Multiscale inference and long‐run variance estimation in non‐parametric regression with time series errors

We develop new multiscale methods to test qualitative hypotheses about the function m in the non‐parametric regression model Yt,T=m(t/T)+ɛt with time series errors ɛt. In time series applications, m represents a non‐parametric time trend. Practitioners are often interested in whether the trend m has certain shape properties. For example, they would like to know whether m is constant or whether it is increasing or decreasing in certain time intervals. Our multiscale methods enable us to test for such shape properties of the trend m. To perform the methods, we require an estimator of the long‐run error variance σ2=Σl=−∞∞cov(ε0,εl). We propose a new difference‐based estimator of σ2 for the case that {ɛt} belongs to the class of auto‐regressive AR(∞) processes. In the technical part of the paper, we derive asymptotic theory for the proposed multiscale test and the estimator of the long‐run error variance. The theory is complemented by a simulation study and an empirical application to climate data.