Repairing the linear-regression approximation for non-monotonic output series

Develop a linear-regression-based approximation for estimating the parameters β and λ in the Jones law of motion dA/A = θ A^{−β} I^{λ} that remains valid when the output series A(t) is non-monotonic or locally flat, thereby avoiding the current requirement that A(t2) > A(t1) over each sampling window and resolving the conflict between choosing large windows to enforce monotonicity and small windows to control input variance.

Background

In the linear-regression approach derived from the integral form of the Jones law of motion, the left-hand side involves log log(A(t2)/A(t1)). This forces A(t) to be strictly increasing over each sampling window; otherwise the expression becomes undefined or diverges to negative infinity.

The authors note that many empirical output series (e.g., TFP) exhibit flat or decreasing periods, making the strict monotonicity requirement unrealistic. While selecting larger sampling windows can enforce A(t2) > A(t1), this clashes with the need for short windows to keep the variance of inputs I(t) low, producing a methodological impasse.

A principled repair of the linear-regression approximation that works when A(t) is not strictly increasing would enable broader applicability of regression-based estimation without resorting to ad hoc window selection.