A Dynamically Consistent Global Nonlinear Solution Method in the Sequence Space and Applications

Lee, Hanbaek (2022): A Dynamically Consistent Global Nonlinear Solution Method in the Sequence Space and Applications.

Abstract

This paper develops a global nonlinear solution method in the sequence space for dynamic stochastic general equilibrium models. The method utilizes the equilibrium’s ergodicity: if a simulated path of the aggregate shock is long enough, all the possible equilibrium outcomes are realized somewhere on the path. Then, the conditional expectation at each period can be completely characterized by identifying the periods of each possible future state realization and combining the corresponding time-specific value (policy) functions without a law of motion or parametrized expectation. I theoretically show that a sufficient statistic can be used for a complex aggregate state, including distributions, when the individual value (policy) functions are strictly monotone in the statistic. Despite its simple implementation, the computation is highly efficient, bypassing fixed-point problems in each iteration, including non-trivial market clearing.

• Item Type: MPRA Paper
• Language: English
• Keywords: Global nonlinear solution, sequence space, dynamic consistency, sufficient statistic, business cycle
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling; E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E32 - Business Fluctuations ; Cycles

References

Ahn, S., G. Kaplan, B. Moll, T. Winberry, and C. Wolf (2018): “When Inequality Matters for Macro and Macro Matters for Inequality,” NBER Macroeconomics Annual, 32, 1–75, eprint: https://doi.org/10.1086/696046.

Auclert, A., B. Bardo ́czy, M. Rognlie, and L. Straub (2021): “Using the Sequence- Space Jacobian to Solve and Estimate Heterogeneous-Agent Models,” Econometrica, 89, 2375–2408.

Azinovic, M., L. Gaegauf, and S. Scheidegger (2022): “DEEP EQUILIBRIUM NETS,” International Economic Review, 63, 1471–1525.

Bloom, N., M. Floetotto, N. Jaimovich, I. Saporta-Eksten, and S. J. Terry (2018): “Really Uncertain Business Cycles,” Econometrica, 86, 1031–1065.

Boppart, T., P. Krusell, and K. Mitman (2018): “Exploiting MIT shocks in heterogeneous-agent economies: the impulse response as a numerical derivative,” Journal of Economic Dynamics and Control, 89, 68–92.

Brumm, J. and S. Scheidegger (2017): “Using Adaptive Sparse Grids to Solve High-Dimensional Dynamic Models,” Econometrica, 85, 1575–1612.

Cai, Y., K. Judd, and J. Steinbuks (2017): “A nonlinear certainty equivalent approximation method for dynamic stochastic problems: Dynamic stochastic problems,” Quantitative Economics, 8, 117–147.

Cao, D., W. Luo, and G. Nie (2023): “Global DSGE Models,” Review of Economic Dynamics, S1094202523000017.

Childers, D. (2018): “Solution of Rational Expectations Models with Function Valued States,” Working Paper, 126.

Den Haan, W. J. (2010): “Assessing the accuracy of the aggregate law of motion in models with heterogeneous agents,” Journal of Economic Dynamics and Control, 34, 79–99.

den Haan, W. J. and A. Marcet (1990): “Solving the Stochastic Growth Model by Parameterizing Expectations,” Journal of Business & Economic Statistics, 8, 31–34.

Den Haan, W. J. and P. Rendahl (2010): “Solving the incomplete markets model with aggregate uncertainty using explicit aggregation,” Journal of Economic Dynamics and Control, 34, 69–78.

Elenev, V., T. Landvoigt, and S. Van Nieuwerburgh (2021): “A Macroeconomic Model With Financially Constrained Producers and Intermediaries,” Econometrica, 89, 1361–1418.

Fair, R. and J. Taylor (1983): “Solution and Maximum Likelihood Estimation of Dynamic Nonlinear Rational Expectations Models,” Econometrica, 51, 1169–85.

