Validity of the time-integrated diminishing-returns estimator

Determine whether the time-integrated estimator that computes the diminishing-returns parameter β by comparing log(Δ log A / I_λ(t1, t2)^λ) across two time windows equals the true β implied by the Jones law of motion, and, if only approximately valid, characterize necessary and sufficient conditions and error bounds under which it provides reliable estimates.

Background

A commonly used variant for estimating diminishing returns replaces instantaneous quantities (\dot A/A and I(t)) with time-aggregated terms, comparing ratios across two intervals to infer β.

The authors explicitly note uncertainty about whether this time-integrated approximation actually holds. Although they later provide conditions under which it can work reasonably well, they do not claim general validity and caution against relying on the method without justification.

Clarifying the precise validity regime and bias of this estimator would inform when it can be safely used as a practical alternative to likelihood-based methods.