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Oracle Inequalities for High Dimensional Vector Autoregressions

Author

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  • Anders Bredahl Kock

    () (Aarhus University and CREATES)

  • Laurent A.F. Callot

    () (Aarhus University and CREATES)

Abstract

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.

Suggested Citation

  • Anders Bredahl Kock & Laurent A.F. Callot, 2012. "Oracle Inequalities for High Dimensional Vector Autoregressions," CREATES Research Papers 2012-16, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2012-16
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    References listed on IDEAS

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    Cited by:

    1. Fengler, Matthias R. & Gisler, Katja I.M., 2015. "A variance spillover analysis without covariances: What do we miss?," Journal of International Money and Finance, Elsevier, vol. 51(C), pages 174-195.
    2. Mehmet Caner & Anders Bredahl Kock, 2016. "Oracle Inequalities for Convex Loss Functions with Nonlinear Targets," Econometric Reviews, Taylor & Francis Journals, pages 1377-1411.
    3. Medeiros, Marcelo C. & Mendes, Eduardo F., 2016. "ℓ1-regularization of high-dimensional time-series models with non-Gaussian and heteroskedastic errors," Journal of Econometrics, Elsevier, vol. 191(1), pages 255-271.
    4. Audrino, Francesco & Camponovo, Lorenzo & Roth, Constantin, 2015. "Testing the lag structure of assets’ realized volatility dynamics," Economics Working Paper Series 1501, University of St. Gallen, School of Economics and Political Science.
    5. Kaufmann, Hendrik & Kruse, Robinson & Sibbertsen, Philipp, 2012. "On tests for linearity against STAR models with deterministic trends," Economics Letters, Elsevier, pages 268-271.
    6. Gianluca Cubadda & Barbara Guardabascio, 2017. "Representation, Estimation and Forecasting of the Multivariate Index-Augmented Autoregressive Model," CEIS Research Paper 397, Tor Vergata University, CEIS, revised 07 Feb 2017.
    7. Ning Xu & Jian Hong & Timothy C. G. Fisher, 2016. "Model selection consistency from the perspective of generalization ability and VC theory with an application to Lasso," Papers 1606.00142, arXiv.org.
    8. Laurent Callot & Johannes Tang Kristensen, 2014. "Vector Autoregressions with parsimoniously Time Varying Parameters and an Application to Monetary Policy," Tinbergen Institute Discussion Papers 14-145/III, Tinbergen Institute, revised 09 Apr 2015.
    9. Matteo Barigozzi & Christian T. Brownlees, 2013. "Nets: Network estimation for time series," Economics Working Papers 1391, Department of Economics and Business, Universitat Pompeu Fabra.
    10. repec:eee:intfor:v:33:y:2017:i:3:p:627-651 is not listed on IDEAS
    11. Christensen, Bent Jesper & Nielsen, Morten Ørregaard & Zhu, Jie, 2015. "The impact of financial crises on the risk–return tradeoff and the leverage effect," Economic Modelling, Elsevier, vol. 49(C), pages 407-418.
    12. MArcelo C. Medeiros & Eduardo F.Mendes, 2012. "Estimating High-Dimensional Time Series Models," Textos para discussão 602, Department of Economics PUC-Rio (Brazil).
    13. Houba Harold & Motchenkova Evgenia & Wen Quan, 2015. "The Effects of Leniency on Cartel Pricing," The B.E. Journal of Theoretical Economics, De Gruyter, pages 351-389.
    14. Laurent Callot & Anders B. Kock & Marcelo C. Medeiros, 2014. "Estimation and Forecasting of Large Realized Covariance Matrices and Portfolio Choice," Tinbergen Institute Discussion Papers 14-147/III, Tinbergen Institute.
    15. Daniel Felix Ahelegbey & Monica Billio & Roberto Casarin, 2016. "Sparse Graphical Vector Autoregression: A Bayesian Approach," Annals of Economics and Statistics, GENES, pages 333-361.
    16. Audrino, Francesco & Camponovo, Lorenzo, 2013. "Oracle Properties and Finite Sample Inference of the Adaptive Lasso for Time Series Regression Models," Economics Working Paper Series 1327, University of St. Gallen, School of Economics and Political Science.
    17. Matteo Barigozzi & Marc Hallin, 2015. "Networks, Dynamic Factors, and the Volatility Analysis of High-Dimensional Financial Series," Papers 1510.05118, arXiv.org, revised Jul 2016.
    18. Baek, Changryong & Davis, Richard A. & Pipiras, Vladas, 2017. "Sparse seasonal and periodic vector autoregressive modeling," Computational Statistics & Data Analysis, Elsevier, pages 103-126.
    19. Kascha, Christian & Trenkler, Carsten, 2015. "Forecasting VARs, model selection, and shrinkage," Working Papers 15-07, University of Mannheim, Department of Economics.
    20. Marcelo C. Medeiros & Eduardo F. Mendes, 2015. "l1-Regularization of High-Dimensional Time-Series Models with Flexible Innovations," Textos para discussão 636, Department of Economics PUC-Rio (Brazil).
    21. Marcelo C. Medeiros & Eduardo F. Mendes, 2015. "Adaptative LASSO estimation for ARDL models with GARCH innovations," Textos para discussão 637, Department of Economics PUC-Rio (Brazil).
    22. Anders Bredahl Kock & Laurent A.F. Callot, 2012. "Oracle Efficient Estimation and Forecasting with the Adaptive LASSO and the Adaptive Group LASSO in Vector Autoregressions," CREATES Research Papers 2012-38, Department of Economics and Business Economics, Aarhus University.
    23. Smeekes, Stephan & Wijler, Etiënne, 2016. "Macroeconomic Forecasting Using Penalized Regression Methods," Research Memorandum 039, Maastricht University, Graduate School of Business and Economics (GSBE).
    24. Bicu A.C. & Lieb L.M., 2015. "Cross-border effects of fiscal policy in the Eurozone," Research Memorandum 019, Maastricht University, Graduate School of Business and Economics (GSBE).

    More about this item

    Keywords

    Vector autoregression; LASSO; Adaptive LASSO; Oracle inequality; Variable selection.;

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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