A general result on the estimation bias of ARMA models

A general result is derived on the finite-sample approximate bias of the Gaussian maximum likelihood estimator of the full vector of parameters in ARMA models when the error term may be nonnormally distributed and exogenous regressors may be included. It is found that there is typically estimation bias for parameters associated with the AR and MA terms and the bias depends only on these parameters themselves and the exogenous regressors. Parameters associated with the exogenous regressors are estimated approximately unbiased. The error variance is estimated with a downward bias, proportional to the number of exogenous regressors and the orders of AR and MA terms in the model. The distributional assumption on the error term does not affect the general bias result. Monte Carlo experiments are provided to illustrate the effectiveness of using analytical bias for the purpose of bias correction.