Ferna ́ndez-Villaverde, J., S. Hurtado, and G. Nun ̃o (2023): “Financial Frictions and the Wealth Distribution,” Econometrica, 91, 869–901.

Guerrieri, L. and M. Iacoviello (2015): “OccBin: A toolkit for solving dynamic models with occasionally binding constraints easily,” Journal of Monetary Economics, 70, 22–38.

Han, J., Y. Yang, and W. E (2021): “DeepHAM: A Global Solution Method for Het- erogeneous Agent Models with Aggregate Shocks,” SSRN Electronic Journal.

Judd, K. L. (1992): “Projection methods for solving aggregate growth models,” Journal of Economic Theory, 58, 410–452.

Judd, K. L. (2002): “The parametric path method: an alternative to Fair–Taylor and L–B–J for solving perfect foresight models,” Journal of Economic Dynamics and Control, 26, 1557–1583.

Judd, K. L., L. Maliar, and S. Maliar (2011): “Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models: Approaches for solving dynamic models,” Quantitative Economics, 2, 173–210.

Juillard, M. (1996): “Dynare: a program for the resolution and simulation of dynamic models with forward variables through the use of a relaxation algorithm,” CEPREMAP Working Papers (Couverture Orange) 9602, CEPREMAP.

Kahou, M. E., J. Fernandez-Villaverde, J. Perla, and A. Sood (2021): “Exploiting Symmetry in High-Dimensional Dynamic Programming,” Working Paper, 51.

Khan, A. and J. K. Thomas (2008): “Idiosyncratic Shocks and the Role of Nonconvexities in Plant and Aggregate Investment Dynamics,” Econometrica, 76, 395–436.

Krusell, P. and A. A. Smith, Jr. (1997): “Income and Wealth Heterogeneity, Portfolio Choice, and Equilibrium Asset Returns,” Macroeconomic Dynamics, 1.

Krusell, P. and A. A. Smith, Jr. (1998): “Income and Wealth Heterogeneity in the Macroeconomy,” Journal of Political Economy, 106, 867–896.

Lee, H. and K. Nomura (2024): “Nonlinear Inflation Dynamics in the Canonical Calvo Pricing Model,” Working Paper, 31.

Lee, H., F. Zanetti, and P. Schnattinger (2024): “State-dependent Nonlinear Search and Matching,” Working Paper, 31.

Maliar, L., S. Maliar, and F. Valli (2010): “Solving the incomplete markets model with aggregate uncertainty using the Krusell–Smith algorithm,” Journal of Economic Dynamics and Control, 34, 42–49.

Maliar, S., L. Maliar, and K. Judd (2011): “Solving the multi-country real business cycle model using ergodic set methods,” Journal of Economic Dynamics and Control, 35, 207–228.

Marcet, A. (1988): “Solving nonlinear stochastic models by parameterizing expectations,” Manuscript. Pittsburgh: Carnegie Mellon Univ.

Pizzinelli, C., K. Theodoridis, and F. Zanetti (2020): “STATE DEPENDENCE IN LABOR MARKET FLUCTUATIONS,” International Economic Review, 61, 1027–1072.

Reiter, M. (2009): “Solving heterogeneous-agent models by projection and perturbation,” Journal of Economic Dynamics and Control, 33, 649–665.

Rendahl, P. (2014): “Inequality Constraints and Euler Equation-Based Solution Methods,” The Economic Journal, 125, 1110–1135.

Winberry, T. (2018): “A method for solving and estimating heterogeneous agent macro models,” Quantitative Economics, 9, 1123–1151.

Winschel, V. and M. Kra ̈tzig (2010): “Solving, Estimating, and Selecting Nonlinear Dynamic Models Without the Curse of Dimensionality,” Econometrica, 78, 803–821.

Young, E. R. (2010): “Solving the incomplete markets model with aggregate uncertainty using the Krusell-Smith algorithm and non-stochastic simulations,” Journal of Economic Dynamics and Control, 34, 36–41.