1932

Abstract

The analysis of dynamic economic models routinely leads to the mathematical problem of determining an unknown function for which no closed-form solution exists. Economists must then resort to methods of numerical approximation when analyzing such models. Among the computational methods that have been successfully applied in economics and finance, one set of techniques stands out due to its flexibility and robustness: projection methods. In this article, we describe the basic steps of these methods for several different applications, surveying many successful applications of projection methods to dynamic economic models. Importantly, we emphasize that the ever-increasing complexity and dimensionality of dynamic models have made the previously used simpler methods obsolete and the applications of projection methods all but mandatory. We closely examine the most recent endeavors in the literature on solving economic models with projection methods.

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2020-08-02
2024-06-16
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Literature Cited

  1. Ai H, Croce MM, Diercks AM, Li K 2018. News shocks and the production-based term structure of equity returns. Rev. Financ. Stud. 31:2423–67
    [Google Scholar]
  2. Ai H, Croce MM, Li K 2012. Toward a quantitative general equilibrium asset pricing model with intangible capital. Rev. Financ. Stud. 26:491–530
    [Google Scholar]
  3. Aldrich EM, Kung H. 2017. Computational methods for production-based asset pricing models with recursive utility Work. Pap Duke Univ Durham, NC:
    [Google Scholar]
  4. Algan Y, Allais O, Den Haan WJ 2010. Solving the incomplete markets model with aggregate uncertainty using parameterized cross-sectional distributions. J. Econ. Dyn. Control 34:59–68
    [Google Scholar]
  5. Anderson GS, Kim J, Yun T 2010. Using a projection method to analyze inflation bias in a micro-founded model. J. Econ. Dyn. Control 34:1572–81
    [Google Scholar]
  6. Aruoba SB, Cuba-Borda P, Schorfheide F 2018. Macroeconomic dynamics near the ZLB: a tale of two countries. Rev. Econ. Stud. 85:87–118
    [Google Scholar]
  7. Aruoba SB, Fernández-Villaverde J, Rubio-Ramírez JF 2006. Comparing solution methods for dynamic equilibrium economies. J. Econ. Dyn. Control 30:2477–508
    [Google Scholar]
  8. Atolia M, Awad B, Marquis M 2011. Linearization and higher-order approximations: How good are they. Comput. Econ. 38:1–31
    [Google Scholar]
  9. Atolia M, Chatterjee S, Turnovsky SJ 2010. How misleading is linearization? Evaluating the dynamics of the neoclassical growth model. J. Econ. Dyn. Control 34:1550–71
    [Google Scholar]
  10. Azacis H, Gillman M. 2010. Flat tax reform: the Baltics 2000–2007. J. Macroecon. 32:692–708
    [Google Scholar]
  11. Azinović M, Gaegauf L, Scheidegger S 2019. Deep equilibrium nets Work. Pap Univ. Zurich Zurich, Switz.:
    [Google Scholar]
  12. Bajari P. 2001. Comparing competition and collusion: a numerical approach. Econ. Theory 18:187–205
    [Google Scholar]
  13. Bansal R, Kiku D, Ochoa M 2016. Price of long-run temperature shifts in capital markets NBER Work. Pap 22529
    [Google Scholar]
  14. Bansal R, Yaron A. 2004. Risks for the long run: a potential resolution of asset pricing puzzles. J. Finance 59:1481–509
    [Google Scholar]
  15. Becker RA. 1980. On the long-run steady state in a simple dynamic model of equilibrium with heterogeneous households. Q. J. Econ. 95:375–82
    [Google Scholar]
  16. Benzoni L, Collin-Dufresne P, Goldstein RS 2011. Explaining asset pricing puzzles associated with the 1987 market crash. J. Financ. Econ. 101:552–73
