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The gut microbiome is a dynamic system that changes with host development, health, behavior, diet, and microbe-microbe interactions. Prior work on gut microbial time series has largely focused on autoregressive models (e.g. Lotka-Volterra). However, we show that most of the variance in microbial time series is non-autoregressive. In addition, we show how community state-clustering is flawed when it comes to characterizing within-host dynamics and that more continuous methods are required. Most organisms exhibited stable, mean-reverting behavior suggestive of fixed carrying capacities and abundant taxa were largely shared across individuals. This mean-reverting behavior allowed us to apply sparse vector autoregression (sVAR)-a multivariate method developed for econometrics-to model the autoregressive component of gut community dynamics. We find a strong phylogenetic signal in the non-autoregressive co-variance from our sVAR model residuals, which suggests niche filtering. We show how changes in diet are also non-autoregressive and that Operational Taxonomic Units strongly correlated with dietary variables have much less of an autoregressive component to their variance, which suggests that diet is a major driver of microbial dynamics. Autoregressive variance appears to be driven by multi-day recovery from frequent facultative anaerobe blooms, which may be driven by fluctuations in luminal redox. Overall, we identify two dynamic regimes within the human gut microbiota: one likely driven by external environmental fluctuations, and the other by internal processes.