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In this paper we consider authentication at the physical layer, in which the authenticator aims at distinguishing a legitimate supplicant from an attacker on the basis of the characteristics of a set of parallel wireless channels, which are affected by time-varying fading. Moreover, the attacker's channel has a spatial correlation with the supplicant's one. In this setting, we assess and compare the performance achieved by different approaches under different channel conditions. We first consider the use of two different statistical decision methods, and we prove that using a large number of references (in the form of channel estimates) affected by different levels of time-varying fading is not beneficial from a security point of view. We then consider classification methods based on machine learning. In order to face the worst case scenario of an authenticator provided with no forged messages during training, we consider one-class classifiers. When instead the training set includes some forged messages, we resort to more conventional binary classifiers, considering the cases in which such messages are either labelled or not. For the latter case, we exploit clustering algorithms to label the training set. The performance of both nearest neighbor (NN) and support vector machine (SVM) classification techniques is evaluated. Through numerical examples, we show that under the same probability of false alarm, one-class classification (OCC) algorithms achieve the lowest probability of missed detection when a small spatial correlation exists between the main channel and the adversary one, while statistical methods are advantageous when the spatial correlation between the two channels is large.