# Oracle Inequalities for High Dimensional Vector Autoregressions

arXiv: https://arxiv.org/abs/1311.0811 · HTML: https://mlsanity.com/paper/1311.0811/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2013.

## Built on by (most cited first, 11 of 11)

- [Criteria for Classifying Forecasting Methods](https://mlsanity.com/paper/2212.03523/) (2022)
- [Neural graphical modelling in continuous-time: consistency guarantees and algorithms](https://mlsanity.com/paper/2105.02522/) (2021)
- [Learning linear dynamical systems under convex constraints](https://mlsanity.com/paper/2303.15121/) (2023)
- [Benign Overfitting in Time Series Linear Models with Over-Parameterization](https://mlsanity.com/paper/2204.08369/) (2022)
- [Sparse Generalized Yule-Walker Estimation for Large Spatio-temporal Autoregressions with an Application to NO2 Satellite Data](https://mlsanity.com/paper/2108.02864/) (2021)
- [On LASSO for High Dimensional Predictive Regression](https://mlsanity.com/paper/2212.07052/) (2022)
- [Concentration inequalities for high-dimensional linear processes with dependent innovations](https://mlsanity.com/paper/2307.12395/) (2023)
- [Econometrics of Machine Learning Methods in Economic Forecasting](https://mlsanity.com/paper/2308.10993/) (2023)
- [Minimax Estimation of Partially-Observed Vector AutoRegressions](https://mlsanity.com/paper/2106.09327/) (2021)
- [Identification of Non-causal Graphical Models](https://mlsanity.com/paper/2410.09480/) (2024)
- [Symmetric Linear Dynamical Systems are Learnable from Few Observations](https://mlsanity.com/paper/2512.05337/) (2025)

## Abstract (arXiv)

This paper establishes non-asymptotic oracle inequalities for the prediction error and estimation accuracy of the LASSO in stationary vector autoregressive models. These inequalities are used to establish consistency of the LASSO even when the number of parameters is of a much larger order of magnitude than the sample size. We also give conditions under which no relevant variables are excluded. Next, non-asymptotic probabilities are given for the Adaptive LASSO to select the correct sparsity pattern. We then give conditions under which the Adaptive LASSO reveals the correct sparsity pattern asymptotically. We establish that the estimates of the non-zero coefficients are asymptotically equivalent to the oracle assisted least squares estimator. This is used to show that the rate of convergence of the estimates of the non-zero coefficients is identical to the one of least squares only including the relevant covariates.