Parameter Identification and Inference in Discretely Sampled or Temporally Aggregated Autoregressions

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). I first characterize the resulting ARMA process followed by observables. Under fairly mild assumptions, I then 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 an algorithm to compute all candidate points and determine their membership in the identified set. My exact but implicit characterization supports the following conjecture that I prove in some settings and verify numerically more broadly: (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 $q$ and identified up to alternating sign for even $q$. My analysis supplements existing inference results that show consistency and asymptotic Normality of the Gaussian Maximum Likelihood estimator conditional on point-identification. Holding the number of observations fixed, I show that its precision does not necessarily decrease with $q$.