Beyond Aggregate Accuracy: A Bayesian Time-Varying Autoregressive Model and Its Multilevel Extension

Autoregressive (AR) models are widely used to study within-person dynamics in intensive longitudinal data (ILD), but they assume constant parameters, an assumption that may not hold when the dynamics change, as during development, intervention, or recovery. The time-varying autoregressive (TV-AR) model accommodates such change by representing the intercept and AR coefficient as smooth functions of time. As originally proposed, however, it was developed for a single individual and evaluated with time-averaged accuracy metrics. We address these limitations by extending it to a Bayesian multilevel formulation with population-average and individual-specific time-varying parameters, and by emphasizing pointwise evaluation of accuracy rather than aggregate accuracy alone. Two simulation studies examined how well the single-level and multilevel models recover time-varying dynamics. Examining bias at each time point, we found that it was largest near the series boundaries and in regions of steep change or high curvature. The time-varying intercept and AR coefficient showed complementary biases that largely cancelled in the implied mean, which was therefore recovered with smaller bias. In the multilevel model, the best allocation between persons and time points depended on the parameter and inferential level: the AR coefficient favored longer series at both levels, whereas the implied mean favored more persons for population-average recovery but longer series for individual recovery. We discuss implications and recommendations for study design, result interpretation, and model selection.