Regularized Autoregressive Approximation in Time Series

Abstract

In applications, the true underlying model of an observed time series is typically unknown or has a complicated structure. A common approach is to approximate the true model by autoregressive (AR) equation whose orders are chosen by information criterions such as AIC, BIC and Parsen's CAT and whose parameters are estimated by the least square (LS), the Yule Walker (YW) or other methods. However, as sample size increases, it often implies that the model order has to be refined and the parameters need to be recalculated. In order to avoid such shortcomings, we propose the Regularized AR (RAR) approximation and illustrate its applications in frequency detection and long memory process forecasting. The idea of the RAR approximation is to utilize a “long" AR model whose order significantly exceeds the model order suggested by information criterions, and to estimate AR parameters by Regularized LS (RLS) method, which enables to estimate AR parameters with different level of accuracy and the number of estimated parameters can grow linearly with the sample size. Therefore, the repeated model selection and parameter estimation are avoided as the observed sample increases.

We apply the RAR approach to estimate the unknown frequencies in periodic processes by approximating their generalized spectral densities, which significantly reduces the computational burden and improves accuracy of estimates. Our theoretical findings indicate that the RAR estimates of unknown frequency are strongly consistent and normally distributed. In practice, we may encounter spurious frequency estimates due to the high model order. Therefore, we further propose the robust trimming algorithm (RTA) of RAR frequency estimation. Our simulation studies indicate that the RTA can effectively eliminate the spurious roots and outliers, and therefore noticeably increase the accuracy. Another application we discuss in this thesis is modeling and forecasting of long memory processes using the RAR approximation. We demonstration that the RAR is useful in long-range prediction of general ARFIMA(p,d,q) processes with p > 1 and q > 1 via simulation studies.