Inference for regression models with errors from a non‐invertible MA(1) process

Abstract

AbstractThis paper considers maximum likelihood estimation in a regression model when the errors follow a first‐order moving average model which is non‐invertible or nearly non‐invertible. The latter corresponds to a moving average parameter θ that is equal to or close to 1. The joint limiting distribution of the maximum likelihood estimators b̂ and $\hat\theta$ of the regression parameter vector b and the moving average parameter θ is described. Unlike the case with standard time series models, the limiting distribution of b̂ depends on whether or not θ is being estimated. Specifically, the limit distribution of b̂ is non‐normal if θ is also being estimated and is normal if θ is unestimated and equal to 1. The asymptotic behavior of the generalized likelihood ratio statistic for testing θ = 1 vs. θ < 1 is also studied and shown to perform well compared to the locally best invariant unbiased test of Tanaka (1990). We also indicate extensions to seasonal moving average models with a unit root. Copyright © 2010 John Wiley & Sons, Ltd.