Parameter Identification in Autoregressions under Discrete Sampling or Temporal Aggregation

Abstract: 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.

• The paper develops a coefficient-based method that expresses sampled AR dynamics and error autocovariances directly in the original parameters, enabling exact identified sets for arbitrary lag order and sampling frequency.
• The error variance is always point-identified, while temporal aggregation point-identifies all parameters and stock sampling identifies them uniquely for odd frequencies or up to alternating coefficient signs for even frequencies.
• The identified set contains at most 2^r q^c candidates for even q and q^c for odd q, with simulations covering 120,000 specifications supporting the conjectured results despite open cases involving complex roots.

Motivation and setting

The paper studies a latent scalar AR($p$) process $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$ with white-noise innovations of variance $v$, observed only every $q$ periods. Two observation schemes are considered: stock sampling, where the econometrician sees snapshots $y_t = x_{tq}$, and flow sampling (temporal aggregation), where $y_t = x_{tq} + \dots + x_{tq-q+1}$. The author calls the resulting observable an AR($p,q$). The question is what can be learned about $(\phi,v)$ from $\{y_t\}$ alone — a long-standing problem initiated by Telser (1967) and extended by Amemiya–Wu (1972), Brewer (1973), and Palm–Nijman (1984), but previously resolved only for small $p$ or $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$0, or only conjecturally.

Four assumptions govern the analysis: weak stationarity ($$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$1 for all roots), actual lag order $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$2 ($$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$3), distinct $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$4-th powers of roots ($$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$5 for $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$6), and Gaussianity of $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$7. Under these, the paper derives exact identified sets for general $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$8 and $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$9, bounds their cardinality, and provides a computational recipe.

Observed dynamics in closed form

The technical foundation is a representation result built on the polynomial

$v$0

where $v$1. A lemma shows that $v$2 contains only powers of $v$3 divisible by $v$4, so there exists a real polynomial $v$5 with $v$6. Multiplying the high-frequency law of motion by the annihilator $v$7 then yields:

• Stock case: $v$8 follows an ARMA($v$9) with $q$0, driven by $q$1 with explicitly computable autocovariances.
• Flow case: with $q$2 where $q$3, $q$4 follows an ARMA($q$5) with $q$6.

The decisive methodological point is that these objects are expressed as closed-form functions of the coefficient vector $q$7 itself, not of its roots. Prior constructions (Telser; Amemiya–Wu; Palm–Nijman) build the annihilator root-by-root, which requires solving for $q$8 first. This distinction is what makes the general-$q$9, general-$y_t = x_{tq}$0 identification analysis tractable.

Identification results

The identification analysis proceeds via the autocovariance generating function. For stock sampling, any observationally equivalent candidate must satisfy $y_t = x_{tq}$1 (after unique re-indexing) together with equality of the constants $y_t = x_{tq}$2 appearing in the partial-fraction expansion of the ACGF. An analogous characterization holds under flow sampling, with constants $y_t = x_{tq}$3 plus one additional moment condition on $y_t = x_{tq}$4. These characterizations are exact but unwieldy; they support three headline conclusions, stated as conjectures and verified numerically:

1. The error variance $y_t = x_{tq}$5 is always point-identified, under both sampling schemes.
2. Under temporal aggregation (flow), $y_t = x_{tq}$6 is point-identified for all $y_t = x_{tq}$7 and $y_t = x_{tq}$8.
3. Under discrete sampling (stock), $y_t = x_{tq}$9 is point-identified for odd $y_t = x_{tq} + \dots + x_{tq-q+1}$0 and identified up to alternating sign for even $y_t = x_{tq} + \dots + x_{tq-q+1}$1, i.e., the identified set is $y_t = x_{tq} + \dots + x_{tq-q+1}$2 or $y_t = x_{tq} + \dots + x_{tq-q+1}$3 with $y_t = x_{tq} + \dots + x_{tq-q+1}$4.

Several results are proved outright. For $y_t = x_{tq} + \dots + x_{tq-q+1}$5, both conjectures are proved exactly: the stock identified set is precisely $y_t = x_{tq} + \dots + x_{tq-q+1}$6, while the flow set is the singleton $y_t = x_{tq} + \dots + x_{tq-q+1}$7. The proof technique is notable: the author shows that two point-identified symmetric polynomials in $y_t = x_{tq} + \dots + x_{tq-q+1}$8 and $y_t = x_{tq} + \dots + x_{tq-q+1}$9 force any candidate's $p,q$0 to equal either $p,q$1 or $p,q$2 globally (ruling out pointwise switching via a finiteness-of-zeros argument), and in the flow case the factor $p,q$3 breaks the reflection symmetry entirely. With real roots, the stock conjecture is proved for all $p,q$4, and the flow conjecture for odd $p,q$5. Additionally, it is shown that $p,q$6 is observationally equivalent to $p,q$7 under stock sampling for any even $p,q$8 — generalizing Palm–Nijman's $p,q$9 sign ambiguity — whereas under flow sampling $(\phi,v)$0 is observationally equivalent to $(\phi,v)$1 for no $(\phi,v)$2 and any even $(\phi,v)$3, via an argument exploiting that $(\phi,v)$4 for $(\phi,v)$5.

