arXiv subjects

Likai Chen

Publications and source records attributed to Likai Chen.

7 recordsLinked to original sources

arXiv2026

Online simultaneous inference for quantiles via smoothed stochastic gradient descent

This paper considers the estimation of quantiles via a smoothed version of the stochastic gradient descent (SGD) algorithm. By smoothing the score function with a bandwidth tied to the learning rate, we obtain estimates that are monotone in the quantile level at every iteration, while retaining the memory and computational efficiency required for streaming data. We establish non-asymptotic tail probability bounds for the smoothed estimate with and without Polyak-Ruppert averaging, which are sub-exponential with a multi-regime structure. For the averaged estimate we further derive a Bahadur representation that is uniform in the quantile level and across coordinates, and a resulting Gaussian approximation by the maximum of Brownian bridges, with the dimension $p$ allowed to grow exponentially in the sample size. This yields simultaneous inference across coordinates and quantile levels. As an alternative that avoids estimating the sparsity function, we propose an online multiplier bootstrap that preserves monotonicity, runs in a single pass and is asymptotically valid. Extending the theory to a localized recursion, we obtain online nonparametric conditional quantile estimates with uniform bands over design points and quantile levels. Simulations confirm accurate finite-sample coverage, and we illustrate the method on conditional value-at-risk curves.

stat.ML↗

arXiv2026

From Vector Autoregressions to AI-based Time Series Forecasting: A Review

Forecasting is a central goal of time-series analysis. This review centers on three major developments in recent AI-based time-series forecasting: transformers, large pretrained models for zero-shot forecasting, and diffusion-based generative forecasters. We connect these methods to the econometric tradition built around the vector autoregression (VAR) through a common object: the conditional distribution of the future given the past. The review is organized around three long-standing challenges: \emph{high dimensionality}, \emph{nonstationarity}, and \emph{nonlinearity}. We argue that modern methods make progress by expanding the classical forecasting template: they allow more flexible dynamics, use larger information sets and training corpora, and represent richer predictive distributions. Yet they often lack the inferential and structural tools that make classical models useful for testing, explanation, and policy analysis. We close by outlining open problems where econometric tools remain important.

econ.EM↗

arXiv2026

High-dimensional inference on jumps in nonparametric time series regression models

We study simultaneous inference on jumps in the conditional mean functions of a high-dimensional collection of heterogeneous nonparametric time series, where the number of series may exceed the sample size and the data may exhibit strong cross-sectional dependence. The jump depends on one specific covariate, and we allow the regression function to vary with additional latent variables. We propose two uniform tests: one for the existence of jumps and one for their homogeneity across series. We derive a simple closed-form approximation to the covariance structure of the jump estimators and establish a high-dimensional Gaussian approximation showing that, owing to the localized construction of the statistics, the maximum of the studentized jumps is approximated by the maximum of independent Gaussians. The cross-sectional dependence is thus asymptotically negligible for critical values, even under strong (e.g., factor) dependence, and the approximation requires estimating only the variance for each series. For pronounced cross-sectional dependence, a dependence-aware refinement restores the off-diagonal covariances, improving finite-sample size and power. Simulations show accurate size and reasonable power under both cross-sectional and serial dependence, and two empirical applications reveal significant non-smooth effects.

econ.EM↗

arXiv2026

Central Limit Theorems for Stochastic Gradient Descent Quantile Estimators

This paper develops asymptotic theory for quantile estimation via stochastic gradient descent (SGD) with a constant learning rate. The quantile loss function is neither smooth nor strongly convex. Beyond conventional perspectives and techniques, we view quantile SGD iteration as an irreducible, periodic, and positive recurrent Markov chain, which cyclically converges to its unique stationary distribution regardless of the arbitrarily fixed initialization. To derive the exact form of the stationary distribution, we analyze the structure of its characteristic function by exploiting the stationary equation. We also derive tight bounds for its moment generating function (MGF) and tail probabilities. Synthesizing the aforementioned approaches, we prove that the centered and standardized stationary distribution converges to a Gaussian distribution as the learning rate $η\rightarrow0$. This finding provides the first central limit theorem (CLT)-type theoretical guarantees for the quantile SGD estimator with constant learning rates. We further propose a recursive algorithm to construct confidence intervals of the estimators with statistical guarantees. Numerical studies demonstrate the effective finite-sample performance of the online estimator and inference procedure. The theoretical tools developed in this study are of independent interest for investigating general SGD algorithms formulated as Markov chains, particularly in non-strongly convex and non-smooth settings.

stat.ML↗

arXiv2025

Estimation of High-dimensional Nonlinear Vector Autoregressive Models

High-dimensional vector autoregressive (VAR) models have numerous applications in fields such as econometrics, biology, climatology, among others. While prior research has mainly focused on linear VAR models, these approaches can be restrictive in practice. To address this, we introduce a high-dimensional non-parametric sparse additive model, providing a more flexible framework. Our method employs basis expansions to construct high-dimensional nonlinear VAR models. We derive convergence rates and model selection consistency for least squared estimators, considering dependence measures of the processes, error moment conditions, sparsity, and basis expansions. Our theory significantly extends prior linear VAR models by incorporating both non-Gaussianity and non-linearity. As a key contribution, we derive sharp Bernstein-type inequalities for tail probabilities in both non-sub-Gaussian linear and nonlinear VAR processes, which match the classical Bernstein inequality for independent random variables. Additionally, we present numerical experiments that support our theoretical findings and demonstrate the advantages of the nonlinear VAR model for a gene expression time series dataset.

math.ST↗

arXiv2023

Simultaneous Inference of a Partially Linear Model in Time Series

We introduce a new methodology to conduct simultaneous inference of the nonparametric component in partially linear time series regression models where the nonparametric part is a multivariate unknown function. In particular, we construct a simultaneous confidence region (SCR) for the multivariate function by extending the high-dimensional Gaussian approximation to dependent processes with continuous index sets. Our results allow for a more general dependence structure compared to previous works and are widely applicable to a variety of linear and nonlinear autoregressive processes. We demonstrate the validity of our proposed methodology by examining the finite-sample performance in the simulation study. Finally, an application in time series, the forward premium regression, is presented, where we construct the SCR for the foreign exchange risk premium from the exchange rate and macroeconomic data.

stat.ME↗

arXiv2023

$\ell^2$ Inference for Change Points in High-Dimensional Time Series via a Two-Way MOSUM

We propose an inference method for detecting multiple change points in high-dimensional time series, targeting dense or spatially clustered signals. Our method aggregates moving sum (MOSUM) statistics cross-sectionally by an $\ell^2$-norm and maximizes them over time. We further introduce a novel Two-Way MOSUM, which utilizes spatial-temporal moving regions to search for breaks, with the added advantage of enhancing testing power when breaks occur in only a few groups. The limiting distribution of an $\ell^2$-aggregated statistic is established for testing break existence by extending a high-dimensional Gaussian approximation theorem to spatial-temporal non-stationary processes. Simulation studies exhibit promising performance of our test in detecting non-sparse weak signals. Two applications, analyzing equity returns and COVID-19 cases in the United States, showcase the real-world relevance of our proposed algorithms.

math.ST↗