1. Population dynamics of insect host–parasitoid systems are important in many natural and managed ecosystems and have inspired much ecological theory. However, ecologists have a limited knowledge about the relative strengths of species interactions, abiotic effects and density dependence in natural host–parasitoid dynamics. Statistical time-series analyses would be more informative by incorporating multiple factors, measurement error and noisy dynamics. 2. We use a novel maximum likelihood and model-selection analysis of a state-space model for host–parasitoid dynamics to examine 21 years of annual census data for woolly bear caterpillars (Platyprepia virginalis) and their locally host-specific tachinid parasitoids (Thelaira americana). 3. Caterpillar densities varied by three orders of magnitude and were driven by density dependence and precipitation from the previous March but not detectably by parasitoids, despite variable and sometimes high (>50%) parasitism. 4. Fly fluctuations, as estimated from per cent parasitism, were affected by density dependence and precipitation from the previous July. There was marginal evidence that host abundance drives fly fluctuations as a generic linear effect but no evidence for classical Nicholson–Bailey coupling. 5. The state-space model analysis includes new methods for likelihood calculation and allows a balanced consideration of effect magnitude and statistical significance in a nonlinear model with multiple alternative explanatory variables.