Expectation-maximization algorithm for autoregressive models with cauchy innovations

Abstract

In this paper, we study the autoregressive (AR) models with Cauchy distributed innovations. In the AR models, the response variable yt depends on previous terms and a stochastic term (the innovation). In the classical version, the AR models are based on normal distribution which could not capture the extreme values or asymmetric behavior of data. In this work, we consider the AR model with Cauchy innovations, which is a heavy-tailed distribution. We derive closed forms for the estimates of parameters of the considered model using the expectation-maximization (EM) algorithm. The efficacy of the estimation procedure is shown on the simulated data. The comparison of the proposed EM algorithm is shown with the maximum likelihood (ML) estimation method. Moreover, we also discuss the joint characteristic function of the AR(1) model with Cauchy innovations, which can also be used to estimate the parameters of the model using empirical characteristic function.