    [Google Scholar]
  17. Bocola L. 2016. The pass-through of sovereign risk. J. Political Econ. 124:879–926
    [Google Scholar]
  18. Boldrin M, Christiano LJ, Fisher JD 2001. Habit persistence, asset returns, and the business cycle. Am. Econ. Rev. 91:149–66
    [Google Scholar]
  19. Branger N, Konermann P, Schlag C 2016. Optimists, pessimists, and the stock market: the role of preferences and market (in)completeness Work. Pap Münster Univ Münster, Ger.:
    [Google Scholar]
  20. Brumm J, Grill M. 2014. Computing equilibria in dynamic models with occasionally binding constraints. J. Econ. Dyn. Control 38:142–60
    [Google Scholar]
  21. Brumm J, Grill M, Kubler F, Schmedders K 2015a. Collateral requirements and asset prices. Int. Econ. Rev. 56:1–25
    [Google Scholar]
  22. Brumm J, Grill M, Kubler F, Schmedders K 2015b. Margin regulation and volatility. J. Monet. Econ. 75:54–68
    [Google Scholar]
  23. Brumm J, Scheidegger S. 2017. Using adaptive sparse grids to solve high-dimensional dynamic models. Econometrica 85:1575–612
    [Google Scholar]
  24. Bungartz HJ, Griebel M. 2004. Sparse grids. Acta Numer 13:147–269
    [Google Scholar]
  25. Burnside C. 1998. Solving asset pricing models with Gaussian shocks. J. Econ. Dyn. Control 22:329–40
    [Google Scholar]
  26. Cai Y, Judd KL, Lontzek TS 2017. The social cost of carbon with economic and climate risks Work. Pap. 18113 Hoover Inst. Econ., Stanford Univ Stanford, CA:
    [Google Scholar]
  27. Caldara D, Fernández-Villaverde J, Rubio-Ramírez JF, Yao W 2012. Computing DSGE models with recursive preferences and stochastic volatility. Rev. Econ. Dyn. 15:188–206
    [Google Scholar]
  28. Calvo GA. 1983. Staggered prices in a utility-maximizing framework. J. Monet. Econ. 12:383–98
    [Google Scholar]
  29. Campbell JY. 2018. Financial Decisions and Markets: A Course in Asset Pricing Princeton, NJ: Princeton Univ. Press
    [Google Scholar]
  30. Campbell JY, Cochrane JH. 1999. By force of habit: a consumption-based explanation of aggregate stock market behavior. J. Political Econ. 107:205–51
    [Google Scholar]
  31. Campbell JY, Shiller RJ. 1988. The dividend-price ratio and expectations of future dividends and discount factors. Rev. Financ. Stud. 1:195–228
    [Google Scholar]
  32. Carroll CD. 2006. The method of endogenous gridpoints for solving dynamic stochastic optimization problems. Econ. Lett. 91:312–20
    [Google Scholar]
  33. Chari VV, Christiano LJ, Kehoe PJ 1994. Optimal fiscal policy in a business cycle model. J. Political Econ. 102:617–52
    [Google Scholar]
  34. Chen Y, Cosimano TF, Himonas AA 2008. Analytic solving of asset pricing models: the by force of habit case. J. Econ. Dyn. Control 32:3631–60
    [Google Scholar]
  35. Christiano LJ, Fisher JD. 2000. Algorithms for solving dynamic models with occasionally binding constraints. J. Econ. Dyn. Control 24:1179–232
    [Google Scholar]
  36. Cochrane JH. 1991. Production-based asset pricing and the link between stock returns and economic fluctuations. J. Finance 46:209–37
    [Google Scholar]
  37. Cochrane JH. 2001. Asset Pricing Princeton, NJ: Princeton Univ. Press
    [Google Scholar]
  38. Colacito R, Croce M, Ho S, Howard P 2018. BKK the EZ way: international long-run growth news and capital flows. Am. Econ. Rev. 108:3416–49
    [Google Scholar]
  39. Croce MM. 2014. Long-run productivity risk: a new hope for production-based asset pricing. J. Monet. Econ. 66:13–31
    [Google Scholar]
  40. Cuñat A, Maffezzoli M. 2004. Neoclassical growth and commodity trade. Rev. Econ. Dyn. 7:707–36
    [Google Scholar]
