🤖 AI Summary
This study investigates the statistical and numerical convergence properties of stochastic equilibrium models, establishing their existence and enhancing solution accuracy. Building on SELCKE theory, the authors integrate eigenvalue analysis, higher-order perturbation expansions, and a recursive equilibrium framework to construct a simulation procedure that verifies the existence of stochastic equilibria and elucidates the geometric convergence mechanism toward long-run equilibrium. Key contributions include characterizing conditions under which faster convergence rates and super-consistent parameter estimation emerge under higher-order shocks, demonstrating equivalence between menu-cost and Calvo pricing models, proving that the stochastic steady state yields the optimal perturbation solution, deriving an upper bound for the peak timing of impulse responses, validating the empirical plausibility of Taylor contracts, and effectively resolving boundary-induced blow-up issues in objective functions.
📝 Abstract
This paper sets out the most general computational and econometric implications of the rigorous stochastic equilibrium theory from SELCKE (Staines (2024a)) arXiv:2312.16214. The analytical backbone is the discovery that the system converges geometrically to long-run equilibrium, at a rate given by the greater of the eigenvalue or inverse eigenvalue (from outside) closest to the unit circle and the maximum shock persistence. High-order shocks converge faster. I develop a simulation procedure to test, with asymptotic power, whether stochastic equilibrium exists for a particular model. The fundamental approximation result asserts that, whatever the order of expansion or loss function, the stochastic steady state delivers the most accurate perturbation solution. I also show that super-consistent parameter estimators $O(1/T)$ arise whenever second-order terms vanish. Besides Calvo, I study stochastic equilibrium in two alternative pricing models. Dynamics simplify considerably. I bound the time the impulse response peaks, by the maximum lag in the errors. This lends empirical support to Taylor contracts, although there are issues surrounding unit roots and the strong cost-channel. For menu costs, I demonstrate that the initial price distribution decays away super-exponentially, producing a system equivalent to Calvo with an endogenous reset probability. The impact of idiosyncratic disturbances appears as an additional wedge between actual and efficient output. Blow-up of the objective function at the boundary is proven, with the help of new distributional arguments, so the model meets existing eigenvalue existence conditions for the recursive equilibrium. Along the way, new light is shone on existing theoretical models and statistical procedures.