Modified LASSO estimators for time series regression models with dependent disturbances


This paper applies the modified least absolute shrinkage and selection operator (LASSO) to the regression model with dependent disturbances, especially, long-memory disturbances. Assuming the norm of different column in the regression matrix may have different order of observation length n, we introduce a modified LASSO estimator where the tuning parameter \(\lambda\) is not a scalar but vector. When the dimension of parameters is fixed, we derive the asymptotic distribution of the modified LASSO estimators under certain regularity condition. When the dimension of parameters increases with respect to n, the consistency on the probability of the correct selection of penalty parameters is shown under certain regularity conditions. Some simulation studies are examined.

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  1. Anderson TW (1971) The statistical analysis of time series. Wiley, Hoboken

  2. Beran J (1992) Statistical methods for data with long-range dependence. Stat Sci 7(4):404–427

  3. Granger CW, Joyeux R (1980) An introduction to long-memory time series models and fractional differencing. J Time Ser Anal 1(1):15–29

  4. Grenander U, Rosenblatt M (1957) Statistical analysis of stationary time series. Wiley, New York

  5. Hallin M, Taniguchi M, Serroukh A, Choy K (1999) Local asymptotic normality for regression model with long-memory disturbance. Ann Stat 27(6):2054–2080

  6. Hannan EJ (1970) Multiple time series. Wiley, Hoboken

  7. Hurvich CM, Tsai CL (1989) Regression and time series model selection in small samples. Biometrika 76(2):297–307

  8. Kaul A (2014) Lasso with long memory regression errors. J Stat Plan Inference 153:11–26

  9. Kock AB, Callot L (2015) Oracle inequalities for high dimensional vector autoregressions. J Econom 180:325–344

  10. Mandelbrot BB, Taqqu MS (1979) Robust R/S analysis of long run serial correlation. Proc 42nd Sess Int Stat Inst (Bull Int Stat Inst) 48(2):69–104

  11. Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10(4):422–437

  12. Medeiros MC, Mendes EF (2016) \(l_1\)-regularization of high-dimensional time-series models with non-Gaussian and heteroskedastic errors. J Econom 191:255–271

  13. Meinshausen N, Bühlmann P (2006) High-dimensional graphs and variable selection with the lasso. Ann Stat 34(3):1436–1462

  14. Osborne MR, Presnell B, Turlach BA (2000) On the lasso and its dual. J Comput Graph Stat 9(2):319–337

  15. Tang L, Zhou Z, Wu C (2012) Efficient estimation and variable selection for infinite variance autoregressive models. J Appl Math Comput 40:399–413

  16. Taniguchi M, Hirukawa J, Tamaki K (2008) Optimal statistical inference in financial engineering. CRC Press, Boca Raton

  17. Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Series B 58:267–288

  18. Wang H, Li G, Tsai CL (2007) Regression coefficient and autoregressive order shrinkage and selection via the lasso. J R Stat Soc Ser B Stat Methodol 69(1):63–78

  19. Yajima Y (1991) Asymptotic properties of the LSE in a regression model with long-memory stationary errors. Ann Stat 19(1):158–177

  20. Yuan M, Lin Y (2007) Model selection and estimation in the Gaussian graphical model. Biometrika 94(1):19–35

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The first author would like to thank Mr. Fujimori for helping understand the high dimensional problem-solving idea. The second author thanks supports by Research Institute for Science and Engineering, Waseda University and JSPS Fundings: Kiban(A)(23244011, 15H02061) and Kiban(S)(18H05290).

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Correspondence to Yujie Xue.

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Xue, Y., Taniguchi, M. Modified LASSO estimators for time series regression models with dependent disturbances. Stat Methods Appl (2020) doi:10.1007/s10260-020-00506-w

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  • Modified LASSO
  • Long-memory disturbances
  • High dimensional regression