Computational Statistics and Data Analysis 52 (2008) 3645–3657 www.elsevier.com/locate/csda Subset selection for vector autoregressive processes using Lasso Nan-Jung Hsu ∗ , Hung-Lin Hung, Ya-Mei Chang Institute of Statistics, National Tsing-Hua University, Taiwan Received 6 January 2007; received in revised form 12 August 2007; accepted 7 December 2007 Available online 8 January 2008 Abstract A subset selection method is proposed for vector autoregressive (VAR) processes using the Lasso [Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society, Series B 58, 267–288] technique. Simply speaking, Lasso is a shrinkage method in a regression setup which selects the model and estimates the parameters simultaneously. Compared to the conventional information-based methods such as AIC and BIC, the Lasso approach avoids computationally intensive and exhaustive search. On the other hand, compared to the existing subset selection methods with parameter constraints such as the top-down and bottom-up strategies, the Lasso method is computationally efficient and its result is robust to the order of series included in the autoregressive model. We derive the asymptotic theorem for the Lasso estimator under VAR processes. Simulation results demonstrate that the Lasso method performs better than several conventional subset selection methods for small samples in terms of prediction mean squared errors and estimation errors under various settings. The methodology is applied to modeling U.S. macroeconomic data for illustration. c  2007 Elsevier B.V. All rights reserved. 1. Introduction Multivariate time series processes are of interest in many fields, such as physical sciences, geophysics, meteorology, social sciences, particularly economics, finance and business. For example, one may be interested in the dynamics between different financial markets and try to understand the impact of changes in one market on the others (see for example Tsay (2002) Chapter 8, and the references therein). As another example, Niu and Tiao (1995) analyzed the satellite ozone data for assessing long-term trends in ozone distributions in which spatial data corrected at many locations over time are treated as a multivariate time series. The vector autoregressive (VAR) models are the most popular and successful models for analyzing multiple time series in the literature, because of their simple specification with easy interpretation for the dynamic relationships between series. In order to find a suitable VAR model and produce good predictions, model selection is the key issue in the statistical analysis. Roughly speaking, there are three types of procedures for determining the best VAR model in previous studies. The first type is the information- based methods such as AIC (Akaike, 1974) and BIC (Schwarz, 1978). Usually, the information-based criteria focus only on selecting the best order instead of exhaustive searching among all possible subset structures due to infeasible computations. For example, suppose the multiple time series are k -dimensional and the largest order considered in the ∗ Corresponding author. E-mail address: njhsu@stat.nthu.edu.tw (N.-J. Hsu). 0167-9473/$ - see front matter c  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2007.12.004