  41. Cuñat A, Maffezzoli M. 2007. Can comparative advantage explain the growth of US trade. Econ. J. 117:583–602
    [Google Scholar]
  42. de Boor C. 1978. A Practical Guide to Splines New York: Springer Verlag
    [Google Scholar]
  43. Den Haan WJ, Judd KL, Juillard M 2010. Computational suite of models with heterogeneous agents: incomplete markets and aggregate uncertainty. J. Econ. Dyn. Control 34:1–3
    [Google Scholar]
  44. Den Haan WJ, Judd KL, Juillard M 2011. Computational suite of models with heterogeneous agents II: multi-country real business cycle models. J. Econ. Dyn. Control 35:175–77
    [Google Scholar]
  45. Den Haan WJ, Marcet A 1990. Solving the stochastic growth model by parameterizing expectations. J. Bus. Econ. Stat. 8:31–34
    [Google Scholar]
  46. Den Haan WJ, Marcet A 1994. Accuracy in simulations. Rev. Econ. Stud. 61:3–17
    [Google Scholar]
  47. Den Haan WJ, Rendahl P 2010. Solving the incomplete markets model with aggregate uncertainty using explicit aggregation. J. Econ. Dyn. Control 34:69–78
    [Google Scholar]
  48. Dergunov I, Meinerding C, Schlag C 2019. Extreme inflation and time-varying consumption growth Discuss. Pap. 16/2019 Deutsche Bundesbank Frankfurt:
    [Google Scholar]
  49. Devereux MB, Siu HE. 2007. State dependent pricing and business cycle asymmetries. Int. Econ. Rev. 48:281–310
    [Google Scholar]
  50. Diamond PA. 1982. Aggregate demand management in search equilibrium. J. Political Econ. 90:881–94
    [Google Scholar]
  51. Doraszelski U. 2003. An R&D race with knowledge accumulation. RAND J. Econ. 34:20–42
    [Google Scholar]
  52. Doraszelski U. 2004. Innovations, improvements, and the optimal adoption of new technologies. J. Econ. Dyn. Control 28:1461–80
    [Google Scholar]
  53. Dou W, Fang X, Lo AW, Uhlig H 2020. Macro-finance models with nonlinear dynamics Work. Pap Becker Friedman Inst. Econ., Univ. Chicago Chicago:
    [Google Scholar]
  54. Drechsler I, Yaron A. 2010. What's vol got to do with it. Rev. Financ. Stud. 24:1–45
    [Google Scholar]
  55. Duffy J, McNelis PD. 2001. Approximating and simulating the stochastic growth model: parameterized expectations, neural networks, and the genetic algorithm. J. Econ. Dyn. Control 25:1273–303
    [Google Scholar]
  56. Epstein LG, Zin SE. 1991. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: an empirical analysis. J. Political Econ. 99:263–86
    [Google Scholar]
  57. Favilukis J, Lin X. 2013. Long run productivity risk and aggregate investment. J. Monet. Econ. 60:737–51
    [Google Scholar]
  58. Favilukis J, Lin X. 2015. Wage rigidity: a quantitative solution to several asset pricing puzzles. Rev. Financ. Stud. 29:148–92
    [Google Scholar]
  59. Fernández-Villaverde J, Gordon G, Guerrón-Quintana P, Rubio-Ramirez JF 2015. Nonlinear adventures at the zero lower bound. J. Econ. Dyn. Control 57:182–204
    [Google Scholar]
  60. Fernández-Villaverde J, Levintal O. 2018. Solution methods for models with rare disasters. Quant. Econ. 9:903–44
    [Google Scholar]
  61. Gallant AR, Jahan-Parvar MR, Liu H 2019. Does smooth ambiguity matter for asset pricing. Rev. Financ. Stud. 32:3617–66
    [Google Scholar]
  62. Gaspar J, Judd KL. 1997. Solving large-scale rational-expectations models. Macroecon. Dyn. 1:45–75
    [Google Scholar]
  63. Gertler M, Karadi P. 2011. A model of unconventional monetary policy. J. Monet. Econ. 58:17–34
    [Google Scholar]
  64. Gertler M, Kiyotaki N. 2010. Financial intermediation and credit policy in business cycle analysis. Handbook of Monetary Economics Vol 3 BM Friedman, M Woodford 547–99 Amsterdam: Elsevier
    [Google Scholar]
