Some remarks on regression with autoregressive errors and their residual processes

Abstract

We consider some linear regression models Y = Σα lfl(z) + X, where X is an autoregressive (AR) process. The residuals estimate the i.i.d. innovations sequence which drives the AR process. We then consider the partial sum process of the residuals and show they converge to Brownian bridges in certain cases. Some remarks are also made on similar processes when differencing is first applied to remove trends. When an AR process is differenced the residual partial sum can be asymptotically a random polynomial.