[Paper Review] Cycling in stochastic general equilibrium

Zhijian Wang, Bin Xu|arXiv (Cornell University)|Oct 30, 2014

Economic theories and models27 references3 citations

TL;DR

This paper introduces a novel method to visualize and quantify cyclic dynamics in stochastic general equilibrium models using angular momentum measurements in 3D phase space. It demonstrates that even in noisy, stochastic environments, the variables output gap (y), inflation (π), and nominal interest rate (r) exhibit persistent clockwise, counterclockwise, and weak cycles when projected onto 2D planes—patterns also found in U.S. macroeconomic data (1960–2013), suggesting that cycles are intrinsic to equilibrium dynamics rather than artifacts of noise.

ABSTRACT

By generalizing the measurements on the game experiments of mixed strategy Nash equilibrium, we study the dynamical pattern in a representative dynamic stochastic general equilibrium (DSGE). The DSGE model describes the entanglements of the three variables (output gap [$y$], inflation [$π$] and nominal interest rate [$r$]) which can be presented in 3D phase space. We find that, even though the trajectory of $π\!-\!y\!-\!r$ in phase space appears highly stochastic, it can be visualized and quantified. It exhibits as clockwise cycles, counterclockwise cycles and weak cycles, respectively, when projected onto $π\!-\!y$, $y\!-\!r$ and $r\!-\!π$ phase planes. We find also that empirical data of United State (1960-2013) significantly exhibit same cycles. The resemblance between the cycles in general equilibrium and the cycles in mixed strategy Nash equilibrium suggest that, there generally exists dynamical fine structures accompanying with equilibrium. The fine structure, describing the entanglement of the non-equilibrium (the constantly deviating from the equilibrium), displays as endless cycles.

Motivation & Objective

• To develop a visual and quantitative method for detecting dynamical cycles in stochastic general equilibrium models, particularly in the context of macroeconomic variables.
• To test whether the same cyclic patterns observed in theoretical DSGE simulations also appear in real-world empirical data (U.S. data, 1960–2013).
• To investigate whether cycles in general equilibrium are a result of inherent dynamical fine structures arising from non-equilibrium deviations, rather than external shocks or model misfit.
• To examine the dependence of cycle strength on noise amplitude in DSGE models, using controlled simulations.

Proposed method

• The study employs a representative Dynamic Stochastic General Equilibrium (DSGE) model with three key variables: output gap (y), inflation (π), and nominal interest rate (r), simulated over 20,000 periods.
• It introduces a phase-space analysis using 3D velocity vector fields and projects them onto three 2D planes: π–y, y–r, and r–π, to visualize directional dynamics.
• The core method is the n-sampling angular momentum (L²) measurement, defined as the signed area of the trajectory in each 2D projection, used to quantify clockwise or counterclockwise cycling.
• The method applies this L² metric to both simulated DSGE trajectories and empirical U.S. data (1961–2013) from the World Bank database.
• Robustness is tested via noise amplitude variation (α = 0.3, 0.5, 0.7), with repeated simulations to assess cycle strength dependence on noise.
• Theoretical evaluation of L² is performed using simplified numerical models to validate simulation results.

Experimental results

Research questions

• RQ1 Do persistent, quantifiable cycles emerge in the phase space of a stochastic general equilibrium model, even under continuous noise?
• RQ2 Do the same cyclic patterns—clockwise, counterclockwise, or weak cycles—appear in empirical U.S. macroeconomic data (1960–2013) for output gap, inflation, and interest rates?
• RQ3 Is the strength of observed cycles in the DSGE model dependent on the amplitude of stochastic shocks?
• RQ4 Can the angular momentum metric (L²) reliably distinguish and quantify the direction and persistence of cyclic motion in multi-variable dynamic systems?

Key findings

• In the DSGE model, the π–y projection exhibits consistently clockwise cycles, with L² > 0 across all simulations.
• The y–r projection shows definitively counterclockwise cycles, with L² > 0 in all six shock scenarios tested.
• The r–π projection does not show consistent cycle direction, as L² values are not definitively positive or negative, indicating weak or inconsistent cycling.
• Empirical U.S. data (1961–2013) displays the same cyclic patterns as the DSGE model, with significant similarity in cycle direction and strength across π–y and y–r planes.
• Cycle strength, measured by L², increases with noise amplitude: median L² values decrease when noise is reduced (α = 0.3 vs. α = 0.7), confirming a positive dependence on noise.
• Theoretical evaluation of L² via simplified numerical models confirms the simulation results, validating the robustness of the angular momentum metric.