  65. Gomes F, Michaelides A. 2007. Asset pricing with limited risk sharing and heterogeneous agents. Rev. Financ. Stud. 21:415–48
    [Google Scholar]
  66. Gottardi P, Kubler F. 2011. Social security and risk sharing. J. Econ. Theory 146:1078–106
    [Google Scholar]
  67. Gourio F. 2012. Disaster risk and business cycles. Am. Econ. Rev. 102:2734–66
    [Google Scholar]
  68. Gourio F. 2013. Credit risk and disaster risk. Am. Econ. J. Macroecon. 5:1–34
    [Google Scholar]
  69. Gräber N, Schumacher M. 2019. Solving DSGE models—when local approximations fail Work. Pap Univ. Münster Münster, Ger.:
    [Google Scholar]
  70. Hagedorn M, Manovskii I. 2008. The cyclical behavior of equilibrium unemployment and vacancies revisited. Am. Econ. Rev. 98:1692–706
    [Google Scholar]
  71. Hall RE. 1971. The dynamic effects of fiscal policy in an economy with foresight. Rev. Econ. Stud. 38:229–44
    [Google Scholar]
  72. Heer B, Maussner A. 2018. Projection methods and the curse of dimensionality. J. Math. Finance 8:317–34
    [Google Scholar]
  73. Hubbard TP, Paarsch HJ. 2009. Investigating bid preferences at low-price, sealed-bid auctions with endogenous participation. Int. J. Ind. Organ. 27:1–14
    [Google Scholar]
  74. İmrohoroğlu A, Tüzel Ş 2014. Firm-level productivity, risk, and return. Manag. Sci. 60:2073–90
    [Google Scholar]
  75. Jermann UJ. 1998. Asset pricing in production economies. J. Monet. Econ. 41:257–75
    [Google Scholar]
  76. Jermann UJ, Quadrini V. 2012. Macroeconomic effects of financial shocks. Am. Econ. Rev. 102:238–71
    [Google Scholar]
  77. Jones LE, Manuelli RE, Siu HE 2005. Fluctuations in convex models of endogenous growth, II: business cycle properties. Rev. Econ. Dyn. 8:805–28
    [Google Scholar]
  78. Judd KL. 1992. Projection methods for solving aggregate growth models. J. Econ. Theory 58:410–52
    [Google Scholar]
  79. Judd KL. 1996. Approximation, perturbation, and projection methods in economic analysis. Handbook of Computational Economics 1 K Schmedders, KL Judd 509–85 Amsterdam: Elsevier
    [Google Scholar]
  80. Judd KL. 1998. Numerical Methods in Economics Cambridge, MA: MIT Press
    [Google Scholar]
  81. Judd KL, Guu SM. 1993. Perturbation solution methods for economic growth models. Economic and Financial Modeling with Mathematica ®, ed. HR Varian 80–103 New York: Springer
    [Google Scholar]
  82. Judd KL, Guu SM. 1997. Asymptotic methods for aggregate growth models. J. Econ. Dyn. Control 21:1025–42
    [Google Scholar]
  83. Judd KL, Kubler F, Schmedders K 2003. Computational methods for dynamic equilibria with heterogeneous agents. Advances in Economics and Econometrics: Theory and Applications, Eight World Congress M Dewatripont, LP Hansen, SJ Turnovsky 243–90 Cambridge, UK: Cambridge Univ. Press
    [Google Scholar]
  84. Judd KL, Maliar L, Maliar S 2010. A cluster-grid projection method: solving problems with high dimensionality NBER Work. Pap 15965
    [Google Scholar]
  85. Judd KL, Maliar L, Maliar S 2011. Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models. Quant. Econ. 2:173–210
    [Google Scholar]
  86. Kaltenbrunner G, Lochstoer LA. 2010. Long-run risk through consumption smoothing. Rev. Financ. Stud. 23:3190–224
    [Google Scholar]
  87. Kollmann R, Maliar S, Malin BA, Pichler P 2011. Comparison of solutions to the multi-country Real Business Cycle model. J. Econ. Dyn. Control 35:186–202
    [Google Scholar]
  88. Krueger D, Kubler F. 2004. Computing equilibrium in OLG models with stochastic production. J. Econ. Dyn. Control 28:1411–36
    [Google Scholar]
  89. Kung H, Schmid L. 2015. Innovation, growth, and asset prices. J. Finance 70:1001–37
