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.
Most existing predictor selection methods in time series are largely based on heuristics, @cite_22 or simply use plain lasso @cite_8 @cite_14 on the entire data and it is non-trivial to provide guarantees for such methods. For the specific case of vector auto regression (VAR) models, propose a grouped penalty based approach that provably identifies relevant lags and predictors in the asymptotic @math (number of time series) and @math (number of lags) regime. Our method is of a fundamentally distinct flavor in that we provide quantifiable improvement over any base predictor selection method, including the method in @cite_7 , even in the finite data (sample or dimension) setting. Moreover our approach also works for the more general VAR-X (VAR with exogenous) model and in general is independent of the base predictor selection mechanism.
@cite_14 @cite_22 @cite_7 @cite_8
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The vector autoregressive (VAR) model is a powerful tool in learning complex time series and has been exploited in many fields. The VAR model poses some unique challenges to researchers: On one hand, the dimensionality, introduced by incorporating multiple numbers of time series and adding the order of the vector autoregression, is usually much higher than the time series length; On the other hand, the temporal dependence structure naturally present in the VAR model gives rise to extra difficulties in data analysis. The regular way in cracking the VAR model is via "least squares" and usually involves adding different penalty terms (e.g., ridge or lasso penalty) in handling high dimensionality. In this manuscript, we propose an alternative way in estimating the VAR model. The main idea is, via exploiting the temporal dependence structure, formulating the estimating problem to a linear program. There is instant advantage of the proposed approach over the lasso-type estimators: The estimation equation can be decomposed to multiple sub-equations and accordingly can be solved efficiently using parallel computing. Besides that, we also bring new theoretical insights into the VAR model analysis. So far the theoretical results developed in high dimensions (e.g., Song and Bickel, 2011 and Kock and Callot, 2015) are based on stringent assumptions that are not transparent. Our results, on the other hand, show that the spectral norms of the transition matrices play an important role in estimation accuracy and build estimation and prediction consistency accordingly. Moreover, we provide some experiments on both synthetic and real-world equity data. We show that there are empirical advantages of our method over the lasso-type estimators in parameter estimation and forecasting. We propose robust methods for inference on the effect of a treatment variable on a scalar outcome in the presence of very many controls. Our setting is a partially linear model with possibly non-Gaussian and heteroscedastic disturbances. Our analysis allows the number of controls to be much larger than the sample size. To make informative inference feasible, we require the model to be approximately sparse; that is, we require that the effect of confounding factors can be controlled for up to a small approximation error by conditioning on a relatively small number of controls whose identities are unknown. The latter condition makes it possible to estimate the treatment effect by selecting approximately the right set of controls. We develop a novel estimation and uniformly valid inference method for the treatment effect in this setting, called the "post-double-selection" method. Our results apply to Lasso-type methods used for covariate selection as well as to any other model selection method that is able to find a sparse model with good approximation properties. The main attractive feature of our method is that it allows for imperfect selection of the controls and provides confidence intervals that are valid uniformly across a large class of models. In contrast, standard post-model selection estimators fail to provide uniform inference even in simple cases with a small, fixed number of controls. Thus our method resolves the problem of uniform inference after model selection for a large, interesting class of models. We illustrate the use of the developed methods with numerical simulations and an application to the effect of abortion on crime rates. We consider the least-square linear regression problem with regularization by the l1-norm, a problem usually referred to as the Lasso. In this paper, we present a detailed asymptotic analysis of model consistency of the Lasso. For various decays of the regularization parameter, we compute asymptotic equivalents of the probability of correct model selection (i.e., variable selection). For a specific rate decay, we show that the Lasso selects all the variables that should enter the model with probability tending to one exponentially fast, while it selects all other variables with strictly positive probability. We show that this property implies that if we run the Lasso for several bootstrapped replications of a given sample, then intersecting the supports of the Lasso bootstrap estimates leads to consistent model selection. This novel variable selection algorithm, referred to as the Bolasso, is compared favorably to other linear regression methods on synthetic data and datasets from the UCI machine learning repository. Extracting useful information from high-dimensional data is an important focus of today's statistical research and practice. Penalized loss function minimization has been shown to be effective for this task both theoretically and empirically. With the virtues of both regularization and sparsity, the L 1 -penalized squared error minimization method Lasso has been popular in regression models and beyond. In this paper, we combine different norms including L 1 to form an intelligent penalty in order to add side information to the fitting of a regression or classification model to obtain reasonable estimates. Specifically, we introduce the Composite Absolute Penalties (CAP) family, which allows given grouping and hierarchical relationships between the predictors to be expressed. CAP penalties are built by defining groups and combining the properties of norm penalties at the across-group and within-group levels. Grouped selection occurs for nonoverlapping groups. Hierarchical variable selection is reached by defining groups with particular overlapping patterns. We propose using the BLASSO and cross-validation to compute CAP estimates in general. For a subfamily of CAP estimates involving only the L 1 and L ∞ norms, we introduce the iCAP algorithm to trace the entire regularization path for the grouped selection problem. Within this subfamily, unbiased estimates of the degrees of freedom (df) are derived so that the regularization parameter is selected without cross-validation. CAP is shown to improve on the predictive performance of the LASSO in a series of simulated experiments, including cases with p » n and possibly mis-specified groupings. When the complexity of a model is properly calculated, iCAP is seen to be parsimonious in the experiments.