In this paper we develop valid inference for high-dimensional time series. We extend the desparsified lasso to a time series setting under Near-Epoch Dependence (NED) assumptions allowing for non-Gaussian, serially correlated and heteroskedastic processes, where the number of regressors can possibly grow faster than the time dimension. We first derive an error bound under weak sparsity, which, coupled with the NED assumption, means this inequality can also be applied to the (inherently misspecified) nodewise regressions performed in the desparsified lasso. This allows us to establish the uniform asymptotic normality of the desparsified lasso under general conditions, including for inference on parameters of increasing dimensions. Additionally, we show consistency of a long-run variance estimator, thus providing a complete set of tools for performing inference in high-dimensional linear time series models. Finally, we perform a simulation exercise to demonstrate the small sample properties of the desparsified lasso in common time series settings.
Many theoretical results for the lasso require the samples to be iid. Recent work has provided guarantees for the lasso assuming that the time series is generated by a sparse Vector Auto-Regressive (VAR) model with Gaussian innovations. Proofs of these results rely critically on the fact that the true data generating mechanism (DGM) is a finite-order Gaussian VAR. This assumption is quite brittle: linear transformations, including selecting a subset of variables, can lead to the violation of this assumption. In order to break free from such assumptions, we derive non-asymptotic inequalities fo…
Many theoretical results on estimation of high dimensional time series require specifying an underlying data generating model (DGM). Instead, along the footsteps of~\cite{wong2017lasso}, this paper relies only on (strict) stationarity and $ β$-mixing condition to establish consistency of lasso when data comes from a $β$-mixing process with marginals having subgaussian tails. Because of the general assumptions, the data can come from DGMs different than standard time series models such as VAR or ARCH. When the true DGM is not VAR, the lasso estimates correspond to those of the best linear predi…
Discussion of "A statistical analysis of multiple temperature proxies: Are reconstructions of surface temperatures over the last 1000 years reliable?" by B.B. McShane and A.J. Wyner [arXiv:1104.4002]
This paper introduces structured machine learning regressions for high-dimensional time series data potentially sampled at different frequencies. The sparse-group LASSO estimator can take advantage of such time series data structures and outperforms the unstructured LASSO. We establish oracle inequalities for the sparse-group LASSO estimator within a framework that allows for the mixing processes and recognizes that the financial and the macroeconomic data may have heavier than exponential tails. An empirical application to nowcasting US GDP growth indicates that the estimator performs favorab…
We improve upon the two-stage sparse vector autoregression (sVAR) method in Davis et al. (2016) by proposing an alternative two-stage modified sVAR method which relies on time series graphical lasso to estimate sparse inverse spectral density in the first stage, and the second stage refines non-zero entries of the AR coefficient matrices using a false discovery rate (FDR) procedure. Our method has the advantage of avoiding the inversion of the spectral density matrix but has to deal with optimization over Hermitian matrices with complex-valued entries. It significantly improves the computation…
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…
We extend the feature selection methodology to dependent data and propose a novel time series predictor selection scheme that accommodates statistical dependence in a more typical i.i.d sub-sampling based framework. Furthermore, the machinery of mixing stationary processes allows us to quantify the improvements of our approach over any base predictor selection method (such as lasso) even in a finite sample setting. Using the lasso as a base procedure we demonstrate the applicability of our methods to simulated and several real time series datasets.
We study the finite sample behavior of Lasso-based inference methods such as post double Lasso and debiased Lasso. We show that these methods can exhibit substantial omitted variable biases (OVBs) due to Lasso not selecting relevant controls. This phenomenon can occur even when the coefficients are sparse and the sample size is large and larger than the number of controls. Therefore, relying on the existing asymptotic inference theory can be problematic in empirical applications. We compare the Lasso-based inference methods to modern high-dimensional OLS-based methods and provide practical gui…
A fundamental challenge in estimations of daily streamflow time series at sites with incomplete records is how to effectively and efficiently select reference or donor gauges from an existing gauge network to infer the missing data. While research on estimating missing streamflow time series is not new, the existing approaches either use a single reference streamflow gauge or employ a set of "ad-hoc" reference gauges, leaving a systematic selection of reference gauges as a long-standing open question. In this work, a novel method is introduced that facilitates systematical selection of multipl…
We present an efficient alternating direction method of multipliers (ADMM) algorithm for segmenting a multivariate non-stationary time series with structural breaks into stationary regions. We draw from recent work where the series is assumed to follow a vector autoregressive model within segments and a convex estimation procedure may be formulated using group fused lasso penalties. Our ADMM approach first splits the convex problem into a global quadratic program and a simple group lasso proximal update. We show that the global problem may be parallelized over rows of the time dependent transi…
Aix Marseille University（艾克斯-马赛大学）；An-Najah National University（纳贾赫国立大学）
In this paper, we consider modelling interaction between a set of variables in the context of time series and high dimension. We suggest two approaches. The first is similar to the neighborhood lasso when the lasso model is replaced by a support vector machine (SVMs). The second is a restricted Bayesian network adapted for time series. We show the efficiency of our approaches by simulations using linear, nonlinear data set and a mixture of both.
