This paper proposes a new robust adaptive processing algorithm that is based on the extended least squares (ELS) method with running spectrum filtering (RSF). By utilizing the different characteristics of running spectra between speech signals and noise signals, RSF can retain speech characteristics while noise is effectively reduced. Then, by using ELS, autoregressive moving average (ARMA) parameters can be estimated accurately. In experiments on real speech contaminated by white Gaussian noise and factory noise, we found that the method we propose offered spectrum estimates that were robust against additive noise.