Modified LASSO estimators for time series regression models with dependent disturbances
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ORIGINAL PAPER
Modified LASSO estimators for time series regression models with dependent disturbances Yujie Xue1
•
Masanobu Taniguchi2
Accepted: 3 January 2020 Ó Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper applies the modified least absolute shrinkage and selection operator (LASSO) to the regression model with dependent disturbances, especially, longmemory 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 k 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. Keywords Modified LASSO Long-memory disturbances High dimensional regression
1 Introduction In actual applications, it is rare that we are firmly convinced of a parameter specification in the face of ever-increasing multitudes of data. That is the reason of developing model selection. The least absolute shrinkage and selection operator (LASSO) which was proposed by Tibshirani (1996) as L1 norm has been widely applied to prediction and variable selection problems because of its computational & Yujie Xue [email protected] Masanobu Taniguchi [email protected] 1
Department of Pure and Applied Mathematics, Waseda University, 3 Chome-4-1, Okubo, Shinjuku-ku, Tokyo, Japan
2
Research Institute for Science and Engineering, Waseda University, 3 Chome-4-1, Okubo, Shinjuku-ku, Tokyo, Japan
123
Y. Xue, M. Taniguchi
feasibility. Wang et al. (2007) extended its application to the regression model with autoregressive disturbances. Tang et al. (2012) showed that the adaptive LASSO based on self-weighted quantile regression enjoys the oracle properties for infinite variance autoregressive models. LASSO has also been discussed in the sense of high dimension. Meinshausen and Bu¨hlmann (2006) discussed that consistency of neighborhood with the LASSO hinges on the choice of the penalty parameter for high-dimensional graphs. Kock and Callot (2015) showed the oracle inequalities for high dimensional vector autoregressions. Medeiros and Mendes (2016) studied the asymptotic properties of adaptive sparse, high-dimensional, multiple linear timeseries models. Kaul (2014) studied the case where regression errors form a long memory moving average process and regressors are known nonrandom. A finite sample oracle inequality for the Lasso solution is established, and when the dimension p of variables is greater than the observation length n, the consistency on the probability of the correct selection of parameters was shown. Also when p is fixed, the consistency (asymptotic normal) of LASSO estimat
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