The AR(1) model Xt = ρXt-1 + ϵt with iid error ϵ_t has been used extensively for the inference of the stochastic process Xt where its key parameter ρ plays an essential role. In particular, the Dickey-Fuller test (DF test) has been extensively used for testing random walk model (or ρ = 1) in the literatures. However, it is well known that the DF test is subject to serious size distortion when errors are correlated. This study proposes the use of an extended MA(∞) model Xt = ∑i=0∞ bi ϵt-i for a more precise inference of Xt by the DF test. We develop and investigate a new persistency parameter b∞=limj→∞ bj from the extended MA(∞) model. It is shown that the DF test serves well for testing the MA(∞) model with the new persistency parameter b∞. Our approach critically addresses the size distortion issues in the literatures.