Oracle Inequalities for High Dimensional Vector Autoregressions

By Anders Bredahl Kock, Laurent Callot. 2012

247 citations recorded by the source corpus.

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. Furthermore, it is shown that under suitable conditions the number of variables selected is of the right order of magnitude and that no relevant variables are excluded. Next, non-asymptotic probabilities are given for the Adaptive LASSO to select the correct sign pattern (and hence the correct sparsity pattern). Finally conditions under which the Adaptive LASSO reveals the correct sign pattern with probability tending to one are given. Again, the number of parameters may be much larger than the sample size. Some maximal inequalities for vector autoregressions which might be of independent interest are contained in the appendix.

Subfields: Control and Systems Engineering, Statistics and Probability

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Citations in the corpus

High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes (2019, methodology): “The parameters of equation 1 are estimated by the Lasso implemented in [4] using the procedure described in [8].”
On LASSO for predictive regression (2018, unknown): “Kock and Callot (2015) discuss LASSO in a vector autoregression (VAR) system.”
Forecasting Macroeconomic Variables Using Neural Network Models and Three Automated Model Selection Techniques (2016, methodology): “(2010, Chapter 14), or Kock and Teräsvirta (2011a). In nonlinear forecasting, the latter method appears to be more common, see for example Stock and Watson (1999) and Marcellino (2002), whereas Teräsvirta et al. (2005) constitutes an example of the former alternative. A systematic comparison of the ”
Machine Learning Advances for Time Series Forecasting (2020, background): “In high-dimensional settings, where the dimension of the series increase with the number of observations, Kock and Callot (2015) and Basu and Michailidis (2015) show oracle bounds and model selection consistency for the LASSO in Gaussian 𝑉𝐴𝑅(𝑝) models, extending previous works.”
Lassoing the HAR Model: A Model Selection Perspective on Realized Volatility Dynamics (2016, unknown): “…also been established in time series econometrics by, for example, Wang et al. (2007), Hsu et al. (2008), Nardi and Rinaldo (2011), Kock (2012), Kock and Callot (2012), Medeiros and Mendes (2012), and Audrino and Camponovo (2013), and applications of the Lasso are also found in Park and…”
Regularized estimation in sparse high-dimensional time series models (2013, background): “pendence among the columns of the error matrix in the multiple regression. Some early attempts to establish consistency of VAR models under high-dimensional scaling include Song and Bickel (2011) and Kock and Callot (2012). Both papers rely on certain regularity assumptions but do not investigate th”
NETS: Network estimation for time series (2019, unknown): “This paper is also related to the literature on the estimation of sparse VARs (see Basu & Michailidis, 2015; Davis et al., 2016; Kock & Callot, 2015; Medeiros & Mendes, 2016; Song & Bickel, 2011).”
CONSISTENT AND CONSERVATIVE MODEL SELECTION WITH THE ADAPTIVE LASSO IN STATIONARY AND NONSTATIONARY AUTOREGRESSIONS (2015, background): “Oracle inequalities for high-dimensional stationary VAR models have been established in Kock and Callot (2015).”
Estimation and Selection of Spatial Weight Matrix in a Spatial Lag Model (2019, background): “In particular, Kock and Callot (2015) demonstrated Oracle properties of LASSO and Adaptive LASSO estimators in the context of high-dimensional vector autoregresions.”
Forecasting Inflation in a Data-Rich Environment: The Benefits of Machine Learning Methods (2019, unknown): “8 For all methods within the LASSO family, the penalty parameter λ is chosen by the BIC as advocated by Kock and Callot (2015) and Medeiros and Mendes (2016).”
Oracle Efficient Estimation and Forecasting with the Adaptive Lasso and the Adaptive Group Lasso in Vector Autoregressions (2012, methodology): “with the sample size we refer to Kock and Callot (2012) who however don’t consider the adaptive group Lasso.”
Modeling and Forecasting Large Realized Covariance Matrices and Portfolio Choice (2017, unknown): “In this paper we will take a different route and will consider the estimation of vast vector autoregressive models for realized measures of covariance matrices as in Kock and Callot (2015).”