High-Dimensional Time Series Modeling and Inference

High-dimensional time series analysis addresses settings where the number of observed variables or series rivals or exceeds the number of time points. Such scenarios arise in macroeconomics, environmental monitoring, neuroscience and genomics, where vast arrays of sensors or indicators evolve in time and interact in complex ways. Traditional models focusing on a handful of series falter when faced with hundreds or thousands of interdependent processes. Modern approaches impose structural assumptions—most notably sparsity—so that only a limited subset of potential interactions or coefficients is nonzero. Penalised estimation techniques, such as the ℓ1-penalty or Lasso, enable consistent selection and accurate parameter estimation under high-dimensional scaling. Extensions to classical vector autoregressions accommodate time-varying coefficients, locally stationary dependencies and network-driven dynamics. Inference in this context requires careful control of bias introduced by regularisation, often achieved via desparsification or post-selection adjustment, coupled with robust variance estimation for serially correlated and heteroskedastic errors. These advances yield reliable forecasts, uncover causal relationships in volatility networks, and support real-time decision-making across diverse domains.

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Recent methodological work has adapted the desparsified Lasso for time series under weak dependence conditions. By combining uniform asymptotic normality with consistent long-run variance estimation, this framework permits valid confidence intervals and hypothesis tests even when the number of predictors grows faster than the sample size. Simulation studies confirm favourable finite-sample performance in forecasting and parameter inference.

Another strand focuses on Granger causality testing in high-dimensional vector autoregressions. A post-double-selection procedure guards against omitted-variable bias induced by initial Lasso screening. The resulting test maintains size and power across various dependence structures, and empirical applications reveal clearer spillover networks in financial volatility than low-dimensional counterparts.

Work on locally stationary vector autoregressive models advances estimation of smoothly time-varying transition matrices. A hybrid of kernel smoothing and ℓ1-regularisation yields estimators that adapt to evolving dependence patterns. Theoretical rates of convergence account for matrix smoothness, while thresholding delivers vanishing error rates in support recovery. Applications to stock and exchange-rate dynamics illustrate improved forecasting and interpretable evolution of interseries links.

High-Dimensional Time Series Modeling and Inference publication trend

The graph below shows the total number of articles in high-dimensional time series modeling and inference across all publications each year (not limited to Nature Index journals).

Technical terms

High-dimensional time series: A sequence of multivariate observations in which the number of variables is comparable to or exceeds the number of time points.

Sparsity: The condition that only a small fraction of model parameters or interactions are nonzero, enabling dimension reduction.

Lasso (ℓ1-penalty): A regularisation technique that shrinks coefficients toward zero and performs variable selection by imposing an ℓ1 norm penalty.

Vector autoregressive (VAR) model: A framework in which each variable is modelled as a linear function of its own past values and those of other variables.

Near-epoch dependence: A weak dependence condition allowing variables to be approximated by functions of an underlying mixing process over finite lags.

Granger causality: A predictive notion of causality where one series is said to Granger-cause another if past values of the first improve forecasts of the second.

References

1. Lasso inference for high-dimensional time series. Journal of Econometrics (2023).
2. Granger Causality Testing in High-Dimensional VARs: A Post-Double-Selection Procedure. Journal of Financial Econometrics (2021).
3. Sparse transition matrix estimation for high-dimensional and locally stationary vector autoregressive models. Electronic Journal of Statistics (2017).
4. Time series modeling on dynamic networks. Electronic Journal of Statistics (2019).