In this paper we develop hypothesis tests for speech waveform nonstationarity based on time-varying autoregressive models, and demonstrate their efficacy in speech analysis tasks at both segmental and sub-segmental scales. Key to the successful synthesis of these ideas is our employment of a generalized likelihood ratio testing framework tailored to autoregressive coefficient evolutions suitable for speech. After evaluating our framework on speech-like synthetic signals, we present preliminary results for two distinct analysis tasks using speech waveform data. At the segmental level, we develop an adaptive short-time segmentation scheme and evaluate it on whispered speech recordings, while at the sub-segmental level, we address the problem of detecting the glottal flow closed phase. Results show that our hypothesis testing framework can reliably detect changes in the vocal tract parameters across multiple scales, thereby underscoring its broad applicability to speech analysis.