Oracle Inequalities for High Dimensional Vector Autoregressions
This work addresses the problem of high-dimensional time series modeling for statisticians and econometricians, offering incremental theoretical guarantees for LASSO-based methods.
The paper establishes non-asymptotic oracle inequalities for LASSO and Adaptive LASSO in high-dimensional vector autoregressive models, showing consistency even when parameters vastly outnumber samples and providing conditions for correct variable selection and asymptotic equivalence to oracle estimators.
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.