A cardinality bound complements these results: writing $(\phi,v)$6 for the number of real roots and $(\phi,v)$7 for conjugate pairs ($(\phi,v)$8), the identified set contains at most $(\phi,v)$9 points if $\{y_t\}$0 is even and $\{y_t\}$1 if $\{y_t\}$2 is odd. This mirrors Hansen–Sargent's resolution of Phillips' continuous-time aliasing problem: the map $\{y_t\}$3 has finitely many preimages, and positivity of the error variance trims the candidate set further. The even/odd-$\{y_t\}$4 distinction has no analogue in the continuous-time setting.

Computation and numerical verification

The paper provides a constructive procedure: enumerate the $\{y_t\}$5 candidate root multisets consistent with $\{y_t\}$6 (real roots may flip sign only when $\{y_t\}$7 is even; each conjugate pair admits $\{y_t\}$8 rotations), then check whether the implied ratios $\{y_t\}$9 (stock) or $p$0 (flow) equal a common positive constant $p$1, and, for flows, whether the level moment matches at $p$2. Candidates passing all checks lie in the identified set.

Using this algorithm with $p$3 lag orders, $p$4 sampling frequencies, and $p$5 random parameter draws per specification — 120,000 specifications in total, with tolerance $p$6 — the author finds no violations of either conjecture. Rejected candidates fail the checks by a clear margin (diagnostics separated by many orders of magnitude on a log scale), so the numerical evidence is not marginal. All checks are invariant to $p$7, so fixing $p$8 is without loss of generality.

Relation to prior work

The results settle a tension in the literature. Telser (1967) claimed point-identification could always be achieved using a residual-variance ratio; Palm–Nijman (1984) refuted this for stock sampling with $p$9 via the alternating-sign counterexample. The present analysis vindicates Telser everywhere else: the counterexample is the only failure mode, and it extends to all even $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$00 under stock sampling but never under flow aggregation. Nijman–Palm's assertion that $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$01 and $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$02 exhaust the identified set for even $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$03 — verified there only in examples, with the authors noting they "cannot exclude" further solutions — is here turned into an exact theorem for $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$04 and, with real roots, for all $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$05. The paper also clarifies the roles of the moment conditions: the AR block $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$06 depends on the roots only through their $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$07-th powers and pins down $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$08 up to $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$09 candidates, while the $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$10 (or $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$11) error-autocovariances supply the remaining identifying restrictions.

An implication worth emphasizing: gathering data at a frequency closer to the model frequency does not necessarily improve identifiability — identification is essentially complete already at low frequencies (up to at most a sign ambiguity), in contrast to the well-documented losses in forecasting accuracy, estimation efficiency, and Granger-causality inference caused by temporal aggregation.

Limitations and open questions

The central conjectures are proved only for $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$12 (both cases), for real roots (stock, all $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$13), and for real roots with odd $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$14 (flow); the remaining claims rest on extensive but finite numerical verification over random draws, so a formal proof for complex roots with general $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$15 remains open. The assumptions exclude repeated roots and roots whose $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$16-th powers coincide; behavior of the identified set at such degeneracies is not characterized. Gaussianity is used to equate observational equivalence with matching autocovariances, and the extension beyond Gaussian errors is not addressed. Finally, the framework covers pure AR processes: the author notes that under MA errors the finiteness of the identified set fails outright (an MA(1) sampled every second period yields white noise, hence a continuum of observationally equivalent parameters), so the approach does not directly extend to ARMA processes — applying the coefficient-based representation to that class is left as an open problem.

Conclusion

The paper delivers an exact, general characterization of parameter identification in AR($$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$17) processes observed every $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$18 periods, for arbitrary lag length and sampling frequency. Its key innovation — expressing the observed process' autoregressive polynomial and error autocovariances in closed form as functions of $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$19 rather than its roots — yields provable results for $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$20 and real-rooted cases, a finite cardinality bound $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$21 (even $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$22) or $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$23 (odd $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$24), and a computable recipe for the full identified set. The emerging picture is sharp: the error variance is always point-identified, temporal aggregation preserves point-identification of $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$25, and discrete sampling costs at most a joint sign flip of odd-lag coefficients when $$x_\tau = \phi_1 x_{\tau-1} + \dots + \phi_p x_{\tau-p} + e_\tau$$26 is even.