequent post-selective inference. Key questions of inferential interest -- …
Dynamic networks models describe a growing number of important scientific processes, from cell biology and epidemiology to sociology and finance. There are many aspects of dynamical networks that require statistical considerations. In this paper we focus on determining network structure. Estimating dynamic networks is a difficult task since the number of components involved in the system is very large. As a result, the number of parameters to be estimated is bigger than the number of observations. However, a characteristic of many networks is that they are sparse. For example, the molecular st…
We propose Bayesian methods for Gaussian graphical models that lead to sparse and adaptively shrunk estimators of the precision (inverse covariance) matrix. Our methods are based on lasso-type regularization priors leading to parsimonious parameterization of the precision matrix, which is essential in several applications involving learning relationships among the variables. In this context, we introduce a novel type of selection prior that develops a sparse structure on the precision matrix by making most of the elements exactly zero, in addition to ensuring positive definiteness -- thus cond…
In many applied fields, such as genomics, different types of data are collected on the same system, and it is not uncommon that some of these datasets are subject to censoring as a result of the measurement technologies used, such as data generated by polymerase chain reactions and flow cytometer. When the overall objective is that of network inference, at possibly different levels of a system, information coming from different sources and/or different steps of the analysis can be integrated into one model with the use of conditional graphical models. In this paper, we develop a doubly penaliz…
We propose the use of a robust covariance estimator based on multivariate Winsorization in the context of the Tarr-Muller-Weber framework for sparse estimation of the precision matrix of a Gaussian graphical model. Likewise Croux-Ollerer's precision matrix estimator, our proposed estimator attains the maximum finite sample breakdown point of 0.5 under cellwise contamination. We conduct an extensive Monte Carlo simulation study to assess the performance of ours and the currently existing proposals. We find that ours has a competitive behavior, regarding the the estimation of the precision matri…
Hard thresholding, LASSO , adaptive LASSO and SCAD point estimators have been suggested for use in the linear regression context when most of the components of the regression parameter vector are believed to be zero, a sparsity type of assumption. Potscher and Schneider, 2010, Electronic Journal of Statistics, have considered the properties of fixed-width confidence intervals that include one of these point estimators (for all possible data values). They consider a normal linear regression model with orthogonal regressors and show that these confidence intervals are longer than the standard co…
This paper addresses the issue of detecting change-points in multivariate time series. The proposed approach differs from existing counterparts by making only weak assumptions on both the change-points structure across series, and the statistical signal distributions. Specifically change-points are not assumed to occur at simultaneous time instants across series, and no specific distribution is assumed on the individual signals. It relies on the combination of a local robust statistical test acting on individual time segments, with a global Bayesian framework able to optimize configurations fr…
This paper studies iteration convergence of Kronecker graphical lasso (KGLasso) algorithms for estimating the covariance of an i.i.d. Gaussian random sample under a sparse Kronecker-product covariance model and MSE convergence rates. The KGlasso model, originally called the transposable regularized covariance model by Allen ["Transposable regularized covariance models with an application to missing data imputation," Ann. Appl. Statist., vol. 4, no. 2, pp. 764-790, 2010], implements a pair of $\ell_1$ penalties on each Kronecker factor to enforce sparsity in the covariance estimator. The KGlass…
We study tools for inference conditioned on model selection events that are defined by the generalized lasso regularization path. The generalized lasso estimate is given by the solution of a penalized least squares regression problem, where the penalty is the l1 norm of a matrix D times the coefficient vector. The generalized lasso path collects these estimates for a range of penalty parameter (λ) values. Leveraging a sequential characterization of this path from Tibshirani & Taylor (2011), and recent advances in post-selection inference from Lee et al. (2016), Tibshirani et al. (2016), we dev…
We consider linear regression in the high-dimensional regime where the number of observations $n$ is smaller than the number of parameters $p$. A very successful approach in this setting uses $\ell_1$-penalized least squares (a.k.a. the Lasso) to search for a subset of $s_0< n$ parameters that best explain the data, while setting the other parameters to zero. Considerable amount of work has been devoted to characterizing the estimation and model selection problems within this approach. In this paper we consider instead the fundamental, but far less understood, question of \emph{statistical sig…
An approximate method for conducting resampling in Lasso, the $\ell_1$ penalized linear regression, in a semi-analytic manner is developed, whereby the average over the resampled datasets is directly computed without repeated numerical sampling, thus enabling an inference free of the statistical fluctuations due to sampling finiteness, as well as a significant reduction of computational time. The proposed method is based on a message passing type algorithm, and its fast convergence is guaranteed by the state evolution analysis, when covariates are provided as zero-mean independently and identi…
The new class of Markov processes is proposed to realize the flexible shrinkage effects for the dynamic models. The transition density of the new process consists of two penalty functions, similarly to Bayesian fused LASSO in its functional form, that shrink the current state variable to its previous value and zero. The normalizing constant of the density, which is not ignorable in the posterior computation, is shown to be essentially the log-geometric mixture of double-exponential densities. This process comprises the state equation of the dynamic regression models, which is shown to be condi…
We model time-varying network data as realizations from multivariate Gaussian distributions with precision matrices that change over time. To facilitate parameter estimation, we require not only that each precision matrix at any given time point be sparse, but also that precision matrices at neighboring time points be similar. We accomplish this with two different algorithms, by generalizing the elastic net and the fused LASSO, respectively. Our main focuses are efficient computational algorithms and convenient degree-of-freedom formulae for choosing tuning parameters. We illustrate our method…
Gaussian graphical regression is a powerful means that regresses the precision matrix of a Gaussian graphical model on covariates, permitting the numbers of the response variables and covariates to far exceed the sample size. Model fitting is typically carried out via separate node-wise lasso regressions, ignoring the network-induced structure among these regressions. Consequently, the error rate is high, especially when the number of nodes is large. We propose a multi-task learning estimator for fitting Gaussian graphical regression models; we design a cross-task group sparsity penalty and a …
The lasso has been studied extensively as a tool for estimating the coefficient vector in the high-dimensional linear model; however, considerably less is known about estimating the error variance in this context. In this paper, we propose the natural lasso estimator for the error variance, which maximizes a penalized likelihood objective. A key aspect of the natural lasso is that the likelihood is expressed in terms of the natural parameterization of the multiparameter exponential family of a Gaussian with unknown mean and variance. The result is a remarkably simple estimator of the error var…
