On Partial-Sum Processes of ARMAX Residuals

In time series analysis and other setups, numerous inference procedures need to use residuals, instead of the unobserved errors. In the present paper, we establish general and versatile results regarding the limit of partial-sum processes of ARMAX residuals. Process limits based on residuals often do not correspond to process limits based on the error terms. To illustrate the generality and versatility of our results, we apply them to misspecified ARMA with correlated errors, nonlinear ARMAX, a range of ARMA processes with infinite-variance errors, ARMA with a unit root and a polynomial time-trend, or with seasonal dummy variables to obtain consistency of kernel estimation of the density of ARMAX errors, the asymptotic distribution of statistics useful for CUSUM tests and change-point problem type of tests, and some local power results.