    [Google Scholar]
  90. Lan H. 2018. Comparing solution methods for DSGE models with labor market search. Comput. Econ. 51:1–34
    [Google Scholar]
  91. Leith C, Liu D. 2016. The inflation bias under Calvo and Rotemberg pricing. J. Econ. Dyn. Control 73:283–97
    [Google Scholar]
  92. Lemoine D, Traeger CP. 2016. Economics of tipping the climate dominoes. Nat. Climate Change 6:514–19
    [Google Scholar]
  93. Levintal O. 2018. Taylor projection: a new solution method for dynamic general equilibrium models. Int. Econ. Rev. 59:1345–73
    [Google Scholar]
  94. Lontzek TS, Narita D. 2011. Risk-averse mitigation decisions in an unpredictable climate system. Scand. J. Econ. 113:937–58
    [Google Scholar]
  95. Lorenz F, Schmedders K, Schumacher M 2020. Nonlinear dynamics in conditional volatility. Work. Pap Univ. Münster, Münster Ger.:
    [Google Scholar]
  96. Lorenz F, Schumacher M. 2018. Downside risks and the price of variance uncertainty Work. Pap Univ. Münster, Münster Ger.:
    [Google Scholar]
  97. Magill MJ. 1977. A local analysis of N-sector capital accumulation under uncertainty. J. Econ. Theory 15:211–19
    [Google Scholar]
  98. Maliar L, Maliar S. 2003. Parameterized expectations algorithm and the moving bounds. J. Bus. Econ. Stat. 21:88–92
    [Google Scholar]
  99. Maliar L, Maliar S. 2015. Merging simulation and projection approaches to solve high-dimensional problems with an application to a new Keynesian model. Quant. Econ. 6:1–47
    [Google Scholar]
  100. Maliar L, Maliar S, Valli F 2010. Solving the incomplete markets model with aggregate uncertainty using the Krusell–Smith algorithm. J. Econ. Dyn. Control 34:42–49
    [Google Scholar]
  101. Maliar S, Maliar L, Judd K 2011. Solving the multi-country real business cycle model using ergodic set methods. J. Econ. Dyn. Control 35:207–28
    [Google Scholar]
  102. Malin BA, Krueger D, Kubler F 2011. Solving the multi-country real business cycle model using a Smolyak-collocation method. J. Econ. Dyn. Control 35:229–39
    [Google Scholar]
  103. Mortensen DT. 1982. Property rights and efficiency in mating, racing, and related games. Am. Econ. Rev. 72:968–79
    [Google Scholar]
  104. Ngo PV. 2014. Optimal discretionary monetary policy in a micro-founded model with a zero lower bound on nominal interest rate. J. Econ. Dyn. Control 45:44–65
    [Google Scholar]
  105. Niemann S, Pichler P, Sorger G 2013. Central bank independence and the monetary instrument problem. Int. Econ. Rev. 54:1031–55
    [Google Scholar]
  106. Papageorgiou C, Perez-Sebastian F. 2004. Can transition dynamics explain the international output data. Macroecon. Dyn. 8:466–92
    [Google Scholar]
  107. Papageorgiou C, Perez-Sebastian F. 2007. Is the asymptotic speed of convergence a good proxy for the transitional growth path. J. Money Credit Bank. 39:1–24
    [Google Scholar]
  108. Petrosky-Nadeau N, Zhang L. 2017. Solving the Diamond–Mortensen–Pissarides model accurately. Quant. Econ. 8:611–50
    [Google Scholar]
  109. Petrosky-Nadeau N, Zhang L, Kuehn LA 2013. Endogenous disasters and asset prices Work. Pap Carnegie Mellon Univ Pittsburgh, PA:
    [Google Scholar]
  110. Petrosky-Nadeau N, Zhang L, Kuehn LA 2018. Endogenous disasters. Am. Econ. Rev. 108:2212–45
    [Google Scholar]
  111. Pichler P. 2011. Solving the multi-country Real Business Cycle model using a monomial rule Galerkin method. J. Econ. Dyn. Control 35:240–51
    [Google Scholar]
  112. Pichler P, Sorger G. 2009. Wealth distribution and aggregate time-preference: Markov-perfect equilibria in a Ramsey economy. J. Econ. Dyn. Control 33:1–14
