Parameter Identification in Autoregressions under Discrete Sampling or Temporal Aggregation

I consider an AR($p$) process that is observed every $q$ periods, either as a snapshot (stock variable) or as a sum over the sampling interval (flow variable). Under fairly mild assumptions, I derive the identified set for general lag lengths $p \in \mathbb{N}$ and sampling frequencies $q \in \mathbb{N}$, I bound its cardinality, and I provide a recipe to compute all candidate points and determine their membership in the identified set. My analysis supports the following conjecture: (i) the error term-variance is point-identified, (ii) under temporal aggregation, the autoregressive parameters are point-identified, and (iii) under discrete sampling they are point-identified for odd sampling frequencies and identified up to alternating sign for even sampling frequencies. I prove this conjecture in some settings and verify it numerically more broadly.