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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 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 that uses a data-weighted 𝓁₁-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.

We consider estimation and inference in panel data models with additive unobserved individual specific heterogeneity in a high-dimensional setting. The setting allows the number of time-varying regressors to be larger than the sample size. To make informative estimation and inference feasible, we require that the overall contribution of the time-varying variables after eliminating the individual specific heterogeneity can be captured by a relatively small number of the available variables whose identities are unknown. This restriction allows the problem of estimation to proceed as a variable selection problem. Importantly, we treat the individual specific heterogeneity as fixed effects which allows this heterogeneity to be related to the observed time-varying variables in an unspecified way and allows that this heterogeneity may differ for all individuals. Within this framework, we provide procedures that give uniformly valid inference over a fixed subset of parameters in the canonical linear fixed effects model and over coefficients on a fixed vector of endogenous variables in panel data instrumental variable models with fixed effects and many instruments. We present simulation results in support of the theoretical developments and illustrate the use of the methods in an application aimed at estimating the effect of gun prevalence on crime rates.

We consider inference post-model-selection in linear regression. In this setting, Berk et al. [Ann. Statist. 41 (2013a) 802–837] recently introduced a class of confidence sets, the so-called PoSI intervals, that cover a certain non-standard quantity of interest with a user-specified minimal coverage probability, irrespective of the model selection procedure that is being used. In this paper, we generalize the PoSI intervals to confidence intervals for post-model-selection predictors.

Understanding complex relationships between random variables is of fundamental importance in high-dimensional statistics, with numerous applications in biological and social sciences. Undirected graphical models are often used to represent dependencies between random variables, where an edge between two random variables is drawn if they are conditionally dependent given all the other measured variables. A large body of literature exists on methods that estimate the structure of an undirected graphical model, however, little is known about the distributional properties of the estimators beyond the Gaussian setting. In this paper, we focus on inference for edge parameters in a high-dimensional transelliptical model, which generalizes Gaussian and nonparanormal graphical models. We propose ROCKET, a novel procedure for estimating parameters in the latent inverse covariance matrix. We establish asymptotic normality of ROCKET in an ultra high-dimensional setting under mild assumptions, without relying on oracle model selection results. ROCKET requires the same number of samples that are known to be necessary for obtaining a √n consistent estimator of an element in the precision matrix under a Gaussian model. Hence, it is an optimal estimator under a much larger family of distributions. The result hinges on a tight control of the sparse spectral norm of the nonparametric Kendall’s tau estimator of the correlation matrix, which is of independent interest. Empirically, ROCKET outperforms the nonparanormal and Gaussian models in terms of achieving accurate inference on simulated data. We also compare the three methods on real data (daily stock returns), and find that the ROCKET estimator is the only method whose behavior across subsamples agrees with the distribution predicted by the theory.

• Heterogeneity has been observed in the responses of Arctic shrubs to climate variability over recent decades, which may reflect landscape-scale variability in belowground resources. At a northern fringe of tall shrub expansion (Yuribei, Yamal Peninsula, Russia), we sought to determine the mechanisms relating nitrogen (N) limitation to shrub growth over decadal time. • We analysed the ratio of 15N to 14N isotopes in wood rings of 10 Salix lanata individuals (399 measurements) to reconstruct annual point-based bioavailable N between 1980 and 2013. We applied a model-fitting/model-selection approach with a suite of competing ecological models to assess the most-likely mechanisms that explain each shrub’s individual time-series. • Shrub δ15N time-series indicated declining (seven shrubs), increasing (two shrubs) and no trend (one shrub) in N availability. The most appropriate model for all shrubs included N-dependent growth of linear rather than saturating form. Inclusion of plant–soil feedbacks better explained ring width and δ15N for eight of 10 individuals. • Although N trajectories were individualistic, common mechanisms of varying strength confirmed the N-dependency of shrub growth. The linear mechanism may reflect intense scavenging of scarce N; the importance of plant–soil feedbacks suggests that shrubs subvert the microbial bottleneck by actively controlling their environment.

