Sparse Parameter Estimation in Overcomplete Time Series Models

Abstract

We suggest a new approach to parameter estimation in time series models with large number of parameters. We use a modified version of the Basis Pursuit Algorithm (BPA) by Chen et al. [SIAM Review 43 (2001), No. 1] to verify its applicability to times series modelling. For simplicity we restrict to ARIMA models of univariate stationary time series. After having accomplished and analyzed a lot of numerical simulations we can draw the following conclusions: (1) We were able to reliably identify nearly zero parameters in the model allowing us to reduce the originally badly conditioned overparametrized model. Among others we need not take care about model orders the fixing of which is a common preliminary step used by standard techniques. For short time series paths (100 or less samples) the sparse parameter estimates provide more precise predictions compared with those based on standard maximum likelihood estimators from MATLAB’s System Identification Toolbox (IDENT). For longer paths (500 or more) both techniques yield nearly equal prediction paths. (2) As the model usually depends on the estimated parameters, we tried to improve their accuracy by iterating BPA several times.