Point identification under discrete sampling of stock variables

Prove that, for a stationary Gaussian AR(p) process observed every q periods as a stock variable, the autoregressive parameters and innovation variance are point-identified when q is odd, while the identified set consists exactly of the original parameters and the alternating-sign transformation when q is even, under the stated nonzero-lag, distinct-qth-power-root, and normality assumptions.

Background

The paper studies a latent stationary AR(p) process observed only every q periods through the snapshot variable y_t = x_{tq}. The observed process is represented as an ARMA process whose autoregressive roots are the qth powers of the latent autoregressive roots, creating a finite root-aliasing problem.

The authors derive an exact characterization of the stock-sampling identified set and prove the conjecture for q=2 and for models with real roots. They also establish the alternating-sign observational equivalence for every even q and numerically verify the broader claim, but the general statement for arbitrary p, q, and complex roots is presented as a conjecture rather than proved.