Many recent developments in the high-dimensional statistical time series literature have centered around time-dependent applications that can be adapted to regularized least squares. Of particular interest is the lasso, which both serves to regularize and provide feature selection. The lasso requires the specification of a penalty parameter that determines the degree of sparsity to impose. The most popular penalty parameter selection approaches that respect time dependence are very computationally intensive and are not appropriate for modeling certain classes of time series. We propose enhanci…
The Vector AutoRegressive (VAR) model is fundamental to the study of multivariate time series. Although VAR models are intensively investigated by many researchers, practitioners often show more interest in analyzing VARX models that incorporate the impact of unmodeled exogenous variables (X) into the VAR. However, since the parameter space grows quadratically with the number of time series, estimation quickly becomes challenging. While several proposals have been made to sparsely estimate large VAR models, the estimation of large VARX models is under-explored. Moreover, typically these sparse…
We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary multivariate Gaussian time series. A sparse-group lasso based frequency-domain formulation of the problem has been considered in the literature where the objective is to estimate the sparse inverse power spectral density (PSD) of the data. The CIG is then inferred from the estimated inverse PSD. In this paper we investigate use of a sparse-group log-sum penalty (LSP) instead of sparse-group lasso penalty. An alternating direction method of multipliers (ADMM) approach for iterative opt…
This paper studies high-dimensional regression models with lasso when data is sampled under multi-way clustering. First, we establish convergence rates for the lasso and post-lasso estimators. Second, we propose a novel inference method based on a post-double-selection procedure and show its asymptotic validity. Our procedure can be easily implemented with existing statistical packages. Simulation results demonstrate that the proposed procedure works well in finite sample. We illustrate the proposed method with a couple of empirical applications to development and growth economics.
University of California, Los Angeles（加州大学洛杉矶分校）；University of Copenhagen（哥本哈根大学）；Aarhus Center for Econometrics (ACE)（奥胡斯计量经济学中心）；Yale University（耶鲁大学）
LASSO Inference for High Dimensional Predictive Regressions
Southern Methodist University（南方卫理公会大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Macau（澳门大学）；The Chinese University of Hong Kong（香港中文大学）
We consider the estimation and inference in a system of high-dimensional regression equations allowing for temporal and cross-sectional dependency in covariates and error processes, covering rather general forms of weak temporal dependence. A sequence of regressions with many regressors using LASSO (Least Absolute Shrinkage and Selection Operator) is applied for variable selection purpose, and an overall penalty level is carefully chosen by a block multiplier bootstrap procedure to account for multiplicity of the equations and dependencies in the data. Correspondingly, oracle properties with a…
We study Granger causality testing for high-dimensional time series using regularized regressions. To perform proper inference, we rely on heteroskedasticity and autocorrelation consistent (HAC) estimation of the asymptotic variance and develop the inferential theory in the high-dimensional setting. To recognize the time series data structures we focus on the sparse-group LASSO estimator, which includes the LASSO and the group LASSO as special cases. We establish the debiased central limit theorem for low dimensional groups of regression coefficients and study the HAC estimator of the long-run…
We consider inference about coefficients on a small number of variables of interest in a linear panel data model with additive unobserved individual and time specific effects and a large number of additional time-varying confounding variables. We allow the number of these additional confounding variables to be larger than the sample size, and suppose that, in addition to unrestricted time and individual specific effects, these confounding variables are generated by a small number of common factors and high-dimensional weakly-dependent disturbances. We allow that both the factors and the distur…
This article proposes an estimation method to detect breakpoints for linear time series models with their parameters that jump scarcely. Its basic idea owes the group LASSO (group least absolute shrinkage and selection operator). The method practically provides estimates of such time-varying parameters of the models. An example shows that our method can detect each structural breakpoint's date and magnitude.
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…
