Point identification under temporal aggregation of flow variables

Prove that, for a stationary Gaussian AR(p) process observed every q periods as an equally weighted flow variable, the autoregressive parameters and innovation variance are point-identified for every sampling frequency q under the stated nonzero-lag, distinct-qth-power-root, and normality assumptions.

Background

The flow observation aggregates the latent AR(p) process over each sampling interval, y_t = x_{tq} + x_{tq-1} + ... + x_{tq-q+1}. Unlike stock sampling, temporal aggregation introduces additional weighting factors involving the latent roots, which can potentially distinguish root aliases.

The paper gives an exact identified-set characterization for general p and q, proves point identification for q=2, proves it for real roots with odd q, and proves that the alternating-sign transformation is not observationally equivalent under even q. The full point-identification assertion for arbitrary complex-root configurations and sampling frequencies remains stated as a conjecture, supported by numerical verification.