Modern coexistence theory (MCT) offers a conceptually straightforward approach for connecting empirical observations with an elegant theoretical framework, gaining popularity rapidly over the past decade. However, beneath this surface‐level simplicity lie various assumptions and subjective choices made during data analysis. These can lead researchers to draw qualitatively different conclusions from the same set of experiments. As the predictions of MCT studies are often treated as outcomes, and many readers and reviewers may not be familiar with the framework's assumptions, there is a particular risk of ‘researcher degrees of freedom’ inflating the confidence in results, thereby affecting reproducibility and predictive power. To tackle these concerns, we introduce a checklist consisting of statistical best practices to promote more robust empirical applications of MCT. Our recommendations are organised into four categories: presentation and sharing of raw data, testing model assumptions and fits, managing uncertainty associated with model coefficients and incorporating this uncertainty into coexistence predictions. We surveyed empirical MCT studies published over the past 15 years and discovered a high degree of variation in the level of statistical rigour and adherence to best practices. We present case studies to illustrate the dependence of results on seemingly innocuous choices among competition model structure and error distributions, which in some cases reversed the predicted coexistence outcomes. These results demonstrate how different analytical approaches can profoundly alter the interpretation of experimental results, underscoring the importance of carefully considering and thoroughly justifying each step taken in the analysis pathway. Our checklist serves as a resource for authors and reviewers alike, providing guidance to strengthen the empirical foundation of empirical coexistence analyses. As the field of empirical MCT shifts from a descriptive, trailblazing phase to a stage of consolidation, we emphasise the need for caution when building upon the findings of earlier studies. To ensure that progress made in the field of ecological coexistence is based on robust and reliable evidence, it is crucial to subject our predictions, conclusions and generalisability to a more rigorous assessment than is currently the trend.

We consider the problem of estimating the conditional distribution of a post-model-selection estimator where the conditioning is on the selected model. The notion of a post-model-selection estimator here refers to the combined procedure resulting from first selecting a model (e.g., by a model selection criterion such as AIC or by a hypothesis testing procedure) and then estimating the parameters in the selected model (e.g., by least-squares or maximum likelihood), all based on the same data set. We show that it is impossible to estimate this distribution with reasonable accuracy even asymptotically. In particular, we show that no estimator for this distribution can be uniformly consistent (not even locally). This follows as a corollary to (local) minimax lower bounds on the performance of estimators for this distribution. Similar impossibility results are also obtained for the conditional distribution of linear functions (e.g., predictors) of the post-model-selection estimator.

We propose a self-tuning $\\sqrt {Lasso} $ 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 {Lasso} $ including prediction norm rate and sparsity. Our analysis is based on new impact factors that are tailored for bounding prediction norm. In order to cover heteroscedastic non-Gaussian noise, we rely on moderate deviation theory for self-normalized sums to achieve Gaussian-like results under weak conditions. Moreover, we derive bounds on the performance of ordinary least square (ols) applied to the model selected by $\\sqrt {Lasso} $ accounting for possible misspecification of the selected model. Under mild conditions, the rate of convergence of ols post $\\sqrt {Lasso} $ is as good as $\\sqrt {Lasso's} $ rate. As an application, we consider the use of $\\sqrt {Lasso} $ and ols post $\\sqrt {Lasso} $ as estimators of nuisance parameters in a generic semiparametric problem (nonlinear moment condition or Z-problem), resulting in a construction of $\\sqrt n - consistent$ and asymptotically normal estimators of the main parameters.

We address the issue of detecting changes of models that lie behind a data stream. The model refers to an integer-valued structural information such as the number of free parameters in a parametric model. Specifically we are concerned with the problem of how we can detect signs of model changes earlier than they are actualized. To this end, we employ continuous model selection on the basis of the notion of descriptive dimensionality (Ddim). It is a real-valued model dimensionality, which is designed for quantifying the model dimensionality in the model transition period. Continuous model selection is to determine the real-valued model dimensionality in terms of Ddim from a given data. We propose a novel methodology for detecting signs of model changes by tracking the rise-up/descent of Ddim in a data stream. We apply this methodology to detecting signs of changes of the number of clusters in a Gaussian mixture model and those of the order in an auto regression model. With synthetic and real data sets, we empirically demonstrate its effectiveness by showing that it is able to visualize well how rapidly model dimensionality moves in the transition period and to raise early warning signals of model changes earlier than they are detected with existing methods.

We show a link between Bayesian inference and information theory that is useful for model selection, assessment of information entropy and experimental design. We align Bayesian model evidence (BME) with relative entropy and cross entropy in order to simplify computations using prior-based (Monte Carlo) or posterior-based (Markov chain Monte Carlo) BME estimates. On the one hand, we demonstrate how Bayesian model selection can profit from information theory to estimate BME values via posterior-based techniques. Hence, we use various assumptions including relations to several information criteria. On the other hand, we demonstrate how relative entropy can profit from BME to assess information entropy during Bayesian updating and to assess utility in Bayesian experimental design. Specifically, we emphasize that relative entropy can be computed avoiding unnecessary multidimensional integration from both prior and posterior-based sampling techniques. Prior-based computation does not require any assumptions, however posterior-based estimates require at least one assumption. We illustrate the performance of the discussed estimates of BME, information entropy and experiment utility using a transparent, non-linear example. The multivariate Gaussian posterior estimate includes least assumptions and shows the best performance for BME estimation, information entropy and experiment utility from posterior-based sampling.