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

Summary We develop new multiscale methods to test qualitative hypotheses about the function m in the non‐parametric regression model Y t , T = m ( t / T )+ ɛ t with time series errors ɛ t . 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 varianceσ 2 = Σ l = − ∞ ∞ cov ( ε 0 , ε l ) . We propose a new difference‐based estimator of σ 2 for the case that { ɛ t } 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.