    [Google Scholar]
  113. Pissarides CA 1985. Short-run equilibrium dynamics of unemployment vacancies, and real wages. Am. Econ. Rev. 75:676–90
    [Google Scholar]
  114. Pohl W, Schmedders K, Wilms O 2018. Higher order effects in asset pricing models with long-run risks. J. Finance 73:1061–111
    [Google Scholar]
  115. Pohl W, Schmedders K, Wilms O 2020. Asset pricing with heterogeneous agents and long-run risk. J. Financ. Econ In press
    [Google Scholar]
  116. Reddien GW. 1980. Projection methods for two-point boundary value problems. SIAM Rev 22:156–71
    [Google Scholar]
  117. Reich G, Wilms O. 2015. Adaptive grids for the estimation of dynamic models Work. Pap Univ. Zurich Zurich, Switz.:
    [Google Scholar]
  118. Reiter M. 2009. Solving heterogeneous-agent models by projection and perturbation. J. Econ. Dyn. Control 33:649–65
    [Google Scholar]
  119. Reiter M. 2010. Solving the incomplete markets model with aggregate uncertainty by backward induction. J. Econ. Dyn. Control 34:28–35
    [Google Scholar]
  120. Reiter M. 2015. Solving OLG models with many cohorts, asset choice and large shocks Work. Pap Inst. Adv. Stud Vienna, Austria:
    [Google Scholar]
  121. Renner P, Scheidegger S. 2018. Machine learning for dynamic incentive problems Work. Pap Univ. Lancaster Lancaster, UK:
    [Google Scholar]
  122. Rotemberg JJ. 1982. Sticky prices in the United States. J. Political Econ. 90:1187–211
    [Google Scholar]
  123. Routledge BR, Zin SE. 2010. Generalized disappointment aversion and asset prices. J. Finance 65:1303–32
    [Google Scholar]
  124. Sánchez-Marcos V, Sánchez-Martín AR. 2006. Can social security be welfare improving when there is demographic uncertainty. J. Econ. Dyn. Control 30:1615–46
    [Google Scholar]
  125. Scheidegger S, Bilionis I. 2019. Machine learning for high-dimensional dynamic stochastic economies. J. Comput. Sci. 33:68–82
    [Google Scholar]
  126. Schumacher M, Żochowski D. 2017. The risk premium channel and long-term growth Work. Pap. 2114 Eur. Cent. Bank Frankfurt:
    [Google Scholar]
  127. Schumaker L. 2007. Spline Functions: Basic Theory Cambridge, UK: Cambridge Univ. Press, 3rd ed..
    [Google Scholar]
  128. Silvester PP, Ferrari RL. 1996. Finite Elements for Electrical Engineers Cambridge, UK: Cambridge Univ. Press, 3rd ed..
    [Google Scholar]
  129. Song Z, Storesletten K, Zilibotti F 2012. Rotten parents and disciplined children: a politico-economic theory of public expenditure and debt. Econometrica 80:2785–803
    [Google Scholar]
  130. Taylor JB, Uhlig H. 1990. Solving nonlinear stochastic growth models: a comparison of alternative solution methods. J. Bus. Econ. Stat. 8:1–17
    [Google Scholar]
  131. Trefethen LN. 2013. Approximation Theory and Approximation Practice Philadelphia: SIAM
    [Google Scholar]
  132. Tuzel S. 2010. Corporate real estate holdings and the cross-section of stock returns. Rev. Financ. Stud. 23:2268–302
    [Google Scholar]
  133. van Zandweghe W, Wolman AL 2019. Discretionary monetary policy in the Calvo model. Quant. Econ. 10:387–418
    [Google Scholar]
  134. Weil P. 1989. The equity premium puzzle and the risk-free rate puzzle. J. Monet. Econ. 24:401–21
    [Google Scholar]
  135. Werner M. 2016. Occasionally binding liquidity constraints and macroeconomic dynamics Work. Pap Univ. Zurich Zurich, Switz.:
    [Google Scholar]
  136. Young ER. 2010. Solving the incomplete markets model with aggregate uncertainty using the Krusell–Smith algorithm and non-stochastic simulations. J. Econ. Dyn. Control 34:36–41
    [Google Scholar]
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