Interaction between deterministic trend and autoregressive process

[1] When both an autoregressive (AR) process (stochastic trend) and a deterministic trend (systematic changes over mean) exist within a time series, there may be some interactions between them. This study investigates whether these two components interact with each other. Only time series consisting of a linear trend and a lag 1 autoregressive AR(1) process are explored, which are commonly used in hydrology. Results indicate that (1) the presence of a deterministic trend will overestimate positive serial correlation and underestimate negative serial correlation, while (2) the existence of an AR(1) process does not affect the estimate of the magnitude of the deterministic trend. However, a positive AR(1) will inflate the variance of the trend and a negative AR(1) will shrink the variance of the trend. For a time series with a deterministic trend the trend has to be removed in order to correctly model the stochastic properties of the time series. In the literature, both differencing and detrending have been proposed to fulfill such a task. Their applicability is based on the assumption that they cannot distort the residual series. However, no evidence has been provided to certify whether this assumption is appropriate. This study examines this issue and it indicates that differencing can remove a deterministic trend from a time series but it can seriously damage the existing AR(1) process. In contrast to differencing, detrending can remove a deterministic trend without distorting the existing AR process.