'Filtering' operations in the time domain are often used to eliminate unwanted short- or long-time scale variations in random data. to suppress long-period variations, authors often subtract a moving average or fitted polynomial from the raw data. Short-time variations are often eliminated by using moving-averaged data. The statistical properties of these processes are derived, and it is shown that they can introduce spurious features in the Fourier spectrum, correlation function, and power spectrum of the data.